Volume 494

Springer Proceedings in Mathematics & Statistics

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Editors

D. Marc Kilgour, Herb Kunze, Roman N. Makarov, Roderick Melnik and Xu Wang

Addressing Modern Challenges in the Mathematical, Statistical, and Computational Sciences

The VI AMMCS International Conference, Waterloo, Canada, August 14–18, 2023

Editors

D. Marc Kilgour

Wilfrid Laurier University, Waterloo, ON, Canada

Herb Kunze

University of Guelph, Guelph, ON, Canada

Roman N. Makarov

Wilfrid Laurier University, Waterloo, ON, Canada

Roderick Melnik

Wilfrid Laurier University, Waterloo, ON, Canada

Xu Wang

Wilfrid Laurier University, Waterloo, ON, Canada

ISSN 2194-1009e-ISSN 2194-1017

Springer Proceedings in Mathematics & Statistics

ISBN 978-3-031-84868-1e-ISBN 978-3-031-84869-8

https://doi.org/10.1007/978-3-031-84869-8

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

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Preface

This volume contains a selection of the papers that were presented at AMMCS-2023, the international conference held in Waterloo, Ontario, Canada from 14 to 18 August 2023. It was the sixth in the series of AMMCS meetings, held biennially at Wilfrid Laurier University beginning in 2011 (except in 2021). The 2023 event continued our tradition of promoting interdisciplinary research and collaboration involving mathematical, statistical, and computational sciences within the broader international scientific and research community, demonstrating how recent advances are meeting modern challenges as they strengthen the disciplines included in Applied Mathematics, Modelling, and Computational Science (AMMCS).

The interdisciplinary focus of the AMMCS conferences has always been crucial to their success. As organizers, our primary aim remains the same: to promote research and collaboration involving new applications within the mathematical, statistical, and computational sciences, and to build bridges from those fields to their areas of application. We hope to create, and benefit, international communities of researchers, practitioners, and students.

Mathematical methods have been fundamental to the development of human knowledge for millennia. Sophisticated mathematical, statistical, and computational tools have made, and are continuing to make, essential contributions to progress in an amazing range of application areas—in the natural and social sciences, engineering, finance, and even the arts. Techniques from mathematics, statistics, and computational and data science are now recognized for their ability to throw new light on problems—both ancient and contemporary—and thereby contribute to human well-being.

Some of today’s most challenging problems arise in the physical sciences and engineering, where mathematics is traditionally applied, but others come from the life sciences, the social sciences, and finance. The power of mathematical techniques and reasoning has been augmented by data collection and analysis on a massive scale, and by computational studies that support analysis and explore new combinations and structures. At the same time, subtle insights have led to stunning advances. New connections have been forged among disciplines that were once seen as widely separated; indeed, the horizons of mathematical and computational modelling are expanding—at an increasing rate.

AMMCS-2023 was a major international forum for the exchange of ideas in an interdisciplinary setting, with a focus on applications of mathematical and computational sciences, modelling and simulation to the natural and social sciences, engineering and technology, industry and finance. It proudly followed the traditions of previous AMMCS events, particularly in its emphasis on discussion, comparison, and synthesis across disciplines. We believe that only through interdisciplinary collaboration will it be possible to meet the complex challenges facing humanity today.

This book presents a representative selection of the current research presented at AMMCS-2023. It illustrates how mathematics, statistics, and modeling are contributing to a range of disciplines. The 47 selected contributions are organized into six parts, as follows:

I.

Advances in Mathematical Modelling and Theory

II.

Innovations in Statistical Machine Learning and Stochastic Models

III.

Mathematical Modelling in Engineering, Physical and Chemical Sciences

IV.

Mathematical and Statistical Modelling in Biological, Medical and Health Sciences

V.

Mathematics and Computation in Finance, Economics, and Social Sciences

VI.

Theory and Computational Methods for Differential Equations

The titles of the parts make the breadth of the topics clear. This wide-ranging selection demonstrates clearly how mathematical, statistical, and computational sciences have established themselves as fundamental tools across a wide range of disciplines.

As editors of this volume, it is our duty and our pleasure to extend thanks to all of the contributors to AMMCS-2023, to all of the attendees, to the Organizing, Scientific, and Technical Committees, and to all of the volunteers, without whom the conference could not have taken place. We are grateful to our sponsors, to the City of Waterloo, and to Wilfrid Laurier University—in particular to Charity Parr-Vasquez and the Laurier Office of Research Services. And, of course, we give special thanks to all of the contributors to this volume, and to the referees whose guidance was essential to us as we evaluated proposed contributions. We also thank Ambrose Berkumans of Springer, who coordinated the production of this book.

As well, we wish to thank explicitly our student volunteers: Ama Ansah, Nate Cullen, Mujahid Elmaki, Aya Khalaf, Daniel Mehta, Kenny Nguyen, Sydney Orlander, Dev Patel, Hina Shaheen, Gavan Singh, Howard Zhu, Akshayveer A., Swadesh Pal, Akepogu Venkateshwarlu, Sangeeta Saha, Manoj Belavadi, Vishesh Vipin Verma, and Agassi Iu. Also Vishesh Verma, who acted as our webmaster.

We are proud of this volume and pleased to acknowledge all those who helped bring it to fruition.

Group photo of participants in the AMMCS-2023 International Conference: Applied Mathematics, Modeling and Computational Science

D. Marc Kilgour

Herb Kunze

Roman N. Makarov

Roderick Melnik

Xu Wang

Waterloo, ON, Canada

August 2024

Part IAdvances in Mathematical Modelling and Theory

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_1

The Study of the Transformation Semigroup of the Abelian and Directed Non-Abelian Sandpiles

Hanna Derets1   and Chrystopher L. Nehaniv1  

(1)

University of Waterloo, Waterloo, Canada

Hanna Derets (Corresponding author)

Email: h2derets@uwaterloo.ca

Chrystopher L. Nehaniv

Email: chrystopher.nehaniv@uwaterloo.ca

Abstract

The sandpile model is a discrete event dynamical system always considered on a fixed graph with certain toppling rules for the stabilization of the configurations of the graph. In this work we first consider the Abelian sandpile on a 2-regular graph: show the structure of the recurrent configurations, idempotent, and generators of the recurrent group, and conclude it is cyclic. Second, we consider the non-Abelian aperiodic sandpile semigroup on a directed rooted tree: show the aperiodic complexity of the semigroup and constructively show the emulation of the sandpile semigroup by a minimum length wreath product of irreducible blocks.

Keywords

Abelian and non-Abelian sandpilesTransformation semigroupsRecurrent group structureWreath product decompositionSelf-organized criticality

1 Introduction

The sandpile model presents a dynamic process in which grains of sand move along a fixed graph, first introduced on two-dimensional grids by Bak et al. [3]. Later, Dhar [6] showed this model is Abelian. Abelian sandpiles were the subject of many different studies: the structure of the recurrent group, for instance, [5], the chip-firing game [9], semigroup structure of the sandpile [2], and others. There are several variations of the model, for example, stochastic sandpiles [4, 8], and non-Abelian sandpiles on directed graphs [1].

In Sect. 2, we consider Abelian sandpile for the simplest graph: a cycle. We first prove several stabilization properties and then describe the structure of the identity element and a generating configuration of the recurrent group. This yields an explicit alternative way of understanding the well-known cyclic structure of the recurrent group of this 2-regular graph. In Sect. 3 we consider non-Abelian sandpile on rooted trees defined in [1]. We construct a minimal embedding of a sandpile monoid in a wreath product of Cartesian products of flip-flops, and conclude that the aperiodic complexity [10] of the sandpile semigroup is the depth of the tree. For background on the transformation semigroups and wreath products see [7].

2 Abelian Sandpile

A detailed description of the Abelian sandpile model can be found in [6]. Here we list several key features, needed for understanding the work.

In the arbitrary finite graph G=(V,E) fix one vertex, reachable from all other vertices, to be the sink—a dissipative vertex. The capacity ci of every non-sink vertex equals its out-degree. The sink vertex is always empty and absorbs all grains that fall into it. A configuration is called unstable if at least one vertex contains more grains than its capacity. Consider the space of all stable configurations with the binary operation of adding configurations vertex by vertex and stabilizing the result, called sandpile semigroup. Stabilization is a finite iterative process of toppling all unstable vertices simultaneously, which eventually leads to a unique stable configuration. Here the toppling of an unstable vertex refers to the operation of passing one grain to each of its neighbors. Every configuration that is reachable from every other configuration is called recurrent, otherwise—transient. All recurrent configurations form the recurrent group—called the sandpile group.

2.1 One-Dimensional Case: 2-Regular Non-directed Graph

Consider a two-regular graph G=(V,E) which has |V|=N+1 vertices and v0 is the only sink vertex. Such graph has circle structure, and in a stable configuration, each vertex can contain at most one sand grain. The sandpile semigroup defined by such a graph is mentioned in [11] as a one-dimensional model. They give an intuitive formulation of the stabilization rule by mirroring unstable vertex. This rule is given without proof, so first we formalize and prove this statement and then use it to describe the sandpile group of a circle graph.

Lemma 1

Consider the Abelian sandpile model on a 2-regular graph G=(V,E), |V|=N+1 with a sink vertex v0. Let the vector a=(a0,a1,...,aN,aN+1) describe a configuration of grains with a0=aN+1=0 corresponding to the sink vertex. Consider adding a grain into the vertex vs such that 0<s<N+1,as=1. Let i<s and j>s be the indices of the two closest empty vertices, that is 0≤i<s<j≤N+1,ai=aj=0,ai+1=⋯=aj-1=1. Then the result of stabilization of configuration a with as=2 is a¯=(a0¯,a1¯,⋯,aN¯,aN+1¯) constructed as follows:

1.

set as¯=1 and ai¯=aj¯=1, then ai+j-s¯=0,

2.

set ak¯=ak for the rest of the vertices, i.e. ∀k such that k∉{i,j,s,i+j-s},

3.

set a0¯=aN+1¯=0 for the sink vertex.

Proof

Consider induction for (j-i). Ensure that in two base cases statement holds:

(j-i=2):...020...→...101...,(j-i=3):...0210...→...1020...→...1101...

where a¯ is stabilized in one or two updates. Assume it is true for all j-i≤M and consider j-i=M+1. W.l.o.g. i<s-1, after the first toppling as=0,as-1=2.

if s+1<j, then as+1¯=2. Using induction hypothesis, first stabilize segment ai¯,⋯,as¯ – get ai+1¯=0, then stabilize segment ai+1¯,⋯,aj¯ – get ai+j-s¯=0:

a¯=(...0i1...120s21...10j...)⟶(...1i01...11s21...10j...)⟶(...1i1...10i+j-s1...11j...)

otherwise s+1=j,aj¯=1 and stabilization of ai¯⋯as¯ results in a stable configuration, with ai+1¯=0, where i+1=i+j-(j-1)=i+j-s as expected.

The third step in the lemma ensures that grains are discarded in the new configuration, if fall into the sink, i.e. if i=0 and/or j=N+1.    □

2.1.1 The Structure of the Sandpile Group

For convenience, let’s denote the maximal configuration by A, and configurations with exactly one filled and one empty vertex by Ci and Bi respectively:

A=(001....10N+1),Bi=(001...10i1...10N+1),Ci=(000...01i0...00N+1),i∈{1,...,N}

Because of the circle structure of the 2-regular graph extend this notion for the negative indices: Bi-k=B(i-k+N+1). Also, notice the equality B0=BN+1=A as both a0 and aN+1 correspond to the sink.

Lemma 2

For any indices 1≤m,k≤N the following holds:

Bm+Ck=Bm-k=BN+1+m-k

Proof

To stabilize addition using Lemma 1 have to establish the closest empty vertices i and j. Consider cases:

1.

k<m, then i=0,j=m, and Bm+Ck=Bm-k

2.

k=m, then Bm+Ck=Bm+Cm=A=B0=BN+1 or equivalently A=Bm-k=BN+1+m-k,

3.

k>m, then i=m,j=N+1, and Bm+Ck=BN+1+m-k

   □

Lemma 3

The addition of two grains of sand to the vertices equidistant from the sink acts as identity transformation on the type A and type B configurations:

∀k<N-k+1:A+Ck+CN-k+1=A,Bm+Ck+CN-k+1=Bm1≤m≤N

Proof

By Lemma 1 we have: (A+Ck)+CN-k+1=BN-k+1+CN-k+1=A. By Lemma 2 we have: Bm+Ck+CN-k+1=Bm-k+CN-k+1=BN+1+m-k-(N+1-k)=Bm .    □

Theorem 1

For the Abelian sandpile on the 2-regular graph G=(V,E) with |V|=N+1 vertices where the only sink is the vertex v0∈V the following hold:

1.

The elements of the sandpile group G=G(G) are the configurations that have at most one empty vertex:

G={A,B1,⋯,BN}

2.

The identity element I=I(G) of the sandpile group G=G(G) has the following structure:

I=A,ifNiseven,Bq,q=N+12,otherwise.

3.

A generator G~=G~(G) of the group can be defined as follows:

G~=B2-ifN=4k+1forsomek,B1-otherwise.

4.

The sandpile group G=G(G) is a cyclic group of size N+1.

Proof

1.

Graph G has N+1 spanning trees, so N+1 recurrent configurations [6]. Maximal configuration A is reachable from any other configuration, so is recurrent by definition. Moreover, by Lemma 1 we have Bi=A+CN+1-i, so Bi is recurrent as well. So, A and Bi are all elements of the sandpile group G=G(G).

2.

For N=2m identity is I=A=∑k=1m(Ck+CN-k+1). Also, for N=2m+1 have I=Bm+1=∑k=1m(Ck+CN-k+1), so Lemma 3 implies that A+I=A,Bi+I=Bi⇒I is the identity element.

3.

First, consider N=2m and show that G~=B1:

As B1=C2+...+CN=∑k=2m(Ck+CN-k+1)+CN⇒Bi+B1=Bi+CN using Lemma 3. Then conclude Bi+B1=Bi+1,1≤i≤N using Lemma 1. So, B1 generates all elements of the group ⟨B1⟩={B1,B2,...,BN,BN+1=A}=G.

Second, consider N=4m+1 and show that G~=B2:

B2=∑k=1,k≠22m(Ck+CN-k+1)+CN-1+C2m+1⇒Bi+B2=Bi+C4m+C2m+1

for any i∈{1,...,N}, using Lemma 3. Then conclude Bi+B2=Bi-4m+N+1+C2m+1=Bi+2+C2m+1=Bi-(2m-1) using Lemma 2. In other words, by adding a configuration B2 to configuration Bi we move the position of the empty vertex i by 2m-1 positions backward in a circle. Repeating additions of B2 consider intermediate moment Bi+k1·B2, then ∃k2:Bi+k1·B2=Bi+(k1+k2)B2 as we have finite number of type B configurations. Suppose configuration repeats after t full turn around the circle, then

(2m-1)k1=(2m-1)(k1+k2)-t(N+1)⇒k2(2m-1)=2t(2m+1)

As 2m-1 and 2m+1 are coprime ⇒k2|(4m+2)⇒k2≥N+1⇒ at least N+1 different elements occur, before the sequence Bi+B2+⋯+B2 has repeated element. So, ⟨B2⟩=G(G).

The case where N=4m+3 is analogous, so omitted.

4.

As G has N+1 elements and one generator, it is cyclic group G=CN+1.

   □

3 Non-Abelian Sandpile on the Rooted Trees

Consider the non-Abelian sandpile model [1] on a rooted tree G=(V,E),|V|=N+1 with the root vertex v0∈V being the only sink and leaves L⊂V,|L|=m. The capacity of every vertex vi∈V is a non-negative integer ci and c0=0. There are m+N generators of the sandpile semigroup:

fvi—add a sand grain to the vertex vi and stabilize, defined for all leaves vi∈L,

gvi—lift a sand grain from the vertex vi (if it has any) to its parent vertex (towards the root) and stabilize, for all vi∈V,vi≠v0 non-root vertices. If vi is empty, nothing happens.

Three features make this model different from the previously considered. First, the operation of adding the grain is restricted to the leaves only. Second, another operation of lifting a sand grain to the parent vertex is introduced. Third, the capacity of every non-sink vertex is an arbitrary non-negative integer. When considering all edges to be directed towards the root, the stabilization process is the same as before. The first two changes result in the semigroup being non-Abelian, as the operations of adding and lifting a grain do not commute. Actually, for an empty leaf, adding a grain and then lifting it up results in an empty leaf, but switching the order of operations results in the leaf containing a grain.

3.1 Embedding of the Sandpile with Equal to One Capacity

For the rooted tree G=(V,E),|V|=N+1 denote D=D(G) to be the depth of the tree. For each level i∈{1,..,D} of the tree denote mi to be the number of the vertices in the tree which are located on the depth i, so M={m1,m2,...,mD},∑i=1Dmi=N.

Theorem 2

The aperiodic complexity of the sandpile semigroup S=S(G) acting on the configurations X=X(G) of the rooted tree G=(V,E) with equal to one capacity ci=1,i∈{1,...,N}, equals the depth D=D(G) of the tree G. Also, (X, S) can be embedded into the wreath product of D blocks of flip-flops, where each ith block is a direct product of mi flip-flops:

(X,S)↪(FF×...×FF)⏟mD≀...≀(FF×...×FF)⏟m1=(Y,W),

(1)

where, D is the minimal number of blocks in the wreath product needed for the embedding (ϕstate,ϕtrans):(X,S)↪(Y,W) to exist.

3.1.1 Base Case: Sandpile of the Path Graph

Consider the path graph G=(V,E),|V|=N+1 with equal to one capacity:

where the vertices are labeled as above, and we refer to the vertex labeled with i as vi. The sandpile semigroup S=S(G) acts on the state space X=X(G) of the size |X|=2N. Each state x∈X is a binary vector, where xi∈{0,1} is the number of grains in the vertex vi and let’s agree on the following order of the coordinates x=(xN,xN-1,⋯,x2,x1). The semigroup has N+1 generators S=⟨Sg⟩,Sg={f,g1,...,gN}, where we denote f=fvN, gi=gvi for convenience.

The wreath product of N flip-flops (FF≀...≀FF)=(Y,W) acts on the state space Y with the transformations w∈W, where each component wi=wi(yi-1,...,y1) is a function of the suffix yi-1,...,y1 of the state y that w is acting on:

∀y=(yN,⋯,y1)∈Yyi∈{a,b},|Y|=2N,∀w=(wN,...,w1)∈W,wi∈{1,Ca,Cb}

Both configurations of the graph x∈X and states of the wreath product y∈Y have binary coordinates, so there is a natural bijection ϕstate:X→Y via binary strings, where ∀x∈X,ϕstate(x)i=a if xi=0 and ϕstate(x)i=1 otherwise. Note that ith component of both x∈X and ϕstate(x)∈Y correspond to the vertex vi∈V.

Denote the mapping ϕtrans for every generator q∈Sg:

ϕtrans:Sg⟶W,ϕtrans(q)=wq(yN,...,y1)=(wNq,...,w1q)∈W

Consider the components wif of ϕtrans(f) to be defined as follows:

wif(yi-1,...,y1)=Cb,ifi=1or∀k∈{1,...,i-1}yk=b,1,otherwise

(2)

That is, it acts on the state space in the following way:

∀y=(.....⏟q,a,b...b⏟k)∈Y,w(y)=(1...1⏟q,Cb,Cb...Cb⏟k)

y·ϕtrans(f)=(.....⏟q,b,b...b⏟k)

where the prefix of the state y∈Y of length q remains unchanged.

Similarly, consider ϕtrans(gj),j∈{1,..,N} to have the following components:

wigj(yi-1,...,y1)=1,ifi≤N-j,Ca,ifi=N-j+1,Cb,ifi>N-j+1andyk=b,i-1≤k≤N-j+1,1,otherwise.

(3)

Then, any state y=(yN,...,y1)∈Y with yi=a acted upon by transformation ϕtrans(gi)=wgi(y) remains unchanged, but for the states y=(yN,...,y1)∈Y with yi=b the result of action ϕtrans(gi)=wgi(y) is defined as follows:

∀y=(.....⏟q,a,b...b⏟p.....⏟N-i)∈Y,p+q+1=i

wgi(y)=(1...1⏟q,Cb,Cb,...Cb,Ca⏟p,1...1⏟N-i),y·ϕtrans(gi)=(.....⏟q,b,b...b,a⏟p.....⏟N-i)

From such a description of transformations and their action on the states, conclude that ϕtrans(f) and ϕtrans(gi) act on the state space of the wreath product in the same way as f and gi act on the configurations of the tree G=(V,E), so they generate a semigroup isomorphic to the sandpile semigroup S.

Corollary 1

Consider the rooted tree G=(V,E), which consists of m path graphs, glued in a root. Each ith path has ni vertices. There is the following embedding of the sandpile semigroup:

(βstate,βtrans):(X,S)↪(FF≀...≀FF⏟n1)×...×(FF≀...≀FF)⏟nm=(Y,W)

As the model only allows the grains to move towards the root, action on one path doesn’t affect others. The above corollary naturally follows from the independence of the root branches but is not a minimal embedding.

3.1.2 Generalization for an Arbitrary Tree

Consider the arbitrary rooted tree G=(V,E) of the depth D=D(G) with the capacity of each vertex equal to one. Denote the depth of the vertex v∈V as d(v), and its parent as p(v). As G is a rooted tree, the parent of any vertex is unique and we can consider a chain of parents of v on the way to the root vertex:

P(v)=(ud(v),⋯,u2,u1),u1=v,ud(v)-root,ui+1=p(ui),1≤i<d(v)

Suppose that all non-sink vertices are labeled with integers from 1 to N, and we refer to the vertex labeled as i with vi. Assume for all labels holds that if vj=p(vi), then j>i. This implies that indices of the vertices in the parent chain are monotonically decreasing. For every state y∈Y define a vector y(v) of size d(v), which consists only of the components of the vector y=(yN,⋯,y1)∈Y corresponding to the parent chain P(v) of the vertex v:y(v)=(yPd(v),⋯,yP1) where Pj=P(v)j is the jth ancestor of v. Every transformation—fvi or gvi is applied to some vertex vi and the important thing to notice is that it doesn’t affect the vertices other than the chain of parents P(vi) of this vertex. Denote w¯f(yi-1...y1)=wif(yi-1...y1) and w¯gj(yi-1...y1)=wigj(yi-1...y1) the mapping defined by Eqs. (2 and 3) for the case of a path graph.

Consider the following mapping ϕtrans:Sg⟶W for generators of S. For adding a grain into leaf v, the ith component of the transformation ϕtrans(fv)=(wNfv,⋯,w1fv) define as

wifv=wfv(yi-1...y1)=w¯if(y(v)j-1...y(v)1),ifvi=P(v)jforsomej1,otherwise.

For lifting a grain up from the vertex v, the ith component of the transformation ϕtrans(gv)=(wNgv,⋯,w1gv) define as

wigv=wgv(yi-1...y1)=w¯jg(y(v)j-1...y(v)1),ifvi=P(v)jforsomej1,otherwise.

On this, the existence of embedding (2) can be considered proven.

3.2 Generalization for a Rooted Tree with Arbitrary Capacity

Similarly one can construct the emulation of the sandpile semigroup in the case of arbitrary capacity of the vertices. We omit the formal description of the ϕtrans and ϕstate mappings and only give the sketch of the construction.

For the graph G=(V,E) consider the operation of replacing all vertices with ci>1 with a chain of ci vertices of capacity ci′=1, as shown in Fig. 1. Let’s call it extended tree G∗, which now has the depth D∗=D(G∗) equal to the maximum sum of capacities on the parent chain among all leaves. As each colored block actually corresponds to one vertex, all permutations of the grains within the block are equivalent, and only the number of grains matters. For this reason, ϕstate is no longer bijective, i.e. Eq. (1) denotes emulation, where every configuration of the graph has several lifts—corresponding states in the wreath product.

For the transformation ϕtrans(fv) of adding the grain to the leaf, construction is exactly the same as for an arbitrary tree with capacity one, when applied to the deepest vertex of the block corresponding to this leaf.

For the transformation ϕtrans(gv) of lifting the grain to the parent vertex the only change will be to skip all vertices of the extended tree that correspond to the same vertex of the original graph when lifting a grain. Then, Theorem 1 is generalized for the tree of any capacity by applying it to the extended tree.

Fig. 1

Example of the extended tree. Labels on the vertices refer to their capacities. Segments of the binary vector, corresponding to the extended vertices match the colors. The example shows the equivalence of the permutations within the blocks

3.3 Aperiodic Complexity

According to [10], the aperiodic complexity C(S) of the aperiodic semigroup S is the minimum number of levels in the wreath product of blocks from flip-flops necessary to emulate the system. The lower bound for aperiodic complexity is the exponent of the semigroup Sϵ:

Sϵ=leastn:sn=sn+1∀s∈S

Clearly, the non-Abelian sandpile semigroup S is aperiodic.¹ The existence of the embedding (1) implies the upper bound of its complexity:

Sϵ≤C(S)≤D

Consider an arbitrary tree G of capacity one and depth D=D(G). Let v be a leaf vertex, such that d(v)=D, and fv be the transformation of adding grain to the leaf v. The exponent of fv is D, as starting from the empty configuration we can apply it exactly D times before all vertices in the parent chain of v are full and further addition becomes a constant transformation. Then Sϵ≥D and

D≤Sϵ≤C(S)≤D

which implies their equality C(S)=Sϵ=D. So, the aperiodic complexity of the sandpile semigroup S=S(G) is the depth of the tree D=D(G) for the tree G of capacity one, or the depth D∗=D(G∗) of the extended tree G∗ in general.

4 Conclusion

In this study, two sandpile models were considered. For the Abelian model, the structure of the sandpile on a graph having a circle structure (the so-called one-dimensional case) was studied. To begin with, the non-iterative stabilization rule was proved, and then the structure of the identity element and the generator of the sandpile group were described.

For the non-Abelian model on trees, a constructive description was completed for the emulation (and embedding) of generators of the sandpile semigroup in a wreath product of flip-flops, which, moreover, is one that contains the fewest levels in the wreath product. Particularly, it is shown that the aperiodic complexity of the sandpile semigroup on a rooted tree is the depth of the tree.

Acknowledgements

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669.

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Ayyer, A., Schilling, A., Steinberg, B., Thiéry, N.M.: Directed nonabelian sandpile models on trees. Commun. Math. Phys. 335, 1065–1098 (2015)

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Babai, L., Toumpakari, E.: A structure theory of the sandpile monoid for directed graphs. The University of Chicago (2010). https://​people.​cs.​uchicago.​edu/​~laci/​REU10/​evelin.​pdf

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Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59(4), 381 (1987)

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Chan, Y., Marckert, J.-F., Selig, T.: A natural stochastic extension of the sandpile model on a graph. J. Comb. Theory, Ser. A 120(7), 1913–1928 (2013)MathSciNet

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Dartois, A., Fiorenzi, F., Francini, P.: Sandpile group on the graph dn of the dihedral group. Eur. J. Comb. 24(7), 815–824 (2003)

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Dhar, D.: Theoretical studies of self-organized criticality. Phys. A: Stat. Mech. Its Appl. 369(1), 29–70 (2006). Fundamental Problems in Statistical Physics

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Domosi, P., Nehaniv, C.L.: Algebraic Theory of Automata Networks, vol. 11, (Monographs in Discrete Mathematics and Applications). SIAM (2005)

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Kim, S., Wang, Y.: A stochastic variant of the abelian sandpile model. J. Stat. Phys. 178(3), 711–724 (2020)MathSciNet

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Klivans, C.J.: The Mathematics of Chip-Firing. CRC Press (2018)

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Nehaniv, C.L.: Complexity of finite aperiodic semigroups and star-free languages. In: Almeida, J., Gomes, G., Silva, P. (eds.) Semigroups, Automata, Languages, pp. 195–209. World Scientific (1996)

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Redig, F.: Course 14—mathematical aspects of the abelian sandpile model. In: Bovier, A., Dunlop, F., van Enter, A., den Hollander, F., Dalibard, J. (eds.) Mathematical Statistical Physics, vol. 83 of Les Houches, pp. 657–729. Elsevier (2006)

Footnotes

1

The aperiodicity property is a new result that follows from the embedding into/division of the wreath product of flip-flops that we prove above and the Krohn-Rhodes Theorem. Particularly, as the members of wreath product decomposition of non-Abelian sandpile semigroup S are flip-flops or direct products of flip-flops, which are not divisible by a simple group, no group divides the semigroup S itself.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_2

On Constructing Finite Automata by Relational Programming

Attila Egri-Nagy1   and Chrystopher L. Nehaniv2  

(1)

Akita International University, Akita, Japan

(2)

University of Waterloo, Waterloo, Canada

Attila Egri-Nagy (Corresponding author)

Email: egri-nagy@aiu.ac.jp

Chrystopher L. Nehaniv

Email: cnehaniv@uwaterloo.ca

Abstract

We consider ways to construct a transducer for a given set of input word to output symbol pairs. This is motivated by the need for representing game playing programs in a low-level mathematical format that can be analyzed by algebraic tools. This is different from the classical applications of finite state automata, thus the usual optimization techniques are not directly applicable. Therefore, we use relational programming tools to find minimal transducers realizing a given set of input-output pairs.

Keywords

Lossless machine learningFinite transducerRelational programmingAutomata minimizationGame representations

1 Introduction

A fully calculated game tree is a perfect solution for a deterministic two-player game of complete information. Working back from the leaves one minimizes the maximum payoff the opponent can achieve, using a mini-max strategy as demonstrated by von Neumann in 1928 [10]. However, the tree may not be the most suitable representation for all purposes. Here, we are interested in the algebraic study of games [8]. Therefore, we prefer a finite state automaton representation, as those define canonically associated transformation semigroups. These, in turn, can be studied by the hierarchical decomposition tools. The question naturally arises: how can we create finite state automata for playing games?

First we will formalize the automaton construction problem as a machine learning task. Then we will describe relational programming solutions for this task and compare them with classical algorithms.

1.1 Playing Go Possible Representations

As an illustration, let’s consider how we would implement a playing engine for the game of Go as an automaton. We would like to have an automaton that can give the next move based on a given board position. Therefore, the input of the automaton is a sequence of symbols describing a board position. The board position can be described in at least two ways, as an image, or as a history.

We can describe the board position as an image, row by row, or by some traversing curve (e.g., a boustrophedon pattern) to have better locality. We need at least 3 symbols for the local state (white, black, empty intersection). The input words have the same length, the number of intersections on the board. However, in Go we have the ko rule to avoid infinite loops in the game, so there cannot be an immediate retaking of a captured stone. Consequently, we would need to use an extra symbol to denote the position forbidden by the ko rule. Deep learning AIs use this image-like board position, but they address the ko problem differently. The previous 8 positions are all part of the input layer [9].

We can also define the input word as the history, the sequence of moves leading to the given board position from the empty board. This representation then would use different lengths of input words, depending on the state of the game.

In both representations, it is easy to see that a simple (but enormous) tree structure can realize a perfect player as an automaton. The question is how compressible is this structure? That is why we are looking for a minimal automaton that can realize all the desired mappings.

1.2 Programming Finite State Automata

Finite state automata (FSA) form a model of computation, in which the input word is read only once, unlike Turing-machines with freely moveable read/write head. We can consider FSA, and thus transformation semigroups, as a low-level programming language [4]. We can still write programs for this mathematical definition of a computer. However, FSA are further away from practical programming languages, since the Turing-machine has the tape as a dedicated memory device. FSA need to implement memory structure in their state sets. They can have more optimized and efficient memory handling tailored for a given task. This flexibility comes at a price: FSA are truly horrible programming languages to solve a problem. Hence, there is a need for algorithms to generate good FSA, instead of manually constructing them.

2 Formalizing the Lossless Machine Learning Problem

We define a computational task as a list of input-output pairs. The goal is to construct an automaton that given an input word produces the corresponding output symbol. Automata that produce output symbols are called transducers. Here we have a special case that produce only a single output symbol for an input word.

This looks like a machine learning problem, so it is important to emphasize the learning here is not statistical. We want to have an error-free reproduction of the outputs for the input words in the training sets. The generalization is not a goal, only exact reproduction.

Task 1

(Lossless machine learning: transducer construction) For an input alphabet I and an output alphabet O, we have a list of (w, r) word-symbol pairs, w∈I+, r∈O. We need to construct a special type of transducer automaton T, such that it will produce r when processing w for all such pairs.

Definition 1

(Single-output transducer) T=(Q,qo,I,O,δ,ω),

Q is the set of states, I is the set of input symbols,

qo is the initial state (thus Q≠∅)

δ:Q×I→Q is the state transition function

ω:Q→O is the partial output function.

Transducer T realizes a partial function f:I+→O, where we do not care about the values not defined in the input-output pairs. There is a fixed initial state, in which the transducer starts reading each word. For an input word w=w1w2⋯wk of length k, wi∈I, the sequence of states t0,…,tk is a trajectory, defined recursively by t0=qo, and ti=δ(ti-1,wi) for 1≤i≤k. Then the output for w is defined by ω(tk), determined by the last state in the trajectory. If r=f(w)=ω(tk), then the transducer produces the correct output for the (w, r) pair in the specification. There can be states in trajectories with undefined output (but not as last state), therefore |Q|≥|I|.

Example 1

The input-output pairs 00↦0, 01↦1,10↦1, 11↦0 define the textbook problem of checking parity. Here is a transducer realizing these pairs.

Q=I=O={0,1}, q0=0, ω is the identity function. The arrow labels are the input symbols. Inside a node representing a state, we have the state itself followed by the corresponding output value.

2.1 Construction Meta-Algorithms

There are several factors to consider when planning to solve this task. First, we do not know in general how many states are needed to realize a given set of input-output pairs. The minimum number of states is often the ultimate question in a problem domain, like for the game playing transducers. What is the complexity of a game? It is the smallest number of states needed for a transducer to be able to produce an optimal move for all board positions.¹ In practice, we can simply start with a single state |Q|=1, and add a state when it becomes clear that there is an insufficient number of states. Another option would be to build an easy to construct transducer with possibly many redundant states, and apply an algorithm to prune the state set.

Another issue to consider, is the order of the input-output pairs in the construction process, and thus the incremental nature of the algorithm. In general, we cannot expect a gradual structure evolution. If we construct a transducer for n-1 input-output pairs, adding the nth pair may require a completely different transducer.

3 Relational Programming Solution

Finding a correct transducer (not to mention a minimal one) for a set of input-output pairs is a daunting task for a human programmer. Therefore, it is a natural idea to automate it.

We use logic programming, or more generally relational programming. The basic idea can be illustrated with an elementary mathematical example. In functional thinking, we consider exponential and logarithmic functions as functions that produce outputs from input values, e.g., f(x)=2x with input 3 we get output 8, since f(3)=23=8. We can also consider its inverse function, where the input 8 gives output 3, since log28=3. In relational thinking there is no input, there is no output. Just three numbers in a relationship: r(2, 3, 8). If any of the numbers missing, we can recover that value: r(2, ?, 8) will give 3, and r(?, 3, 8) will give 2.

We can have the same distinction for the problem of transducer construction. Functionally speaking, a transducer produces an output symbol from a sequence of input symbols. Relationally speaking, we have a relationship between input, output and the transducer. Therefore, if the input-output pairs are given, we can infer the transducer. The inference is done by a search algorithm for finding variable combinations satisfying the constraints of the relation.

3.1 Implementation

We used the miniKanren [2] family of programming languages for relational programming. It has a minimalistic approach famously described by only two pages of source code in [5]. This simplicity also means that it is easy to implement, thus there are several implementations in many programming languages. In particular, we use the core.logic library written in Clojure [6]. The code for the transducer construction algorithms below can be found in our kigen package [3]. In this project, we treated the logic engine as a black box, i.e., we did not consider any internal parameters of the search. We simply defined the logic variables and their constraints.

3.2 Logic Variable Considerations & Setup

When using a logic search engine, the main question is ‘What are the logic variables?’. The constraints are the (wi,ri) pairs, the unknown part is the transducer itself. We define the state set to be non-negative integers Q={0,1,2,…,n-1}. This is fixed because the states are used as indices internally in the state transition table. The initial state is set to q0=0. If we leave the output map ω flexible, then this fixed choice does not affect the resulting transducer. The input and output symbols can be any distinct symbols without restrictions (anything the underlying language permits). We also need the flexibility for the output function. We cannot fix the outputs to be the first n states as that would impose an additional constraint with the fixed initial state.

The development of the software implementation proceeded from the hard-coded representation (states, input and output symbols represented by non-negative integers) to a more flexible, symbolic hash-map based data structure, as it became clear that fixed choices interfere with satisfiability with a given number of states.

The logic variables are associated with the entries of the state transition table Q×I under δ, and with the output values of the states under ω. Thus, the number of logic variables is |Q|(|I|+1). It also became inevitable during the development to allow for partial δ and ω. Whenever a state transition is not defined, the value nil is used. This indicates that the mapping is not needed. For ω, there need to be at least |O| entries defined. However, adding another possible value for the logic variables increases their number. Thus, it is more efficient to let the logic engine come up with a solution where the functions are totally defined and then simulate the transducer on all input words and record which mappings are not used. The number of defined maps in the state transition table is also a measure for resource usage. If there are many entries undefined, it is likely that task can be solved by a smaller number of states.

Another variation is also considered. Each word w defines a trajectory q0,t1,…,tk, a sequence of states starting with the initial state. We can define logic variables for each ti state in all trajectories. The constraints are the properties of functions, i.e., the state transitions labeled by the input symbols and extracted from the trajectories should define a valid δ function. In other words, if an input symbol takes state qi to qj in one place, then all other occurrences of qi in all trajectories should go to qj under the same input symbol. This method gives partial solutions directly (no need for simulations). However, the number of logic variables grows with the total length of the input words, and finding solutions quickly becomes computationally unfeasible.

3.3 Examples

Here we describe some examples of the constructed transducers.

Example 2

(Signal Locator 9-3) I={0,1}, the input words are bitstrings of length 9, all zeroes except one symbol. The task is to find in which third of the input number 1 (the signal) occurs. For instance, 000010000 ↦ 2. This can be done with 5 states (see Fig. 1), but not with 4 states. The size of the search space is 510·35 for the 15 logic variables. The task can be generalized to n-k, locating the signal in n-bit sequences partitioned into k parts.

Fig. 1

Signal Locator 9-3. The transducer for locating 1 among the zeroes in a bitstring of length 9. The node named nil is a sink state, representing partial transformations. We can see what allows the state transition to remain undefined. State 1 represents the case when the signal is in the first third, thus no more ones can be read. To detect the signal in the second third, we need to reach state 4, then return to the initial state and back to 4 ‘in time’. To locate the signal in the last third, we do the same, just not getting back to 4. This transducer would be easy to design manually, but it is also interesting enough to study the algorithm of this minimal solution

Example 3

(Zeroes or ones) The input words are bitstrings of a given length and the transducer has to decide whether the number of zeroes is equal to, less, or more than the number of ones. The task is similar to the majority function in Boolean logic. The solution is essentially a bi-directional counter (see Fig. 2). Search space size is 510·35 for the 15 logic variables.

Fig. 2

Zeroes or ones. A solution for the ‘Zeroes or ones’ problem for bitstrings of length 4. Left: minimal one by the logic engine, right: minimized trie construction. The difference is clear: the minimized trie has only forward edges

Example 4

(Palindromes) Recognizing palindromes (words that read the same backwards) is the prime example of a problem that cannot be solved by FSA. The language of palindromes is context-free, but not regular. However, for a given fixed length, we can construct a transducer for deciding this property. For length 4 binary strings the search space size is 510·25, and it is easy to find a minimal solution (Fig. 3). However, the logic engine solution does not scale well. Length 5 is not an immediately finishing calculation.

Fig. 3

Palindromes. Transducers for detecting palindromic bitstrings of length 4; minimal and classic trie-based version

Example 5

(Word classification) The task is to sort natural language words into three groups: English (eruption, erudite, oriental, topology, serendipity), Japanese (eki, origami, arigato), and Hungarian (asztal, mester). The search space size is 354. There is a solution with three states.

4 Comparison with Classical Minimization Algorithms

A remarkable result of automata theory is that every FSA has a unique minimal (regarding the number of states) equivalent automaton. This seminal result suggests that we have a simple solution for our transducer construction problem. We could simply construct any transducer satisfying the input-output constraints and minimize that automaton. However, satisfying the input-output constraints only is a more general problem on two accounts. First, recognizing words is a special case of classifying them. Second, the crucial point here, we have no constraints on subwords of input words, and on words others than those specified in the input-output pairs. Therefore, we do not get a minimal transducer by the classical methods.

For applying the classical minimization algorithms, we need to establish that there is no fundamental difference between an automaton used as a recognizer and as a single-output transducer. The latter case can be seen as a more nuanced recognizer. The output symbols can be seen as different classes of acceptance. Or, the other way around, a recognizer is a single output transducer with only two output symbols (accept and reject).

To construct a working transducer we created a search trie, which is a tree encoding the words in the branches with the shared prefixes represented by a single path. Then we used the standard minimization algorithm based on the Myhill-Nerode Theorem [7]. The algorithm constructs an increasingly finer partition of the state set. Two states are equivalent if they yield the same output symbol for the same input word. Initially, we have the equivalence classes for the input symbols at the leaf nodes of the trie, and a class for all the undefined output for internal nodes. When we split an equivalence class, the states in other classes need to be checked for whether this new distinction can tell them apart. When there is no more splitting, the equivalence classes define the states of the minimal transducer.

This classical trie-minimization method is lot faster than the search algorithm in the logic engine, but in terms of number of states used it is far from the minimal values. Here is a comparison for the examples above.

Task

Minimal

Trie

Minimized

Signal Locator 9-3

5

45

24

Signal Locator 8-4

6

36

23

Zeroes and ones 4

5

31

13

Palindrome 4

5

31

12

Word classification

3

68

57

The main reason for the difference is that the trie method does not have the flexibility for assigning the output values to states. The word classification demonstrates that the prefixes are shared by the trie construction, and by optimization some word endings can be identified too. However, it is still about tracing the individual words letter by letter, hence the large number of states. Since we do not care about classifying subwords, the relational programming search method can find a minimal possible transducer.

5 Conclusion and Future Work

We applied relational programming to the problem of constructing minimal transducers for a given set of pairs consisting of input words and their corresponding output values. We compared the method with the classical method of minimizing a transducer constructed by a trie. In summary, the relational method produces a truly minimal transducer, but it is computationally intensive and does not scale well. In contrast, the traditional method does not yield a minimal transducer. This does not contradict the fundamental theorem about the unique minimal automaton for any deterministic finite state automaton, since we have flexibility in defining the output function, and we do not care about values associated with input words not in the training set.

There are at least two different ways for trying to further advance solution of this non-statistical machine learning problem:

1.

Developing an incremental construction algorithm using relational programming;

2.

Adopting more advanced classical approaches (e.g., [1]) to the flexible transducer construction.

Acknowledgements

This project was funded by the Kakenhi grant 22K00015 by the Japan Society for the Promotion of Science (JSPS), titled ‘On progressing human understanding in the shadow of superhuman deep learning artificial intelligence entities’ (Grant-in-Aid for Scientific Research type C, https://​kaken.​nii.​ac.​jp/​grant/​KAKENHI-PROJECT-22K00015/​). This research was also supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée en partie par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669.

References

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Aho, A.V., Corasick, M.J.: Efficient string matching: an aid to bibliographic search. Commun. ACM 18(6), 333–340 (1975). JuneMathSciNet

2.

Byrd, W.: Relational Programming in miniKanren: Techniques, Applications, and Implementations. Ph.D. thesis, Indiana University (2009)

3.

Egri-Nagy, A.: kigen Computational Semigroup Theory Software System written in Clojure (2023). https://​github.​com/​egri-nagy/​kigen

4.

Egri-Nagy, A, Dini, P., Nehaniv, C.L., Schilstra, M.J.: Transformation semigroups as constructive dynamical spaces. In: Colugnati, F.A.B., Lopes, L.C.R., Barretto, S.F.A. (eds.) Digital Ecosystems, pp. 245–265. Springer Berlin Heidelberg, Berlin, Heidelberg (2010)

5.

Friedman, D.P., Byrd, W.E., Kiselyov, O.: The Reasoned Schemer. The MIT Press (Jul. 2005)

6.

Hickey, R.: A history of Clojure. Proc. ACM Program. Lang. 4(HOPL) (Jun. 2020)

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Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Publishing Company (1979)

8.

Rhodes, J.: Applications of Automata Theory and Algebra via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games. World Scientific Press (2009). Foreword by Hirsch, M.W., edited by Nehaniv, C.L. (Original version: UC Berkeley, Mathematics Library, 1971)

9.

Silver, D., et al.: Mastering the game of Go with deep neural networks and tree search. Nature 529(7587), 484–489 (2016). January

10.

von Neumann, J.: Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100, 295–320 (1928)

Footnotes

1

Alternatively, one can define it as the Krohn-Rhodes complexity of the minimal automaton (see [8]).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_3

Skeleton Key: Subduction Classes in Finite Transformation Semigroups and Green’s Relations

Attila Egri-Nagy1   and Chrystopher L. Nehaniv2  

(1)

Akita International University, Akita, Japan

(2)

University of Waterloo, Waterloo, Canada

Attila Egri-Nagy (Corresponding author)

Email: egri-nagy@aiu.ac.jp

Chrystopher L. Nehaniv

Email: cnehaniv@uwaterloo.ca

Abstract

We establish key connections between Green’s J- and L-relations on a finite semigroup and the subduction relation defined on the image sets of an action of the same semigroup when it acts faithfully on a finite set. The construction of the skeleton order, the partial order on equivalence classes of the subduction relation, is shown to depend in a functorial way on transformation semigroups and surjective morphisms, and to factor through the Green’s ≤L-order and ≤J-order on the semigroup and through the inclusion order on image sets. For right regular representations, the correspondence between the J-class order and the skeleton order is one of isomorphism. Finally, we characterize the relationship between natural subsystems of a transformation semigroup, permutator groups and the H-relation.

Keywords

Finite transformation semigroupGreen’s relationsPartial orderSubduction pre-orderFunctoriality

1 Introduction and Mathematical Preliminaries

The analysis of how a transformation semigroup acts on the subsets of the state set which occur as images under the semigroup action is a natural and fruitful direction of exploration.¹ For a transformation semigroup, the study of this action on image sets is shown here depend on the fundamental Green’s relations on the semigroup, and thus is of broad intrinsic interest in the theory of finite semigroups and their faithful actions. (See below for precise definitions.)

One of the main tools of this analysis is the subduction preorder relation defined on the set of images of the members of the semigroup considered as mappings. Green’s preorders give ample information about the semigroup’s internal structure, while subduction captures details of the semigroup action. Therefore, the natural question arises: What is the connection between the Green’s relations and the subduction relation? More generally, by aiming to describe the connection between a semigroup action and the internal structure of semigroup itself we may yield insight and understanding into what transformation representations are possible for an abstract semigroup and the fine details of their structure. Lemma 3 and the theorems proved in this paper are new, elucidating the key relationships of concepts well-known in classical semigroup theory to those used in holonomy methods, such as the skeleton.

For clarity, we include some important mathematical preliminaries that will be used throughout this manuscript: (X, S) is a transformation semigroup with S acting faithfully on the state set X if S is a subsemigroup of the (right) full transformation semigroup T(X) of all mappings on a set X. For x∈X,s∈S, the result of the action is written xs. The action can be extended to subsets of X, if P⊆X and s∈S then Ps={xs∣x∈P}. The image of a transformation s is defined by λ(s)=Xs, and we can also say that λ(s) is the image of X under s. S1 is the monoid obtained by adjoining the identity on X to the semigroup S, if it is not a member of S, otherwise S1=S.

(A,≤) is a preorder (sometimes called a ‘quasi-order’) if ≤ is a reflexive and transitive relation on the set A. For a preorder, there exists an equivalence relation (A,≡) defined by a≡b⟺a≤b and b≤a, and an induced partial order on the equivalence classes (A/≡,≤). The surjective map A↠A/≡ is denoted by η.

The classical Green’s relations ≤R, ≤L, ≤J and ≤H, on any semigroup S are the preorders: t≤Rs⟺tS1⊆sS1, t≤Ls⟺S1t⊆S1s, t≤Js⟺S1tS1⊆S1sS1, and ≤H is the intersection of the ≤L and ≤R relations. Then R,L,J and H denote the equivalence relations arising from the preorders ≤R, ≤L, ≤J and ≤H, respectively, and in each case the equivalence classes carry the induced partial order. The equivalence relation D on S is the composite of L and R, which commute. In the finite case, the J and D relations coincide. (For elementary properties of Green’s preorders and their applications, see standard references for instance [1, 9, 10]). Here we only consider finite transformation semigroups.²

2 Subduction Relation

For a transformation semigroup (X, S) the set I(X)={λ(s)∣s∈S1} is the image set of the semigroup action. Note X=λ(1) is always in I(X).

Definition 1

(Subduction Pre-order) Given a transformation semigroup (X, S), for image sets P and Q in I(X)

P⊆SQ⟺∃s∈S1suchthatP⊆Qs.

That is, P is a subset of Q, or it is a subset of some image of Q under the semigroup action. We say that P is subduction below of Q, or alternatively, Q subduces P.

Lemma 1

1. (I(X),⊆S) is a preorder.

2. If P⊆SQ and Q⊆SP then |P|=|Q|.

Proof

Obviously ⊆S is reflexive, since P⊆P1. It is transitive, since if P⊆Qs1 and Q⊆Rs2 then P⊆Qs1⊆(Rs2)s1=Rs2s1, whence P⊆SR. To see (2), there exists s∈S1 with P⊆Qs, so Q has cardinality at least that of P. By symmetry, it follows that P and Q have the same cardinality.    □

Therefore, we can naturally define by mutually subduced subsets an equivalence relation on I(X), denoted by ≡S, leading to a partial order.³

Definition 2

(Skeleton Order) The skeleton ordering (I(X)/≡S,⊆S) for transformation semigroup (X, S) is the partial order on subduction equivalence classes of I(X).

3 J-Classes and Skeleton Classes

We can establish connection between the induced classes of two preorders through a given preorder preserving map. First, we describe the situation for preorders in general.

Let (A1,≤1) and (A2,≤2) be preorders. Then, a function f:A1→A2 respects preordering, if for all a,a′∈A1, a≤1a′ implies f(a)≤2f(a′).

For a pre-order (A,≤), the equivalence class of a∈A, is [a]={a′∈A:a≤a′anda′≤a}. The set of equivalence classes A/≡ {[a]:a∈A} carries the induced partial order (A/≡,≤) given by [a]≤[a′] if a≤a′.

The natural quotient map is the function η(a)=[a], which respects preordering and maps onto the induced partial order: η:(A,≤)→(A/≡,≤).

Lemma 2

Given a function f:(A1,≤1)→(A2,≤2) that respects pre-ordering, we have that

1.

f induces an order-preserving map f¯:(A1/≡1,≤1)→(A2/≡2,≤2), and the following diagram commutes,

where η1 and η2 are the natural quotients

2.

The equivalence relation induced on A1 by the composite map f¯∘η1=η2∘f is not finer than ≡1.

Proof

(1) Let f¯ denote the function taking the ≡1-class [a]1 of any a∈A1 to the ≡2-class [f(a)]2 of f(a). If a≡1a′ then by definition a≤1a′ and a′≤1a. Since f respects preordering, f(a)≤2f(a′) and f(a′)≤2f(a), therefore f(a)≡2f(a′). It follows that f¯, given by f¯([a]1)=[f(a)]2, is well-defined and order preserving. (2) Since f¯ is a function, the inverse image B of an ≡2-class is a set of ≡1-classes. Hence the inverse image of B in A1 is the union of these ≡1-equivalence classes.    □

Remark: It is important to notice that a<1a′ does not exclude the possibility of f(a)≡2f(a′). Moreover, even if neither a≤1a′ nor a′≤1a holds one might still have f(a)<2f(a′) or f(a)≡2f(a′).

Now we have two preorders: ≤J on S and subduction ⊆S on I(X). Next we show that the surjective function λ respects preordering. For the weaker case, it is a basic fact that aLb⇒λ(a)=λ(b). However, J-related elements can have different images. For instance, in the full transformation semigroups on n points, all constant maps are J-equivalent to each other.

Lemma 3

For any transformation semigroup (X, S) and any a,b∈S, we have

a≤Lb⇒λ(a)⊆λ(b).

a≤Jb⇒λ(a)⊆Sλ(b).

That is, λ maps the L-preorder to the inclusion partial order and maps the J-preorder to the subduction preorder. Moreover, λ induces a surjective map from S1 in each case.

Proof

The first assertion is well-known: If a≤Lb then a=sb for some s∈S1. Thus λ(a)=Xa=Xsb=(Xs)b⊆Xb=λ(b).

For the second, if a≤Jb then there exist s,t∈S1 such that a=sbt,

λ(a)=λ(sbt)=λ(sb)t⊆λ(1b)t=λ(b)t,

therefore λ(a)⊆Sλ(b). Obviously λ maps S1 surjectively onto I(S)={λ(s):s∈S}, hence onto the preorder (I(X),⊆S) which has the same underlying set.    □

Theorem 1

For a transformation semigroup (X, S), there is a surjective order-preserving map λ¯S from the partial order of J-classes (S1/J,≤J), onto the partial order of subduction classes (I(X)/≡S,⊆S). The inverse image of a subduction equivalence class is a union of J-classes.

Proof

λ is a surjective, and is a preorder morphism from the Green’s J preorder to the subduction preorder by Lemma 3, therefore by using Lemma 2(1), the induced map λ¯S is a surjective order-preserving map. By Lemma 2(2), the inverse image of a subduction class corresponds to a union of J-classes.    □

Similarly, generalizing the basic fact mentioned above, we have

Theorem 2

For a transformation semigroup (X, S), there is a surjective order-preserving map λ¯ from the L-class order for S1 onto the inclusion partial order on I(X). The inverse image of an image set is a union of L-classes.

Putting these facts together, it follows that

Theorem 3

For any transformation semigroup (X, S), there is a commutative diagram of surjective order-preserving morphisms:

Corollary 1

For the right regular representation (S1,S):

1.

The J-class order and the subduction order are isomorphic.

2.

The L-class order and the inclusion order on image sets I(X) are isomorphic.

Proof

(1) By Lemma 3, it suffices to show that λ(a)⊆Sλ(b)⇒a≤Jb. By definition of subduction λ(a)⊆λ(b)t for some t∈S1. Since X=S1 we can write λ(a) as (S1)a, or by shifting notation from semigroup action to semigroup multiplication, simply as S1a. Therefore,

S1a⊆S1bt⇒S1aS1⊆S1btS1⊆S1bS1⇒a≤Jb.

It follows that, if λ(a)≢Sλ(b) then aJb does not hold. Thus λ¯S is injective, hence bijective.

(2) More simply for the L-order, λ(a)⊆λ(b) in the case of the right regular representation is just (S1)a⊆(S1)b, i.e., S1a⊆S1b, which is the definition of a≤Lb. Hence, λ(a)⊆λ(b) implies a≤Lb. By Lemma 3 for the ≤L-preorder, the converse holds. It follows that if λ(a)≠λ(b) then it cannot be that aLb, hence λ¯ is injective, and hence bijective as well.    □

In the case of the right regular representation this says that the horizontal mappings in Theorem 3 are order isomorphisms.

Both the J-class order and the skeleton capture information about the structure of the semigroup, therefore surjective homomorphisms should respect them.

Theorem 4

(Functoriality) Suppose φ:(X,S)↠(Y,T) is a surjective morphism of transformation semigroups such that if 1∈S then φ(1) is the identity on Y. Then φ induces a natural mapping of the commutative diagram for (X, S) as in Theorem 3, to the commutative diagram for (Y, T).

Proof

A surjective map of semigroups induces a surjective map of the ≤L and ≤J pre-orders and orderings (as well as for ≤R and ≤H). φ also induces a surjective map from I(X) onto I(Y), and subduction in the source implies subduction in the target since P⊆Qs implies φ(P)⊆φ(Qs)=φ(Q)φ(s), hence the subduction relation is respected, and the result follows.    □

4 Examples

We present a few examples to illustrate the connection between the J-class order and the skeleton. The partial orders are displayed as Hasse diagrams. Shaded clusters of J-classes are mapped to a single subduction class.

Example 1

(Simple collapsing of a chain) Let X={1,2,3}, t1=123133, t2=123313, t3=123333 and M the monoid {1,t1,t2,t3}, so (X, M) is a transformation monoid on 3 points. The principal two-sided ideals are: M1M=M, Mt1M={t1,t2,t3}, Mt2M={t2,t3}, Mt3M={t3}, therefore t3<Jt2<Jt1<J1 and all elements form a singleton J-class on their own. I(X)={{1,2,3},{1,3},{3}} defines the subduction classes.

A simple linear order is mapped to a shorter linear order, since λ(t1)=λ(t2)={1,3}.

Example 2

More general collapsing (a usual motif) for a transformation monoid on 3 points, M={1,123113,123323,123313,123333}. The right regular transformation representation of M can be encoded as

M′={1,1234522445,1234535355,1234545455,1234555555}.

Its skeleton is isomorphic to the J-class order of M.

Example 3

M monoid generated by a=1234521114 and b=1234512234. In M, a and b are ≤J-incomparable, but λ(a)⊂Sλ(b). This is so as there is no solution for the equation b=sat or a=sbt for s,t∈M, although λ(a)⊂λ(b).

This shows that the subduction order may contain new relations beyond those induced by collapsing nodes of the J-order diagram. Consequently, the length of a longest J-chain is not an upper bound for the height of the skeleton.

So far the J-class orders were all lattices, but this is not true in general, therefore we have to look at a monoid with more inner structure.

Example 4

(Nonlinear, non-lattice skeleton) Let a=1234522124, b=1234535232, c=1234535454 and M=⟨a,b,c⟩. |M|=31, |I(X)|=16, number of D-classes is 13, and the number of skeleton classes is 9. On the left the D-class picture is drawn. On top of each L-class (drawn vertically) the corresponding image is displayed. H-classes with an idempotent are shaded. The grey background blobs indicate D-classes that are collapsed into one subduction class. On the right the skeleton order is drawn. It is nonlinear and it is not a lattice. The boxes indicate subduction equivalence classes.

The skeleton also contains nonsingleton subduction equivalence classes.

5 Conclusion

We clarified the connection between the J-classes of a semigroup and the subduction classes of a transformation representation of the same semigroup. We showed how the partial order of J-classes constrains the image relations in the possible (faithful) actions of the semigroup. Therefore, these results may also be useful for investigating or enumerating the possible action representations of a semigroup, and for investigating structures used in decompositions that are respected under surjective maps.

For calculating and checking the examples we used the Gap [6] packages Semigroups [11], SgpDec [4] and SgpViz[2].

Acknowledgements

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée en partie par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669.

Appendix: Natural Subsystems, Permutators, H-Classes

If s∈S1 and P⊆Ps, then Ps=P by finiteness and so s permutes P. The permutator of P denoted Perm(P) is the group of all such permutations of P, and consists of restrictions of all such s to P. If P=Xe for an idempotent e, and Ge is the unique largest subgroup of S containing e, then (P,Ge) is called a natural subsystem of (X, S).

Theorem 5

A natural subsystem (P,Ge) is a faithful permutation group, and

1.

(P,Ge)≅(P,Perm(P)), where the isomorphism from Ge to Perm(P) is given by restriction to P. Moreover Ge=H(e), the H-class of e.

2.

Also, for J-equivalent idempotents e1 and e2, (Xe1,Ge1)≅(Xe2,Ge2), and for L-equivalent idempotents, the state sets coincide.

The proof relies on some facts:

Fact 1

If e2=e and s∈S permutes Xe, then s, ese, es, and se agree as transformations on Xe.

Proof

Take any z=x·e∈Xe. Then e acts as the identity on Xe, since z·e=(x·e)·e=x·e2=z. Since s maps Xe to Xe, it follows that z·es=(z·es)·e=z·ese=(z·e)·se=z·se=(z·s)·e=z·s.    □

Fact 2

For e2=e and s∈S permutes P. Let w>1 such that (ese)w is the unique idempotent power of ese. Then (1) (ese)w=e and (2) ese∈Ge.

Proof

By Fact 1, ese permutes Xe and agrees with s on Xe. Obviously Xese=(Xes)e⊆Xe. Now (ese)w is idempotent and so fixes its image, which is Xe, since e fixes Xe, s permutes Xe and then e fixes the result, for w iterations. Take x∈X. Now, x·(ese)w=(x·e)·(ese)w=x·e. Thus the two transformations e and (ese)w agree for all x∈X. Since (X, S) is faithful, s=(ese)w. Since w>1, (ese)w-1(ese)=(ese)w=(ese)(ese)w-1, and as the element in middle is e, we have shown that (ese)w-1 is a multiplicative inverse of ese with respect to e. Therefore ese is contained in a group ⟨ese⟩ having two-sided identity element e. Therefore ese∈Ge, the unique maximal subgroup of S containing e.    □

Proof of Theorem 5. That (P,Ge) is a permutation group is is well-known and can be shown as follows: To see that (P,Ge) is faithful, suppose that g,g′∈Ge have the same action on all of P=Xe. Let x∈X, then x·g=x·e.g.=(x·e)·g=(x·e)·g′=x·g′, since x·e∈P. Thus g and g′ have the same action on X, whence g=g′ since (X, S) is faithful. Since e acts as the identity on P=Xe, it is easy to check that Ge acts on P by permutations, with each g-1 acting as the inverse of the mapping on Xe due to g∈Ge.

Facts 1 and 2 together show that any permutation of Xe realized by some s∈S can be realized by a member of Ge, namely ese. That is, for all elements s of S permuting Xe, there is a transformation among the elements of the single group Ge having the same restriction to domain Xe. Indeed, s↾Xe=ese↾Xe. The maximal subgroup Ge of S with identity e is H(e), the H-class of an idempotent (cf. e.g. [9]). This proves Theorem 5(1), while Theorem 5(2) follows from Lemma 3 that J-equivalent elements are subduction equivalent and L-equivalent elements have the same image. Since subduction equivalent image sets have isomorphic permutator groups (e.g. [5, 13]), the proof is complete.    □

References

1.

Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. 1, 2nd ed., no. 7 in Mathematical Surveys. American Mathematical Society (1967)

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Delgado, M., Morais, J.: GAP package SgpViz0.999.5 (2022). https://​gap-packages.​github.​io/​sgpviz/​

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Dömösi, P., Nehaniv, C.L.: Algebraic theory of finite automata networks: an introduction, vol. 11 of SIAM Series on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (2005)

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Egri-Nagy, A., Nehaniv, C.L., Mitchell, J.D.: SgpDec—Software Package for Hierarchical Composition and Decomposition of Permutation Groups and Transformation Semigroups, Version 1.1.0 (2024). https://​gap-packages.​github.​io/​sgpdec/​

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Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press (1976)

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The GAP Group: GAP—Groups, Algorithms, and Programming, Ver. 4.13.0 (2024)

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Ginzburg, A.: Algebraic Theory of Automata. Academic Press (1968)

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Holcombe, W.M.L.: Algebraic Automata Theory. Cambridge University Press (1982)

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Howie, J.M.: Fundamentals of Semigroup Theory, vol. 12 of London Mathematical Society Monographs New Series. Oxford University Press (1995)

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Lallement, G.: Semigroups and Combinatorial Applications. Pure and Applied Mathematics. Wiley, New York (1976)

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Mitchell, J.D., et al.: Semigroups—GAP Package, Version 5.3.7 (Mar. 2024)

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Wells, C.: A Krohn-Rhodes theorem for categories. J. Algebr. 64, 37–45 (1980)MathSciNet

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Zeiger, H.P.: Cascade synthesis of finite state machines. Inf. Control 10(4), 419–433 (1967). Erratum: 11(4): 471 (1967)

14.

Zeiger, H.P.: Cascade decomposition using covers. In: Arbib, M.A. (ed.) Algebraic Theory of Machines, Languages, and Semigroups, chapter 4, pp. 55–80. Academic Press (1968)

Footnotes

1

It leads, for example, to the holonomy decomposition algorithm [3, 5, 7, 8, 12–14] in Krohn-Rhodes theory, a widely applied wreath product decomposition theorem for finite transformation semigroups.

2

Nevertheless, we shall refer the J-ordering and J-classes, rather than denoting J- by D- as some authors working with finite semigroups do, since by definition D is an equivalence relation that does not necessarily come from a preorder in the general setting of all (finite and infinite) semigroups (see e.g. [10] for examples).

3

This structure provides the scaffolding for a holonomy decomposition since subduction equivalent subsets have isomorphic holonomy permutation groups, so only one copy of these groups is needed per class in the decomposition [3, 5, 7, 8, 13, 14]. For the holonomy decomposition the skeleton order is extended by using the extended image set I+(X)=I(X)∪{{x}:x∈X} which includes any singletons that do not occur as images. This could potentially result in additional minimal equivalence classes for these singletons being adjoined to the skeleton.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_4

The Attractor-Cycle Notation for Finite Transformations

Attila Egri-Nagy1   and Chrystopher L. Nehaniv2  

(1)

Akita International University, Akita, Japan

(2)

University of Waterloo, Waterloo, Canada

Attila Egri-Nagy (Corresponding author)

Email: egri-nagy@aiu.ac.jp

Chrystopher L. Nehaniv

Email: cnehaniv@uwaterloo.ca

Abstract

We describe a new notation for finite transformations. This attractor-cycle notation extends the orbit-cycle notation for permutations and builds upon existing transformation notations. How the basins of attraction of a finite transformation flow into permuted orbit cycles is visible from the notation. It gives insight into the structure of transformations and reduces the length of expressions without increasing the number of types of symbols.

Keywords

Operator on a finite setDiscrete dynamical systemAttractor-cycle notationContext-free grammarComputer algebrageneralization of orbit-cycle notation

1 Introduction

What is the correct generalization of the orbit-cycle notation for permutations to the case of arbitrary total functions f:X→X on a finite set X? The discrete dynamical system that f gives on X can be visualized as digraph with nodes x∈X and directed edges (x, f(x)). Iterating f maps a point to x to f(x), to f(f(x)), and so on, until eventually some fm(x)=fm+k(x). Taking the least m≥0 for which this happens, and the least positive k for that m, shows that x eventually must enter a (possibly degenerate) periodic orbit from which it never leaves. Points x and x′ eventually entering the same periodic cycle are said to be in the same basin of attraction or generalized cycle. Drawn as digraphs, transformations may have several such disjoint basins of attraction, each consisting of a cycle of points with incoming trees (connected acyclic subgraphs). Figure 1 gives an example with four generalized cycle components in our notation, which is formally introduced in Sect. 3 with further examples. For a permutation, this digraph is a set of disjoint cycles. However, for a transformation, the points in a cycle can have incoming edges from outside the cycle, so a notation for transformations has to deal with these tree structures. The trees, which could also be degenerate, are directed toward the cycle, and there may be a tree of points coming into any given point of the cycle.

We call each basin of attraction associated to f, a generalized cycle or a component, since it is a connected component of the digraph of f. We may restrict our attention to a single component only, since we can write f by concatenating our notation for what f does on each of its components.

The basins of attractions have been studied in the context of dynamical systems as they describe the global dynamics of a temporally evolving system [10]. It is important to note that this is the only dynamics in systems where the only action is the unidirectional passage of discrete time.¹

Various previous notations have been developed for representing such transformations on a set (Sect. 2). The aim of these notations is to give useful information about the transformation without drawing the corresponding digraph. Readability is not an easily measurable quantity, but length and the number of distinct symbols used are influencing factors. The real importance of an efficient notation for transformations lies in the growing use of computer algebra systems in finite semigroup theory, as well as in mathematical calculations with transformations.

Fig. 1

A simple discrete dynamical system. A transformation g:X→X with four components, i.e., basins of attraction (generalized cycles), is visualized as a digraph with arrows (x, g(x)) for each x∈X={1,…,17}. In our canonical form of the attractor-cycle notation, g is denoted [[[1|3,2]|5,4],6]([[7|8,9],10],11,[14,13,12])(16,17). Note: Singleton components are not written, just as in orbit-cycle notation for permutations. Full, formal details for the notation are given in Sect. 3

2 Existing Notations

Several ways for generalizing cyclic notation have been developed based on the particular purposes of each situation in which they were defined.² Here we give a quick summary of previous suggestions. Throughout this paper, writing x↦f(x) denotes as usual that f maps x to f(x). Our running example, whose digraph is visualized in Fig. 2, with just one nontrivial component (basin of attraction) will be the transformation typically written in so-called Cayley notation as

1234521233.

This common notation denotes the function on the set {1,2,3,4,5} taking an element x in the first row to the element written f(x) immediately below it on the second row.

Fig. 2

An irreversible transformation on 5 points with just one basin of attraction. Here a non-degenerate tree collapses in two iterations into a cyclic permutation (1, 2)

Caveat: As it often happens in mathematics, the notations surveyed in this section associate different meanings to the same symbols.

2.1 Path Notation for Partial Symmetries and Partial Transformations

In addition to parentheses (,) for permutation orbits, path notation introduced by S. Lipscomb [7, 8] uses square bracket ] following elements with no images in partial permutations. For partial transformations the paths connect into other paths and ultimately into the cycles. This is denoted by the symbol > , may be used as a visual indication of being funneled into a cycle. The example written in path notation is

(1,2)(4,3,2>(5,3,2> ,

which redundantly represents the paths going into point 2.

2.2 Factorization Notation

In order to avoid path redundancy, Ayik et al. [1] introduced a different notation. Instead of decomposing the transformation into paths and cycles, this notation decomposes it into a very particular type of generalized cycle, i.e. into a product of transformations given by the trajectory of a single point. The example written in path notation is

[4,3,2,1|2][5,3|3] =1234521233,

where the element after the | shows where the path connects to itself to form a cycle. This notation is excellent for describing factorizations in the full transformation semigroups, but it introduces maps that might appear to move things in ways that are not present in the original transformation, since branches connecting to an already described path are terminated with an artificial self-loop. For instance, 3↦3.

Also, the decomposition is not unique.

2.3 Linear Total Transformations

In [5] a new notation was introduced by O. Ganyushkin and V. Mazorchuk, aiming to provide a natural extension of the cyclic notation of permutations. Instead of decomposing trees into paths, linear notation describes the trees explicitly. Trees with one level of branching are denoted by [preimages of root ; root ], where preimages are separated by commas and all map to the root. If a preimage element also has incoming edges from other points, then the same square bracket structure is applied again recursively to specify the tree leading to that element. Although called ‘linear’ as a one-line notation, the notation describes recursive tree structure.

Parentheses are used to indicate the existence of a nontrivial permutation of the roots of the trees (which may include degenerate trees consisting of a single point). Basically the linear notation is the usual cyclic notation used for permutations but the elements in the cycle describe their incoming tree information. The running example is written as

([[4,5;3];2],1).

One of the most useful features of linear notation is that when restricted to permutations it is identical to orbit-cycle form. Also, by looking for parentheses we can easily spot the existence of nontrivial permutations even in large examples. The only drawback is that describing a transformation given by following a simple path, a “line” of maps, requires many square brackets. For instance,

becomes [[[[1;2];3];4];5]. With the attractor-cycle notation we aim to alleviate this particular problem.

2.4 Specialized Notation for Fast Iterations

For special purposes one can develop specialized notations. For example in [9], for the purpose of the efficient calculation of fm(x), the example would be first decomposed into components ((1,2),(4,3),(5)), then the orbits concatenated (1,2,4,3,5) in order to assign indices to points. This of course lacks the information on how to put together the orbits, therefore an auxiliary sequence is needed to encode where (which position) the last elements of the orbit components connect to: f(2) is in position 1, f(3) in 2 and f(5) in 4. So the full information is encoded in these strings:

((1,2),(4,3),(5)), (1,2,4,3,5), (1,2,4).

This illustrates that efficient algorithmic representations do not necessarily provide human readable formats.

3 The Attractor-Cycle Notation

Informally, the attractor-cycle notation makes use of the following conventions and symbols. Formal semantics of the notation are in the next subsection. Here we illustrate the concepts for points:

conveyor belt

Left-to-right comma-separated points. For example, 1,2,3 reads as 1↦2, 2↦3.

cycle

Putting parentheses around a conveyor belt connects the last point to the first , i.e., (1,...,n) adds the map n↦1.

tree operator

Square brackets group points and do not specify the image of the last point. Based on context, this is resolved into one of 3 possibilities.

1.

[...n],k...defines the map n↦k, connecting into a conveyor belt;

2.

(m,...[...n]) defines n↦m , closing a cycle;

3.

[...n] corresponds to n↦n, defining a loop edge.

In short, [...n] is a tree “flowing” into point n, and the context of n defines what happen to n itself.

parallel paths

A vertical bar | suggestive of logical OR is written to separate two or more points (respectively, subtrees) that go to the same point. For instance, 1|2|3,4 means 1↦4, 2↦4, and 3↦4.

3.1 Examples

Any permutation in the attractor-cycle notation is the same as the cyclic notation. In particular, the identity is simply ().

A constant map to n is written as [1|2|...|n-1,n], while a simple path trajectory from 1 to n is denoted by [1,2,...,n]. This compact notation for a path is also better matching the cycle notation for permutations.

The transformation whose digraph is in Fig. 3 on 9 points is written as

([[8|9,5]|6|7,1],[4,3,2]).

The parentheses immediately tell that this transformation has a non-trivial permutation component. Collapsing the tree operators to their roots, we can see that the cycle is (1,2).

Fig. 3

Digraph for an example transformation with a ‘conveyor belt’ trajectory [4,3,2] and a branching tree [[8|9,5]|6|7,1] collapsing into a transposition (1,2). In attractor-cycle notation the whole transformation is denoted ([[8|9,5]|6|7,1],[4,3,2])

3.2 Syntax

The following context-free grammar defines the language of attractor-cycle notations. The terminal symbols are [,],(,),,,| and the symbols for the n points. The nonterminal symbols are C for components, F for in-flows, T for trees and P for points. Note the difference between | and |. The bold symbol | is used as part of the notational grammar defined here, while | is part of the metasyntax (Backus-Naur Form (BNF) for context-free grammars) for describing the structure of the new notation. Similarly, parentheses, i.e., (, ), and also +, are part of the metasyntax, while bold parentheses, i.e., ( and ), are terminal symbols of the attractor-cycle notation.

S→C+∣()

(1)

C→((T,)+T)∣F

(2)

F→[(T,)+P]∣[(T|)+T,P]

(3)

T→F∣P

(4)

P→1∣2∣3∣…∣n

(5)

Rule (1) states that the attractor-cycle notation denotes a positive number nontrivial components (generalized cycles, i.e. basins of attraction), or we can have just an empty cycle for the identity transformation. Rule (2) says that a component can be a in-flow (where something flows into a point) or a permutation cycle (with incoming trees to its members). Rule (3) describes the two cases of an in-flow: either a tree with multiple branches leading to the same root (parallel branches) or a path (conveyor belt). Rule (4) defines a tree to be an in-flow. It allows a point to be a tree. Rule (5) specifies the points.

An important additional constraint in the notation is that each point can only occur at most once.³

3.3 Semantics

We define the semantics by recursively interpreting the valid attractor-cycle notation words as a collection of individual maps of points. We write such a map p↦q as a pair (p, q). We will put the total function f together from these pieces. Using the parse tree of a well-formed word w in the notation, we denote its interpretation by I(w). This determines a unique function f from {1,…,n} to itself, by identifying f with the set of pairs I(w). If w=(), then I(w) is the identity transformation. Otherwise w is derived using S→C1…Ck, and we define

I(S):=⋃i=1kI(Ci)∪{(p,p)∣f(p)isnotdefinedinanyI(Ci)}.

(6)

We label subtrees to define an auxiliary function r that gives the root of a tree, so we let

r(p)=r([T1|…|Tk,p])=p∈{1,…,n}.

andr([T1,…,Tk])=r(Tk).

Then the interpretation of a component derived from a nonterminal symbol C is:

I(C):=I(F)ifC→F⋃i=1kI(Ti)ifC→(T1,…,Tk)∪(r(Ti),r(Ti+1)):1≤i<k∪(r(Tk),r(T1))

(7)

A tree gives a nonempty set of pairs, if it is a nontrivial in-flow. A point (degenerate tree) does not yield a map.

I(T):=I(F)ifT→F∅ifT→P

(8)

A nontrivial tree is an in-flow and it gives a set of pairs as follows:

I(F):=⋃i=1kI(Ti)∪(r(Ti),p):1≤i≤kifF→[T1|…|Tk,p]⋃i=1kI(Ti)ifF→[T1,…,Tk]∪(r(Ti),r(Ti+1)):1≤i<k

(9)

The notation w thus clearly determines a unique well-defined transformation f=I(w) on {1,…,n} to itself, since each element p appears at most once in w. The transformation is total (i.e., not partial, but fully defined) by Eq. 6. Moreover, every f can be written in this attractor-cycle notation in a canonical form.

3.4 Canonical Form

The price to pay for the short length is the loss of uniqueness. Both [1,2,[3|4,6]] and [[1,2]|3|4,6] denote the same transformation. One can simply choose between a conveyor belt (comma-separated list of elements in a trajectory) or the parallel branches (using |). However, a simple recursive algorithm that starts from the point(s) of each component’s cycle can produce a canonical form. All we need to do is to examine the cardinality of the preimage set of each cycle element from outside the cycle, i.e. the number of incoming arrows.

1.

If there is no incoming arrow, then we have a leaf of the tree and recursion stops here.

2.

If there is only one preimage, then a conveyor belt is built by traversing the tree as long as there is only one preimage. Then recursion is done on the first element of the conveyor belt built so far.

3.

In case there is more than one element in the preimage then parallel branches need to be used with recursion on all parallel elements.

For each transformation f on n points one can now obtain a completely canonical expression in this notation by additionally requiring the C’s to appear in order according to their least elements, and that the trees separated by a | are ordered by their least element, and the cycles start with their least point (or tree root) first.

For instance, here are the conjugacy class representatives of the full transformation semigroup T4 on four points in canonical form:

[1|2|3,4]

[[1,2]|3,4]

[1|2,3]

[[1|2,3],4]

[1,2,3,4]

[1,2,3]

[1,2][3,4]

[1,2]

[1,2](3,4)

()

(1,2)

([1,2],3)

(1,2,3)

([1|2,3],4)

([1,2],[3,4])

([1,2,3],4)

(1,2)(3,4)

([1,2],3,4)

(1,2,3,4)

Looking at the list, it is easy to spot the existence of nontrivial cycles and idempotents. Idempotents, other than the identity (), are exactly those transformations given by concatenating a number of conveyor belts of length 2, [x,y], and single-level parallel branches [x1|...|xk,xk+1]. Also, a common feature of all the notations discussed here is that conjugation by a permutation is just relabelling of points according to the permutation, e.g. (1,2,3)-1([1,2],3,4)(1,2,3)=([2,3],1,4).

4 Computer Algebra Implementation

We implemented the attractor-cycle notation in our semigroup decomposition computer algebra package SgpDec [3, 4] available for the GAP system [6].

To verify the correctness, we take a transformation, construct the corresponding notation string, then parse this string back to a transformation, and finally we check whether the input is identical to the output. We conducted this test systematically for all transformations on up to 8 points (88 transformations), and then for random transformations on up to 220 points (more than one million).

These tests also indicate the practical usability. Parsing context-free grammars are known to be efficient (polynomial [11]), and constructing expressions in the notation is essentially the reverse process of the recursive parsing algorithm.

5 Conclusion

By the spreading use of computer algebra systems for investigating transformations and discrete dynamical systems, efficient notation has become a necessity. The orbit-cycle has been described in numerous group theory textbooks. Transformations can have the same status as well.

In the attractor-cycle notation we tried to blend the best features of previous notations and also considered the computational experience to derive what we have found to be a useful and readable notation, giving insight into the structure of transformations.

Acknowledgements

This work reported in this article was funded in part by the EU project BIOMICS, contract number CNECT-318202, and by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. This support is gratefully acknowledged.

References

1.

Ayik, G., Ayik, H., Howie, J.M.: On factorisations and generators in transformation semigroups. Semigroup Forum 70, 225–237 (2005)MathSciNet

2.

Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 14, 143–172 (1961)

3.

Egri-Nagy, A., Nehaniv, C.L., Mitchell, J.D.: SgpDec—Software Package for Hierarchical Composition and Decomposition of Permutation Groups and Transformation Semigroups, Version 1.1.0 (2024). https://​gap-packages.​github.​io/​sgpdec/​

4.

Egri-Nagy, A., Mitchell, J.D., Nehaniv, C.L.: Sgpdec: cascade (de)compositions of finite transformation semigroups and permutation groups. In: Mathematical Software ICMS 2014, vol. 8592 of LNCS, pp. 75–82. Springer (2014)

5.

Ganyushkin, O., Mazorchuk, V.: Classical Transformation Semigroups. Algebra and Applications. Springer (2009)

6.

The GAP Group: GAP—Groups, Algorithms, and Programming, Ver. 4.13.0 (2024)

7.

Howie, J.M.: Review of “Symmetric Inverse Semigroups (Mathematical Surveys and Monographs 46) by Stephen Lipscomb: 166 pp., US\$49.00, ISBN 0 8218 0627 0 (American Mathematical Society, 1996)”. Bull. Lond. Math. Soc. 29(6) (1997)

8.

Lipscomb, S.: Symmetric Inverse Semigroups, vol. 46 of Mathematical Surverys and Monographs. American Mathematical Society (1996)

9.

Tsaban, B.: Decompositions of graphs of functions and fast iterations of lookup tables. Discrete Appl. Math. 155(3), 386–393 (2007). FebruaryMathSciNet

10.

Wuensche, A., Lesser, M.: Global Dynamics of Cellular Automata: An Atlas of Basin of Attraction Fields of One-dimensional Cellular Automata. Santa Fe Institute Studies in the Sciences of Complexity: Reference Volumes. Avalon Publishing (1992)

11.

Younger, D.H.: Recognition and parsing of context-free languages in time n3. Inf. Control 10(2), 189–208 (1967)

Footnotes

1

This is in contrast to more general discrete dynamical systems which are generated by several such transformations on a finite set X, as studied in transformation semigroup theory.

2

Somewhat ironically, we also developed yet another version (called compact notation), before settling on the attractor-cycle notation presented here.

3

As the intersection of a regular language with a context-free one is still context-free [2], we may constrain the strings generated by the above context-free grammar to have no repeated points (i.e., by intersecting with the regular language of all strings w over all the symbols we use such that w does not contain any repeated point symbol from among 1,2,3,…,n ) and obtain a context-free language for the attractor-cycle notation without any repeated points.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_5

Algebraic Applications in Investigation of Musical Symmetry

Olga Ibragimova1   and Chrystopher L. Nehaniv1  

(1)

University of Waterloo, Waterloo, Canada

Olga Ibragimova (Corresponding author)

Email: oibragim@uwaterloo.ca

Chrystopher L. Nehaniv

Email: cnehaniv@uwaterloo.ca

Abstract

This study explores how mathematical group theory can reveal the structure of musical compositions, particularly focusing on transformations such as transposition, inversion, retrograde, and translation in musical works. By reducing melodies to their pitch classes and applying algebraic operations, we uncover new insights into the symmetrical properties of music. This method simplifies the connection between advanced mathematical theories and practical music analysis, enabling the detection of complex patterns and symmetries that traditional techniques may overlook. The results explicitly reveal the duality between tonal and positional symmetry aspects of melodic transformations, acting on the space of abstract melodies including repeated notes and rests. We give an explicit form for a melody of the given length L resulting from application of a symmetry separating the positional and tonal aspects, and explicit recipes for construction and enumeration of symmetric melodies using this duality of structure.

Keywords

Musical symmetryGroup theoryPitch-class transformationsDihedral groupsPermutation groups

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée en partie par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669.

1 Introduction

This study examines algebraic symmetries in musical melodies, drawing from seminal literature linking mathematics and music theory. Schuijer provides a comprehensive overview analyzing chromatic music through pitch-class set theory, which facilitates the understanding of post-tonal compositions [1]. Hook [2] and Peck [3] developed these ideas by applying wreath products to characterize chord transformations. Benson situated tone row operations like transposition and inversion within group theory, exploring compositional strategies in twelve-tone music [4]. Hunter and von Hippel used permutation groups to demonstrate how specific transformations contribute unique structural and aesthetic properties to Viennese serial works, noting their statistical rarity [5]. Fripertinger and Lackner’s group-theoretic classification of tone rows expands analytical tools for serial compositions [6]. Yust explores how discrete Fourier transforms reveal tonal/atonal dimensions, influencing perceptual tonality in twelve-tone music [7]. Inspired by Lewin’s transformational theory [8], this study solidifies connections between algebraic symmetries and music analysis.

This study explores algebraic symmetries in melodies, focusing on transformations like transposition, inversion, retrograde, and translation. Analyzing the action of these operations on melodies algebraically aims to deepen the understanding of structural symmetry in musical compositions, offering new insights into the interaction of these transformations.

2 Musical Symmetry and Group Theory

Exploring the mathematical aspects of musical compositions, we focus on the fundamental melodic structures of music theory. In this context, Group Theory becomes a valuable tool for analyzing and reassembling melodic elements.

2.1 The Algebra of Melodies and Symmetry Operations

In the algebraic framework used in this study, ‘melodies’ are abstractly defined at an intermediary level between conventional melodies and pitch-class rows, focusing solely on pitch classes derived from a twelve-tone chromatic scale. This abstraction disregards specific pitches, durations, loudness, and timbre, ensuring that regardless of transformations such as transposition, inversion, retrograde, or translation, the integrity of the pitch class is consistently preserved. This approach aligns with Lewin’s principles in transformational theory, emphasizing the structural rather than the perceptual qualities of music [8].

The pitches can be denoted both musically and numerically:

N={C,D♭,D,E♭,E,F,G♭,G,A♭,A,B♭,B},

(1)

corresponding to the integers:

N={1,2,3,4,5,6,7,8,9,10,11,12}.

(2)

Fig. 1

a Chromatic Scale on a Piano Keyboard, with each key assigned a corresponding integer. b Color-mapped Chromatic Scale Inversion (inspired by Eugenia Cheng’s Math in Music [9])

Figure 1a depicts the chromatic scale on a piano keyboard, with each key assigned a corresponding integer, with notes in different octaves repeating the same numbering modulo 12.

In the study of musical structure, various symmetry operations can be applied to a melody to transform it while retaining its inherent properties. These operations include transposition, translation, inversion, and retrograde. Each operation alters the melody in a distinct manner, providing a plethora of possibilities for musical variation and development.

Transposition is the operation of shifting a collection of notes up or down by a specified number of semitones. Given a melody M=(m1,m2,...,mL) consisting of notes mi from the set N, transposition can be defined algebraically. For a transposition by n semitones, where n∈Z, the transposed melody M′ is obtained by M′=(m1+n,m2+n,...,mL+n), where addition is modulo 12.

Translation in music can be thought of as shifting the position of each note in a melody. For a melody M, a translation operation moves each note’s position by an amount n, where n∈Z+, within the melody’s sequence. This operation effectively cycles the positions of the notes: M′=cycle(M,n)=(m1+n,m2+n,...,mL+n), where addition is modulo L, for the melodies of length L.

In the realm of music theory, the concept of Inversion is multifaceted, possessing varied interpretations contingent upon the context of its application within musical compositions. Within the scope of this work, inversion is redefined as a systematic transposition of each pitch to its diametrically opposed counterpart, as explicated in Fig. 1b where each note is mapped to the other note of the same color, or, numerically, the note with number m is mapped to 13-m, 1≤m≤12, i.e., Inv(m)=13-m.

Then extending to melodies

Inv(M)=(Inv(m1),Inv(m2),...,Inv(mL)).

The Retrograde operation reverses the order of notes in a melody, effectively playing it from end to beginning. Applying retrograde twice will yield the original melody. For a melody M, the retrograde R(M) is:

R(M)=reverse(M)=(mL,…,m1).

Historically, transformations such as transposition, inversion, and retrograde originated not merely in theoretical studies but as practical tools for understanding composition. Bach’s application of these techniques in his canons and fugues, such as retrograde in BWV 1087 [4], represents their use in exploring complex musical ideas and variations. The presented definitions of these transformations align with those in the literature and awareness of them enriches our understanding of compositional strategies in music [4–6].

2.2 Investigation of the Group Structure

In this section, we will scrutinize the structural properties of the group of permutations G, which operates on the state space of melodies X, defined as a Cartesian product of sets consisting of the notes. We shall focus on melodies of length 8, though the general results will be formulated for arbitrary length L. The group G is generated by four operations: transposition, translation, inversion, and retrograde, formally denoted as

G=⟨Transposition,Translation,Inversion,Retrograde⟩.

Iterating each operation gives raise to a corresponding cyclic group: transposition to C12, translation to C8, and inversion and retrograde each to C2. Upon examining the geometric nature of octagons and dodecagons, we deduce that the subgroup of G generated by transposition and inversion is isomorphic to the dihedral group D12, and similarly, the subgroup generated by translation and retrograde is isomorphic to D8.

We posit that G is isomorphic to the Cartesian product of D12 and D8, symbolized as G≅D12×D8. To illustrate this, let us consider the subgroups of tonal symmetries G1=⟨Transposition,Inversion⟩≅D12 and position symmetries G2=⟨Translation,Retrograde⟩≅D8 within G.

A melody x of length L in the state space X is represented by a sequence of notes:

x=x1,x2,…,xL,

(3)

which, under the action of g1, transforms as follows:

x∗g1=x1∗g1,x2∗g1,…,xL∗g1,

(4)

that is, (x∗g1)i=xi∗g1 for each i in 1,…,L.

For a given melody x and permutation action by g2, the tth note will move to the ith position, represented as:

xt=(x∗g2)i=xi∗g2-1,∀i∈{1,…,L},

(5)

where t is the original position of the note that g2 maps to the ith position, satisfying t∗g2=i and inversely t=i∗g2-1. This leads to the transformed melody:

x∗g2=(x1∗g2-1,x2∗g2-1,…,xL∗g2-1).

(6)

Taking a melodic sequence x∈X, we shall demonstrate the commutativity of G1 with G2, such that for all x∈X, g1∈G1, g2∈G2: x∗g1∗g2=x∗g2∗g1. The following result confirms this commutativity, and moreover leads to an exact formula for the result of applying an arbitrary symmetry to x.

Theorem 1

(Position-Contents Decomposition of Melodic Symmetries) The group G of melodic symmetries of melodies of length L is the internal direct product of two dihedral groups, the tonal content symmetries G1=⟨Transposition,Inversion⟩≅D12 and position symmetries G2=⟨Translation,Retrograde⟩≅DL. That is, we may write any symmetry g∈G as g1g2=g2g1 for a unique g1∈G1 (tonal symmetry) and unique g2∈G2 (position symmetry). Moreover, for all melodies x, the result of applying g∈G is expressed by:

x∗g=x∗g1g2=((x∗g1)1∗g2-1,(x∗g1)2∗g2-1,⋯,(x∗g1)L∗g2-1).

Proof

Applying (6) to the melody x∗g1 of (4) yields:

(x∗g1)∗g2=((x∗g1)1∗g2-1,(x∗g1)2∗g2-1,⋯,(x∗g1)L∗g2-1).

(7)

On the other hand, acting on both sides of (6) by g1 and using (4) yields:

(x∗g2)∗g1=((x1∗g2-1)∗g1,(x2∗g2-1)∗g1,⋯(xL∗g2-1)∗g1,).

(8)

Now since xi∗g1=(x∗g1)i holds for every position index, this gives:

(x∗g2)∗g1=((x∗g1)1∗g2-1,(x∗g1)2∗g2-1,⋯,(x∗g1)L∗g2-1).

(9)

Let’s notice that we get the same results in the Eqs. (7) and (9). Therefore, (x∗g2)∗g1=(x∗g2)∗g1. To conclude, the actions on melodies of the tonal and positional symmetry groups G1 and G2 commute. This also yields the fact that both of these groups are normal in G. Since G1∩G2 is a trivial group, it follows easily that G=G1G2≅G1×G2 (by, e.g., Theorem 2.29, An Introduction to the Theory of Groups by Rotman [10]), establishing the theorem.    □

Melodies of length L=12 with no repeated notes (pitch classes) are permutations of the set N and are called 12-tone rows, and they play an important role in chromatic atonal music of composers Schoenberg, Webern and the Vienna school. Specific results on their symmetry structure obtained by Schuijer [1], by Hunter and von Hippel [5], and by Hook [11, pp. 371–374] rely on the coincidence that 12-tone rows may be encoded as permutations of the 12 pitch-classes and use multiplication within the symmetric group on twelve objects. With the formalism here these limitations are removed: similar results can be derived, and one is not limited to length 12, nor to the 12-tone scale, nor to having no repeated notes nor rests. We next work with melodies of length L=8, where we have that their symmetry group is isomorphic to D12×D8 and we give explicit constructions for all melodies with any desired symmetry.

3 Characterization of Invariant Melodies

3.1 Structural Symmetries in Melodies

A melody of length L exhibits symmetry if it adheres to the symmetry equation x∗g=x, for a certain g=(g1,g2) within the group G≅D12×DL. Then, since the groups G1 and G2 commute, using Eqs. (4) and (5), the symmetry equation that relates content to position can be written in the following form:

xt∗g2=xt∗g1.

(10)

Let us characterize the ensemble of notes that may constitute a symmetrical melody, considering the actions of g1 and g2.

Theorem 2

(Position-Contents Condition for a Symmetry) Melody x exhibits melodic symmetry g=(g1,g2)∈D12×DL if and only if

xt∗g1C=xt,whereC=|Orbg2(t)|,foreachpositiont∈{1,..,L}.

(11)

Proof

Let’s consider the note xt at position index t from 1 to L. Given that g2 cyclically permutes indices within a melody, the index t is part of a cycle of positions.

Note that t∗g2C=t for some C since g2 is an element of the dihedral group DL. Let’s take minimal C, such that g2C=e, the identity permutation. In other words, C=|Orbg2(t)|, i.e., C is the length of orbit of position t under action of g2∈DL. Therefore,

xt∗g2c=xt∗g1c,∀c∈{0,…,C-1},C=|Orbg2(t)|.

(12)

The indices on the left-hand side of Eq. (12) are distinct, forming the orbit of index t, denoted by {t,t∗g2,…,t∗g2C-1}, and their note values are determined by the righthand side as the respective xt∗g1c.    □

To conclude, to understand melodies with a given symmetry, for each position orbit of the symmetry, we may choose any note that solves the fixed-point Eq. (11) and the rest of the notes in the position’s orbit are determined by (12).

This leads us to a formula for calculating the number of symmetrical melodies for a given transformation g=(g1,g2).

Corollary 1

(Number of Melodies Exhibiting a Symmetry) For a melodic symmetry g=(g1,g2), the number of symmetric melodies under g is

∏orbits ofg2|A(g1,g2,Orbg2(t)|,

(13)

where A(g1,g2,Orbg2(t)) denotes the set of notes invariant under the permutation g1C (its fixed points) with C the length of an orbit (cycle) Orbg2(t) of position t under action of g2.

To comprehend the symmetrical nature of melodies, we now harness the position-contents Eq. (11) of Theorem 2.

To derive the symmetrical melodies, we first discern the structure and orders of all elements g2 in the positional symmetry group D8, which can be achieved by hand or through the use of GAP software [12] as presented in Table 1.

Continuing our quest, we seek to enumerate all relevant powers of g1, for each element within the tonal symmetry group D12. This requires calculating the iterations of g1 for its 1st, 2nd, 4th and 8th powers. From these computations, we extract fix points as in (11) for each action on the melodic sequences. Similarly, by manual computation or by using GAP, the results for all possible operations acting on melodies are summarized in Table 2.

This systematic approach allows for the construction of symmetrical melodies by identifying fixed points that remain invariant under a given symmetry.

Table 1

Position Symmetries for Eight Note Melodies. Enumeration of elements of the group G2 acting on position indices, their orders, and the elements’ orbits structures. G2 is generated by cyclic translation R and retrograde F symmetries, and is isomorphic to the dihedral group D8,

Position symmetry g2

Position index orbits

C=Order(g2∈D8)

Orbit structure

R

(1, 2, 3, 4, 5, 6, 7, 8)

8

One orbit of size 8

FR

(2, 8) (3, 7) (4, 6)

2

Three orbits of size 2

and two orbits of size 1

R2

(1, 3, 5, 7) (2, 4, 6, 8)

4

Two orbits of size 4

FR2

(1, 2) (3, 8) (4, 7) (5, 6)

2

Four orbits of size 2

R3

(1, 4, 7, 2, 5, 8, 3, 6)

8

One orbit of size 8

FR3

(1, 3) (4, 8) (5, 7)

2

Three orbits of size 2

and two orbits of size 1

R4

(1, 5) (2, 6) (3, 7) (4, 8)

2

Four orbits of size 2

FR4

(1, 4) (2, 3) (5, 8) (6, 7)

2

Four orbits of size 2

R5

(1, 6, 3, 8, 5, 2, 7, 4)

8

One orbit of size

FR5

(1, 5) (2, 4) (6, 8)

2

Three orbits of size 2

and two orbits of size 1

R6

(1, 7, 5, 3) (2, 8, 6, 4)

4

Two orbits of size 4

FR6

(1, 6)(2, 5) (3, 4)(7, 8)

2

Four orbits of size 2

R7

(1, 8, 7, 6, 5, 4, 3, 2)

8

One orbit of size 8

FR7

(1, 7) (2, 6) (3, 5)

2

Three orbits of size 2

and two orbits of size 1

1(=R8)

()

1

Eight orbits of size 1

F(=FR8)

(1, 8) (2, 7) (3, 6) (4, 5)

2

4 orbits of size 2

Table 2

Computing Tonal Values Compatible with Positional Symmetry on a Positional Orbit. The group G1 of tonal symmetries is generated by upward transposition r by a semitone and inversion of notes f, and is isomorphic to the dihedral group D12. To find melodies with a melodic symmetry g1g2 in the symmetry group G1×G2. A melody with a given tonal symmetry g1 is compatible with positional symmetry g2 on the orbit of position t if and only for all positions t, the note xt is fixed by g1C(t) where C(t) is the size of the orbit of t under g2. One can check the orbit using this table where a note xt is fixed by g1C(t), where for eight note melodies C(t)∈{1,2,4,8}. For given note xt the values of the successive notes in the orbit of t are determined as shown in the column for g1. This column also shows notes fixed by g1 as a singleton orbits, or ∅ if none, or all notes. Subsequent columns show powers g12, g14 and g18 orbit structures, and either fix no notes or are the trivial permutation () fixing all notes

Tonal

g1 Tonal cycles & fixed points

g12 Tonal cycles

g14 Tonal cycles

g18 Tonal cycles

Symmetry g1

r

(C,D♭,D,E♭,E,F, G♭,G,A♭,A,B♭,B)∅

(C,D,E,G♭,A♭,B♭) (D♭,E♭,F,G,A,B)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

fr

(D♭,B)(D,B♭)(E♭,A) (E,A♭)(F,G) (C)(G♭)

()

()

()

r2

(C,D,E,G♭,A♭,B♭) (D♭,E♭,F,G,A,B)∅

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

fr2

(C,D♭) (D, B) (E♭,B♭) (E, A) (F,A♭) (G♭,G)∅

()

()

()

r3

(C,E♭,G♭,A) (D♭,E,G,B♭) (D,F,A♭,B)∅

(C,G♭) (D♭,G) (D,A♭) (E♭,A) (E,B♭) (F, B)

()

()

fr3

(C, D) (E♭,B) (E,B♭) (F, A) (G♭,A♭)(D♭)(G)

()

()

()

r4

(C,E,A♭)(D♭,F,A) (D,G♭,B♭)(E♭,G,B)∅

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

fr4

(C,E♭) (D♭,D) (E, B) (F,B♭) (G♭,A) (G,A♭)∅

()

()

()

r5

(C,F,B♭,E♭,A♭,D♭, G♭,B,E,A,D,G)∅

(C,B♭,A♭,G♭,E,D) (D♭,B,A,G,F,E♭)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

fr5

(C, E) (D♭,E♭) (F, B) (G♭,B♭) (G,A)(D)(A♭)

()

()

()

r6

(C,G♭) (D♭,G) (D,A♭) (E♭,A) (E,B♭) (F,B)∅

()

()

()

fr6

(C, F) (D♭,E) (D,E♭) (G♭,B) (G,B♭) (A♭,A)∅

()

()

()

r7

(C, G, D, A, E, B,  G♭,D♭,A♭,E♭,B♭,F)∅

(C,D,E,G♭,A♭,B♭) (D♭,E♭,F,G,A,B)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

fr7

(C,G♭) (D♭,F) (D, E) (G, B) (A♭,B♭)(A)(E♭)

()

()

()

r8

(C,A♭,E)(D♭,A,F) (D,B♭,G♭)(E♭,B,G)∅

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

fr8

(C, G) (D♭,G♭) (D, F) (E♭,E) (A♭,B) (A,B♭)∅

()

()

()

r9

(C,A,G♭,E♭) (D♭,B♭,G,E) (D,B,A♭,F)∅

(C,G♭) (D♭,G) (D,A♭) (E♭,A) (E,B♭) (F, B)

()

()

fr9

(C,A♭) (D♭,G) (D,G♭) (E♭,F) (A,B)(E)(B♭)

()

()

()

r10

(C,B♭,A♭,G♭,E,D) (D♭,B,A,G,F,E♭)∅

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

fr10

(C, A) (D♭,A♭) (D, G) (E♭,G♭) (E, F) (B♭,B)∅

()

()

()

r11

(C,B,B♭,A,A♭,G, G♭,F,E,E♭,D,D♭)∅

(C,B♭,A♭,G♭,E,D) (D♭,B,A,G,F,E♭)

(C,A♭,E) (D♭,A,F) (D,B♭,G♭) (E♭,B,G)

(C,E,A♭) (D♭,F,A) (D,G♭,B♭) (E♭,G,B)

fr11

(C,B♭) (D♭,A) (D,A♭) (E♭,G) (E,G♭)(B)(F)

()

()

()

1(=r12)   

()    (ALL NOTES)

()

()

()

f(=fr12)

(C, B) (D♭,B♭) (D, A) (E♭,A♭) (E, G) (F,G♭)∅

()

()

()

3.2 Application Example

In this section, we will demonstrate how to use the theory discussed earlier and show how to construct a melody with a given symmetry. We choose a transformation example of translation two steps to the right and transposition up by three semitones.

We define this transformation pair (g1,g2) as (r3,R2) in the group D12×D8. Our objective is to find a melody x that is invariant under a specific transformation, satisfying the equation xt∗g1C=xt. Here, C represents the size of the orbit of t in the group generated by g2, with t ranging from 1 to 8. To find such a melody, we follow these steps: 1. Calculate the orbits Orbg2(t) for each element t in the melody. 2. Find xt that makes the equation xt∗g1C=xt true for each orbit.

The structure of the transformations is such that there exist two orbits Orbg2(t) of size four, explicitly (1, 3, 5, 7) (odd positions) and (2, 4, 6, 8) (even positions), as shown in Table 1. Within this framework, we seek solutions to the equation xt∗g1C=xt, where C is the cardinality of the orbit Orbg2(t). In this example, for both orbits, C=4. As defined in Sect. 3.1, A represents the set of fixed notes under permutation g1C=g14=(). This yields the equivalence:

g14=()⟺Ar3,R2,(1,3,5,7)=Ar3,R2,(2,4,6,8)={allnotes}.

(14)

Given the freedom in the selection of the initial notes, we assign x1=1 and x2=3, corresponding to the notes “C” and “D”, respectively. With these choices, the invariant melody M commences with these two notes, followed by undetermined pitches: M=(C,D,?,?,?,?,?,?). The task then becomes to determine the subsequent notes that satisfy the invariance condition under the specified transformations. Applying transposition g1, we find x3=x1∗g1=4 (“E♭”), x5=x3∗g1=7 (“G♭”), and x7=x5∗g1=10 (“A”). A sanity check, x1∗g13=13mod12, ensures x1 correctly returns to “C”. Note that one is free to select repeated notes or notes in this orbit for x2. Analogous calculations are carried out for the cycle (2, 4, 6, 8) , starting with x2=3, which is chosen to be the note “D” initially. Proceeding in a similar fashion, we deduce x4=6, x6=9, and x8=12.

The melody, therefore, is invariant to the transformations of a two-step translation and a three-semitone transposition. This invariance is encapsulated in the melody M, which comprises the sequence (1, 3, 4, 6, 7, 9, 10, 12) and translates musically to the sequence (C,D,E♭,F,G♭,A♭,A,B), which is also shown in Fig. 2 in standard musical notation. The result is a melody that maintains its identity despite the transformative operations applied to it, reflecting the underlying symmetry and structure of the musical piece.

Fig. 2

An illustration of a melody exhibiting symmetry under the specified transformations

Given that orbit 1 yields 12 solutions, and orbit 2 also presents 12 distinct solutions (Eq. 14), it follows that the number of melodies invariant to this transformation, according to the Eq. (13), will be 12×12=144.

4 Extensions of Melodic Symmetry

Exploring further, we extend the concept of melodic symmetry beyond the chromatic spectrum to include the diatonic C major scale and melodies with rests. Straightforward variants of Theorems 1 and 2 and Corollary 1 hold in these cases.

4.1 Symmetries in the C Major Scale

Transitioning to the C major scale, we must adapt melodic transformations to fit this diatonic setting, essential to Western music.

The C major scale (NCmaj={C,D,E,F,G,A,B}) omits chromatic notes, prompting a restricted approach to transposition and inversion, ensuring transformations remain within the scale’s bounds. There is a unique non-trivial tonal symmetry in G1 mapping the C major scale to itself, namely, ι=fr5 with orbits (D)(C, E)(B, F)(A, B), i.e., the inversion reflecting across a line though D and A♭ of Fig. 1b. The dynamical system for the set of XCmaj of all C major melodies (with all notes in NCmaj), forms a sub-permutation group (XCmaj,GCmaj), with GCmaj≅C2×DL. This facilitates discovery of the symmetrical C major melodies: In particular, a position t may contain any desired note xt, thus determining all other notes at positions in the orbit of t under g2 as in (12), except that xt must be the note D if g1=ι and t is fixed by g2. This characterizes all C major melodies with any given symmetry (g1,g2)∈C2×DL.

4.2 Melodies with Rests

We introduce the rest (⋆) to our twelve-note set, adding a new layer to melody construction where a rest remains invariant under tonal musical transformations. This extension enables the exploration of melodies with rests, extending the applicability of our symmetry analysis. The note set becomes Nrest={C,D♭,D,E♭,E,F,G♭,G,A♭,A,B♭,B,⋆}, or {1,2,3,4,5,6,7,8,9,10,11,12,⋆}, when represented numerically with ⋆ for rest. We can apply Transposition, Translation, Inversion, and Retrogradation extending the group action to melodies with (zero or more) rest elements. The system is modeled by a permutation group (Xrest,D12×DL), where melodies may contain any number of rests, extending (X,D12×DL), with the requirement that when rest is inverted or transposed, it results in rest. All operations within Grest guarantee at least one symmetrical melody, and we define a fully invariant “silence” melody as silence=(⋆,⋆,⋆,⋆,⋆,⋆,⋆,⋆).

4.3 Conclusion

This study applies the discrete dynamic systems approach of group theory to explore symmetries in melodies, describing how transformations such as transposition, translation, inversion, and retrograde combine and preserve structural integrity in melodic lines within a unified framework. Key results demonstrate duality and unique factorization of symmetries generated by these transformations into positional and tonal constituent symmetries acting on melodic phrases.

Natural next steps for exploration will be extensions using wreath products at a finer level of hierarchical structure to include details of duration (of notes and rests) and pitch (frequency within a pitch-class). Another direction is to allow non-invertible transformations of melody, such as omitting a final note or replacing it by the tonic note or by an arpeggio of a key. This will entail the use of discrete transformation semigroups rather than permutation groups.

These findings and directions suggest significant theoretical and practical implications for future research in the algebraic music theory of melodic transformations.

References

1.

Schuijer, M.: Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. University of Rochester Press (2008)

2.

Hook, J.: Uniform triadic transformations. J. Music. Theory 46(1/2), 57–126 (2002)

3.

Peck, R.W.: Wreath products in transformational music theory. Perspect. New Music 47(1), 193–210 (2009)MathSciNet

4.

Benson, D.: Music: A Mathematical Offering. Cambridge University Press (2007)

5.

Hunter, D.J., von Hippel, P.T.: How rare is symmetry in musical 12-tone rows? Am. Math. Mon. 110(2), 124–132 (2003)MathSciNet

6.

Fripertinger, H., Lackner, P.: Tone rows and tropes. J. Math. Music 9(2), 111–172 (2015)MathSciNet

7.

Yust, J.: Dimensions of atonality: a response and extension of von Hippel and Huron. Empir. Music. Rev. 15(1–2), 119–127 (2020)

8.

Lewin, D.: Transformational techniques in atonal and other music theories. Perspect. New Music 21(1/2), 312–371 (1982)

9.

Cheng, E.: Math in Music. Series of Short Video Lectures. WFMT. https://​www.​wfmt.​com/​math-in-music/​. Accessed 10 May 2024

10.

Rotman, J.J.: An Introduction to the Theory of Groups. GTM, vol. 148, Springer (2012)

11.

Hook, J.: Exploring Musical Spaces. Oxford University Press (2023)

12.

The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.11.0 (2020). https://​www.​gap-system.​org

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_6

Analysis of the Semigroup Related to the Petri Net of a Traffic Roundabout

Marta Zheplinska1   and Chrystopher L. Nehaniv1  

(1)

University of Waterloo, Waterloo, Canada

Marta Zheplinska (Corresponding author)

Email: mzheplin@uwaterloo.ca

Chrystopher L. Nehaniv

Email: cnehaniv@uwaterloo.ca

Abstract

Petri nets have proven effective in modeling discrete dynamic systems whose state evolves over configuration space in discrete time steps. In this paper, we construct a Petri net that models the flow of cars on a roundabout with n entrances, symmetric about its centre, and analyze the semigroup related to the constructed Petri net. The paper focuses on investigating the complexity of the model in Krohn–Rhodes terms using the holonomy decomposition of the roundabout transformation semigroup and the general properties of such Petri nets for different n. Analysis of some components in the Petri net has given us insights into their role in defining the complexity of the model. We study the natural subsystems of the state space and the permutator groups acting on them, some of which are non-abelian or non-solvable. We catalog these dynamical patterns of changes in the number and location of vehicles on the traffic roundabout described by permutator groups.

Keywords

Petri netsTransformation semigroupsHolonomy decompositionDiscrete dynamical systemsKrohn-Rhodes theoryNatural subsystemsMaximal subgroups

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669.

1 Introduction

Petri nets, invented by Carl Adam Petri in 1962, have proven effective in modeling discrete dynamical systems [9]. Petri net models consist of components (places P, transitions T, and directed arcs connecting nodes of different types) that interact with each other. The dynamics of the model reproduce those of natural or engineered systems to understand, analyse, design, and simulate them better. The basic structure of Petri nets is as follows.

Definition 1

Petri net is a five-tuple ⟨P,T,I,O,I¯⟩, with

P

={p1,p2...,pn},n≥0—a set of places. Places P are characterized by capacity, i.e. the maximum number of objects, namely tokens, that a place can store.

T

={t1,t2....,tm},m≥0—a set of transitions. Transitions T consume tokens from the input places and produce tokens in the output places.

I:

T→P is a multivalued input function. The function I assigns the input places to the transitions T. A transition is enabled when its input places contain tokens.

O:

T→P is a multivalued output function. The function O assigns the output places to the transitions T. A transition is enabled when its output places have enough free space.

I¯:

T→P is a multivalued inhibition function. The function I¯ defines the dependence of transitions on specific places that, when containing tokens, prevent transitions from firing.

Petri nets can be canonically converted into semigroups [3]. A semigroup (X, S) is an algebraic structure consisting of a set of states X and a set of transformations S that acts on the set X, closed under composition. The set of a Petri net states X is defined by its places. A state (marking) of a Petri net is an assignment of tokens to places. Transitions make the dynamics in a Petri net, changing the number and distribution of tokens. They act on the set of states X. The semigroup (X, S) related to a Petri net is generated through transformations representing transitions’ action on set X. For details about the relation between semigroups and Petri nets, see [3].

According to the Krohn–Rhodes theory (see [1] for background and notation), for each semigroup, a cascade of simpler transformation semigroups can be constructed that is capable of computing everything the original one could (emulation) [6]. For this purpose, we use computer algebra resources to find building blocks¹ decomposed into a hierarchical structure [4]. This decomposition by levels (holonomy decomposition) allows us to understand the initial semigroup, evaluate its complexity, and discover natural subsystems (maximal subgroups acting by permutations on the image of their idempotents) [8, 10].

In the paper, we model a traffic roundabout with Petri Nets and analyze the obtained model using the GAP computer algebra system [5]. We convert a modeled Petri net into its transformation semigroup [3, 11], which is further decomposed [2, 12] into the building blocks to elucidate dynamics and complexity [4].

2 Modeling a Petri Net of a Roundabout

Inspired by a Petri net example depicting car movement at an intersection [13], we developed a model for traffic at a roundabout². We aim to create a computationally accessible model with minimal components while reflecting a roundabout’s essential dynamics.

Assuming right-hand traffic, we divide the roundabout into sections matching the number of entrances. Vehicles move from one section to the next, following one-way traffic. A vehicle in section Xk can transition only to Xk+1. Each section, uniquely labeled, features an exit and entrance, and allows a movement into the next section unless occupied. We consider the vehicles’ movement priority, requiring cars at entrances to yield to those already on the roundabout coming from the left.

A Roundabout Petri Net. The roundabout Petri net contains places pX, directed and inhibitor arcs, and transitions tXin, tXout, tX→Y. Tokens represent vehicles in the roundabout, with places corresponding to roundabout sections. Each place pX has a capacity of one, modeling a single-lane roundabout. Transitions, categorized into three types, create dynamics by moving vehicles within the roundabout, and modeling them entering and exiting.

tYin

These transitions add a token to the place pY, symbolizing a vehicle entering an unoccupied section Y. Denoted as tYin, they connect to pY with an incoming directed arc, where tYin is the source, and pY is the target. To model priority of vehicles already in the roundabout, the preceding place pX connects to tYin with an inhibitor arc, where movement from pX to pY is possible. Firing tYin results in an additional token in the net.

tYout

Transitions tYout remove a token from the place pY, indicating a vehicle’s exit from the roundabout. They connect to pY with an outgoing directed arc, where pY is the source and tYout is the target. Firing the transition decreases the number of tokens in the net.

tY→Z

Transitions tY→Z move tokens from the place pY to the place pZ representing vehicle movement within a roundabout. They connect pY to pZ in a one-way direction, so there is no corresponding tZ→Y. For tY→Z to fire, pY must have a token, and pZ must be empty.

Fig. 1

The Petri net of a roundabout with 3 sections (left) and 4 sections (right). Inhibitory arcs ensure cars entering the roundabout yield to vehicles occupying the section to the left

The model of a traffic roundabout is rotationally symmetric. For illustration of a Petri net with three and four places, we refer the reader to Fig. 1.

The Roundabout Semigroup Rn. Let us define the semigroup associated to the roundabout Petri net for clarity. We use strict firing conditions, i.e., a transition cannot fire if it would exceed capacity of a place. There is no effect on the Petri net state when transition is not enabled, i.e., the state maps to itself. Let pXi,Xj...,Xk denote a state (marking) of the roundabout Petri net where Xi,Xj...,Xk are the distinct sections containing tokens listed in order i<j<⋯<k, and all other sections are empty. For convenience, the transformation that refers to the action of a transition tw, where w∈{Xin,Xout,X→Y}, has the same name as that transition. Let Rn=(X,S) be the transformation semigroup of a traffic roundabout with n places, where X is the set of all possible states pXi,Xj...,Xk and S is the set of transformations generated by all tXin,tXout,tX→Y.

3 Computer Algebraic Analysis of the Semigroup with GAP

We study the dynamics of (2n,Rn) through the Krohn–Rhodes theory³ Due to computational constraints, we first focus on Petri nets with three and four places, revealing exponential growth in roundabout semigroup sizes. For instance, (8,R3) has order of 68,389, while (16,R4) exceeds 8 million. In the following, we explore the complexity and dynamics of holonomy decompositions for R3 and R4. For the details of the permutator groups in the holonomy decomposition of R3, refer to Table 1. The 13-level decomposition includes non-commutative symmetric (3,S3) and dihedral (4,D4) groups. According to the Krohn–Rhodes theory, an upper complexity limit for R3 is 8, representing the number of levels with non-trivial groups.

Table 1

The depths of the holonomy decomposition of R3 where nontrivial holonomy groups arise [2], structure description (n, G) of the permutator groups, their generators, and the states of the natural subsystems they act on, where n denotes an n-element set of markings which is the image of the idempotent e2=e in a maximal subgroup G

Depth

Structure Description

Generators of the Permutator Group

Natural Subsystem’s State Set

3

(5,C2)

tB→C·tA→B·tC→A

{_,pB,pA,pAB,pABC}

4

(5,C2)

tC→A·tA→B·tB→C

{_,pC,pAC,pCB,pABC}

7

(5,C2)

tB→C·tA→B·tC→A

{_,pB,pA,pAB,pABC}

9

(5,C2)

tB→C·tC→A·tA→B

{_,pB,pBC,pAB,pABC}

10

(4,D4)

1: tAin·tCout·tB→C·tA→B

{_,pB,pC,pBC}

2: tC→A·tB→C·tA→B·tAin·tCout·tB→C

11

(3,C2)

tB→C·tC→A·tA→B

{pBC,pAB,pABC}

12

(3,S3)

1: tBin·tC→A·tAout·tC→A·tB→C·tBin·tC→A·tB→C

{pA,pC,pAC}

2: tA→B·tC→A·tB→C

13

(2,C2)

tC→A·tA→B·tCin·tBout

{_,pC}

The holonomy decomposition of R4 consists of 29 levels, 18 containing at least one nontrivial group, so in the Krohn–Rhodes complexity terms, an upper bound of the complexity of R4 is 18. For an examination of the holonomy decomposition of R4 we refer the reader to Table 2.

Table 2

The holonomy decomposition of R4

Depth

Structure description

Generators of the permutator group

Natural subsystem’s state set

4

(9,C3)

tB→C·tA→B·tD→A·tC→D

{_,pD,pB,pBD,pA,pAD,pAB,pABD,pABCD}

5

(9,C3)

tD→A·tA→B·tB→C·tC→D

{_,pD,pCD,pBD,pBCD,pAD,pABD,pACD,pABCD}

7

(7,C2)

tB→C·tD→A·tA→B·tC→D

{_,pD,pB,pBD,pBCD,pABD,pABCD}

8

(6,C2)

tC→D·tB→C·tA→B·tD→A·tA→B

{_,pB,pC,pBC,pABC,pABCD}

9

(6,C2)

tD→A·tA→B·tB→C·tC→D·tB→C

{_,pD,pCD,pBCD,pACD,pABCD}

9

(6,C2)

tC→D·tB→C·tA→B·tD→A·tA→B·tC→D·

{_,pC,pBC,pAC,pABC,pABCD}

tB→C·tA→B·tD→A·tC→D·tD→A·tA→B·

tB→C·tA→B·tD→A

13

(6,C2)

tB→C·tD→A·tA→B·tC→D

{pD,pB,pBD,pBCD,pABD,pABCD}

14

(8,C3)

tB→C·tC→D·tD→A·tA→B

{pB,pBD,pBC,pBCD,pAB,pABD,pABC,pABCD}

14

(5,C2)

tC→D·tB→C·tA→B·tD→A·tA→B

{pC,pB,pBC,pABC,pABCD}

15

(8,C2×A4)

1: tAin·tDout·tC→D·tB→C·tA→B·tD→A·

{_,pC,pD,pCD,pB,pBD,pBC,pBCD}

tC→D·tB→C·tA→B·tD→A·tC→D·tB→C·

tA→B 2: tD→A·tC→D·tB→C·tBin·tAout

3: tD→A·tC→D·tB→C·tA→B

15

(7,C3)

tC→D·tD→A·tA→B·tB→C

{pCD,pBC,pAC,pBCD,pACD,pABC,pABCD}

16

(7,C3)

tC→D·tCin·tBout·tBin·tAout·tD→A

{_,pC,pBC,pA,pAC,pAB,pABC}

19

(5,C2)

tD→A·tA→B·tB→C·tC→D·tB→C·

{pD,pCD,pBD,pBCD,pABCD}

tD→A·tA→B·tB→C·tC→D·tD→A·

tC→D·tB→C·tA→B·tB→C·tC→D

19

(4,C3)

tD→A·tC→D·tB→C·tA→B

{pCD,pBD,pBCpBCD,pABCD}

20

(6,S6)

1: tA→B·tD→A·tC→D·tCin·tBout·tA→B·

{pD,pCD,pA,pAD,pAC,pACD}

tAin·tDin·tB→C·tC→D·tDout·tC→D·tCin·

tBin·tAout·tD→A·tC→D·tB→C·tA→B·

tD→A·tC→D·tB→C 2: tA→B·tD→A·tC→D·

tB→C·tA→B·tD→A·tDin·tCout·tB→C·tBin·

tAin·tC→D·tD→A·tAout·tAin·tDin·tD→A·

tAout·tAin·tC→D·tB→C

20

(5,C2)

tC→D·tD→A·tA→B·tB→C·tD→A

{pC,pAC,pACD,pABC,pABCD}

22

(4,C2)

tA→B·tB→C·tC→D·tB→C·tD→A

{pA,pACD,pABC,pABCD}

23

(4,C2)

1: tA→B·tB→C·tC→D·tD→A·tA→B·tB→C·

{pAD,pAC,pACD,pABCD}

tC→D·tD→A·tB→C·tA→B·tD→A·tB→C·

tC→D·tD→A·tB→C

25

(5,S5)

1: tB→C·tBin·tAout·tD→A·tC→D·tCin·

{pA,pAC,pABC,pACD,pABCD}

tA→B·tAin·tA→B·tB→C·tCout·tB→C·tBin·

tD→A·tDout·tC→D 2: tB→C·tA→B·tD→A·

tDin·tB→C·tBin·tCout·tB→C·tA→B·tAin·

tA→B·tBout·tBin·tD→A·tC→D

25

(4,C2)

tA→B·tC→D·tB→C·tD→A

{pAC,pABC,pACD,pABCD}

26

(4,C3)

tA→B·tB→C·tC→D·tD→A

{pABC,pABD,pACD,pABCD}

27

(4,S4)

1: tAin·tBin·tAout·tDin·tD→A·tBin·tDin·

{_,pD,pC,pCD}

tCout·tA→B·tBout·tA→B·tB→C

2: tAin·tBin·tAout·tDin·tD→A·tDin·tCin·

tA→B·tB→CtCout·tB→C·tAout

27

(3,C2)

tB→C·tA→B·tC→D·tD→A

{pABC,pABD,pABCD}

28

(3,S3)

1: tAout·tAin·tBin·tAout·tD→A· tDin·tB→C·

{_,pA,pAD,pACD}

tC→D·tBout·tD→A 2: tAout·tAin·tA→B·

tAin·tDin·tD→A·tC→D·tB→C

29

(2,C2)

tDin·tCout·tCin·tC→D·tDout

{pA,pAC}

Within the decomposition, we encountered symmetric groups, denoted as Sn, that operate on up to 6 elements. At the 25th level, a non-solvable permutator group (5,S5) emerges. This group includes the smallest simple non-abelian group A5 as a subgroup. The presence of A5 indicates that R4 is finitary computationally universal by the theorem of Maurer and Rhodes [7], giving rise to the following theorem.

Theorem 1

The roundabout Petri net with 4 places is finitarily computationally universal.

The theorem entails that any function f:X→Y, where X and Y are finite sets, can be implemented by encoding it into parallel operations performed by multiple instances of R4. See [8] for a discussion of such computational constructions.

4 Factors Influencing the Complexity of Rn

This section focuses on the examination of the factors that have an impact on the complexity of Rn transformation semigroups. We examine them by analyzing the consequences of their exclusion from the model.

Inhibitor arcs. To investigate the role of inhibitor arcs, we consider semigroups Wn associated with inhibition-free roundabout Petri nets, as depicted in Fig. 2. This model reflects the dynamics created by reckless drivers that do not yield to traffic in the roundabout. Without inhibitor arcs, the size of W3 is 665 as opposed to 68389 in R3, followed by a reduction in the upper complexity bound from 8 to 2. The holonomy decomposition of W3 presents only C2 group components; for W4, we observe C2 and C3 groups. We shall see below that Cn-1 is always a subgroup of Wn (and Rn). A question arises from our observations when comparing Wn and Rn.

Open Problem 1

Are all subgroups of Wn represented by cyclic groups (k,Ck), where 2≤k≤n-1 and n≥3?

The role of the inhibitor arcs in determining the internal reversibility of a semigroup has been discussed in [3].

Fig. 2

Petri nets of a roundabout without inhibitor arcs (to the left) and with n-2 blocked exits and entrances (to the right)

Experiment—blocking some of the exits and entrances. Let us block n-2 exits and entrances in the roundabout Petri net (see Fig. 2). The corresponding semigroup Bn is a subsemigroup of Rn generated by transformations of the type tX→Y, and only two transformations each tXin,tYin and tXout,tYout, where places pX and pY are adjacent. A study of upper and lower bound of complexity and reversible group components of B3 and B4 revealed us no difference from R3 and R4, respectively. Despite fewer vehicle actions when removing n-2 exits and entrances, the model retains the same holonomy groups as the original at least for n≤4.

Open Problem 2

Does blocking all but two adjacent exits and entrances affect Rn’s complexity?

5 Petri Nets of a Roundabout with n Places

Based on the study of R3 and R4, we explore the dynamics of roundabout Petri nets with n places for higher values formulated in the following theorems.

Lemma 1

The “full” state of the roundabout Petri net is not reachable.

Proof

Consider a marking of a Petri net, where all places are occupied except one labeled as j. The only means of introducing a token to the “full” state is by filling the empty place pj, which necessitates the firing of transition tjin. Given that all other places are already occupied and transition tjin is inhibited by an arc connected to one of them, it remains incapable of firing. Hence, the “full” state is unreachable.    □

We study the Rn’s subsemigroups to identify the role of certain types of transitions in determining complexity. Let Rinn be a semigroup generated by the transformations tXin. Let Routn be a semigroup generated by the transformations tXout.

Proposition 1

The semigroups Rinn and Routn are aperiodic.

Proof

Rinn: Assume the roundabout Petri net contains k tokens. When any transition tXin fires, the number of tokens increases to k+1. Returning to the state with k tokens is not possible, since the only transitions are of type tXin. Therefore, Sin is aperiodic. Routn: Similarly, the number of tokens can only decrease.    □

The Move Subsemigroup Rmoven. Let Rmoven be the subsemigroup of Rn (and Wn) generated by just the transformations of type tX→Y that move vehicles around the roundabout, without entering or exiting. We aim to study sequences of transitions that give rise to nontrivial subgroups of the semigroup Rmoven. Let g denote the transformation of markings given by the sequence of transformations t1→2·t2→3·...·tn-1→n·tn→1, where n is a number of places and g∈Rmoven.

Observation 1

Transformation g does not change the number of tokens in the roundabout Petri net.

Proof

Composed of the transformations tX→Y that do not change the number of tokens in the Petri net, g likewise preserves the token quantity.    □

We may identify a marking M with the set of distinct indices of its non-empty places, i.e. {x1,x2,...xr|1≤xi≤n,1≤i≤r}, where xi is an index of a place containing token. If some place is empty in the marking M, the set does not contain its index. We examine the effect of transformations on the marking M, solely considering the occupation of specific places identified by their indexes, ignoring particular tokens and their displacement.

Let X(k) denote the subset of markings with kth place occupied. Clearly, |X(k)|=|X(j)| for all 1≤k,j≤n. For Y⊆X,s∈S, define Ys=Y·s={y·s|y∈Y}.

Proposition 2

For any marking M={x1,x2,...,xr|1≤xi≤n,1≤i≤r}∈X(1), let M·g={y1,y2,...,yr|1≤yi≤n,1≤i≤r} be the transformed marking consisting of indices of occupied places in ascending order after applying the transformation g. The expression yi=xi-1(2≤i≤n) holds, and yi=n if xi=2. In either case the 1st place remains occupied, M·g∈X(1).

Proof

The transformation g shifts the tokens before the next occupied place, starting with the token in the first place. Each token is mapped to the place before the first non-empty position. Therefore, the new index corresponds to the index of the next non-empty place minus 1. In other words, each occupied place with index m guarantees that the place with index m-1 will be non-empty after the transformation g. We will show that if an occupied place has index 2, then this ensures that the place with index n will be filled. Since the first place always has a token, this token stays on the first position as the second place is occupied. Since the first place remains filled, the token in the non-empty place with the highest index moves to the place with index n and stops there. Therefore, occupation of the second place guarantees occupation of the nth place after the application of transformation g.    □

Proposition 3

For a marking M∈X(1), with the 1st place occupied, M·gn-1=M.

Proof

To demonstrate that the (n-1)-fold application of the transformation g to the marking M leaves M unchanged, we aim to show that the indices for non-empty places remain the same. Let m range from 2 to n as the index of any non-empty place except the first one in the roundabout Petri net in the state M. Applying transformation m-2 times, in accordance with Proposition 2, yields a new index yi=xi-(m-2)=m-m+2=2 due to m-2 index shifts. After one more application of g, the new index becomes zi=n as yi=2. Subsequently, transforming the marking (n-m) times results in di=zi-(n-m)=n-n+m=m. In conclusion, applying the transformation m-2+1+n-m=n-1 times restores the initial set of indices.    □

Proposition 4

For X the set of all markings of the roundabout Petri net with n places, Xg is the set of markings X(1) with the first place occupied.

Proof

Let M∈X be a marking containing k tokens randomly placed in n places, 1≤k≤n. Let M·g be a transformed marking. Let j be the largest index of an occupied place in M. The first part of g, t1→2·...·tj-1→j, does not move the token on the jth place. Then tj→j+1·...·tn-1→n shifts this token to the nth place, and the last transition tn→1 places it in the 1st place unless it is already occupied. Thus, M·g must have the first place occupied. Conversely, by Proposition 2, each such M is in Xg.    □

Proposition 5

The transformation g generates a permutator group (Xg,Cn-1).

Proof

By Propositions  3 and  4, it suffices to show that g has an orbit of length n-1. Consider a marking M={1,x2,...,xk}, where M∈Xg represents a set of occupied indices. If xi and xi+1 represent non-empty places (with no occupied places in between) in the marking M, xi,xi+1≠1, the number of unoccupied places between them is (xi+1-xi-1)mod(n-1). Apply the transformation g, and the new indices become xi-1 and xi+1-1. The number of empty places between these corresponding indices (xi+1-1-(xi-1)-1)mod(n-1) remains unchanged. As g preserves this value, its effect is apparent only up to the shift of the entire configuration by one position clockwise (ignoring the first place). Taking, for instance, M with all but one place occupied, the set {M,M·g,...,M·gn-2} must consist of distinct markings, comprising an orbit of size n-1 on which the cyclic group Cn-1=⟨g⟩ acts.    □

Theorem 2

The semigroup Rmoven contains a natural subsystem (Xg,Cn-1), generated by a word g=t1→2·...·tn-1→n·tn→1, where Xg is the set of the roundabout Petri net states with the 1st place occupied.

Proof

Following Proposition  5, the permutator group Cn-1=⟨g⟩ is a subgroup of Rmoven. We aim to demonstrate that (Xg,Cn-1) is a natural subsystem of Rmoven, i.e., a maximal subgroup acting on the image of its idempotent.

Let s∈Rmoven be a transformation which permutes the subset Xg=X(1). Choose a minimal length word w=w1·w2⋯wN-1·wN in the generators of Rmoven realizing the transformation s, where wj is of the form ta→(a+1)modn(1≤a≤n). Observe that wj≠wj+1 for 1≤j≤N-1, since (ta→(a+1)modn)2=ta→(a+1)modn would contradict minimality of w. Let w(k)=w1·w2⋯wk be a prefix of the word w (k≤N). Since w permutes X(1), |X(1)|=|X(1)·w(k)|. We now show by induction on k≥0 up to N that the following assertion holds:

X(1)·w(k)=X(k+1modn),andifk≥1,thentheletterwkistk→(k+1)modn.

Base case. For k=0, w(0) is an empty prefix and X(1)·w0=X(1). Inductive step. Assume the assertion holds up to k-1. So X(1)·w(k-1)=X(1)·w1·w2⋯wk-1=X(kmodn) and wj is tj→(j+1)modn for 1≤j≤k-1. We have

X(1)·w(k)=X(1)·w1·w2⋯wk-1·wk=(X(1)·w1·w2⋯wk-1)·wk=X(k)·wk.

Suppose wk=ta→(a+1)modn(1≤a≤n). Consider two markings M1={k,a} and M2={k,a+1} from the set X(k). We observe that M1·wk=M2 and M2·wk=M2, if a≠k-1 or a≠k, contradicting |X(1)|=|X(1)·w(k)|. Thus, wk must be t(k-1)→kmodn or tk→(k+1)modn. In the first case, t(k-1)→kmodn acts as the identity function on X(k) leaving all markings unchanged since position k is occupied, and contradicting minimality of w. Hence, wk=tk→(k+1)modn. Moreover, X(k)·tk→(k+1)modn⊆X(k+1), since transformation tk→(k+1)modn sends tokens from kth place to (k+1)th place unless it is already occupied. Since |X(k)·tk→(k+1)modn| cannot be smaller than |X(k+1)|, equality holds. This completes the induction.

We conclude that w(k)=t1→2·t2→3⋯tk→(k+1)modn. Finally, the last letter of w must be tn→1, as w is a permutation of Xg. It follows that w=gr for some r≥1.    □

Let g¯=tn→1·tn-1→n·...·t1→2 be a transformation constructed of the transitions tX→Y in the reversed order to that of g. We examine the dual dynamics, shifting attention to the empty places within the roundabout Petri net instead of those occupied by tokens, as was the case in the study of transformation g.

Theorem 3

The semigroup Rmoven contains a natural subsystem (Xg¯,Cn-1), generated by a word g¯=tn→1·tn-1→n·...·t1→2, where Xg¯ is the set of the roundabout Petri net states with the 1st place unoccupied.

The proof follows the previous one, and we delegate it to the reader. We posit that these group components represent inherent symmetries of the original Petri net with and without inhibitory arcs.

Corollary 1

For n≥3, Rmoven is not aperiodic. Hence neither are Rn and Wn, since they contain Rmoven as a subsemigroup.

6 Conclusions

The studied Petri nets are characterized by an exponential increase in complexity and size with the number of places. The emergence of unsolvable groups in R4 shows that roundabout dynamics is finitarily computationally universal. We studied the effect of inhibitory arcs on the complexity and behavior of the subgroups, indicating their role in determining the dynamics of these models. Despite the fact that the dynamics of Petri nets is in general non-commutative, we observe only commutative permutator groups in the W3 and W4 semigroups related to the Petri nets with no inhibitor arcs. There are still open questions related to the fundamental features of roundabout Petri nets, particularly regarding the role of blocked exits and entrances.

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Egri-Nagy, A., Mitchell, J.D., Nehaniv, C.L.: SgpDec: cascade (de)compositions of finite transformation semigroups and permutation groups. In: Mathematical Software (ICMS 2014), pp. 75–82. Springer (2015)

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Egri-Nagy, A., Nehaniv, C.L.: Algebraic properties of automata associated to Petri nets and applications to computation in biological systems. Biosystems 94(1–2), 135–144 (2008)

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Egri-Nagy, A., Nehaniv, C.L.: Computational Holonomy Decomposition of Transformation Semigroups (2015). https://​doi.​org/​10.​48550/​arXiv.​1508.​06345

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Maurer, W., Rhodes, J.: A property of finite simple non-abelian groups. Proc. Am. Math. Soc. 16(3), 552–554 (1965)

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Nehaniv, C.L., Rhodes, J., Egri-Nagy, A., Dini, P., Rothstein Morris, E., Horváth, G., Karimi, F., Schreckling, D., Schilstra, M.: Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 373, 20140223 (2015)

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Footnotes

1

The building blocks are groups (augmented by constant maps). The most common among them are cyclic groups ([n],Cn), dihedral groups ([n],Dn), symmetric groups ([n],Sn), and alternating groups ([n],An), each acting on n elements. Sometimes, the size of the set can be other than n.

2

Also referred to as a “traffic circle.”

3

This theory measures the minimal number of groups required when decomposing a finite semigroup into a wreath product of groups and aperiodic semigroups. Computations are done using the SgpDec package for GAP [12].

Part IIInnovations in Statistical Machine Learning and Stochastic Models

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_7

From Interval-Valued Neurons to Convex-Polygon-Valued Neurons, with Sparsity and Entropy Criteria

B. Boreland1  , H. Kunze2   and K. Levere2  

(1)

Deloitte, Toronto, Canada

(2)

Department of Mathematics and Statistics, University of Guelph, Guelph, Canada

B. Boreland

Email: brysonboreland@gmail.com

H. Kunze (Corresponding author)

Email: hkunze@uoguelph.ca

K. Levere

Email: klevere@uoguelph.ca

Abstract

In a typical neural network, the neurons are real-valued or complex valued. In some recent work, the notion of neurons being (closed) interval-valued has been developed. Such a formulation can be useful in applications where data are interval-valued. In this paper, we develop the idea of convex-polygon-valued neurons. When exploring examples of these two types of networks, we also include sparsity and entropy criteria, which are conflicting, with the goal of demonstrating that including such criteria with appropriately small coefficients can improve network performance.

Keywords

Neural networksSet-valued neuronsSparsityEntropyMulti-criteria decision making

1 Introduction

A neuron can be viewed as an information processing unit in a neural network, with each neuron processing some input and providing some output. All neural networks are comprised of layers that contain some number of neurons. For this paper, we can avoid discussing types of layers and different network architectures (and their individual usefulness in particular applications). In any case, when a neuron receives input vectors, it produces output vectors through a relatively simple process. If the output vector y is generated by the input vectors zj, j=0,…,J, each input vector is multiplied by a weight vector wj, and a bias bj is added. Thereafter, an activation function f is applied in order to restrict the output range and to introduce nonlinearity. The mathematical relationship at the individual neuron level for one of its outputs is given by

y=f∑kzj,kwj,k+bj.

This output becomes an input to neurons in subsequent layers. An output y of the final (output) layer can be viewed as a function of the weights w and the inputs x to the first (input) layer, y=f(x,w). The training of a neural network involves using a dataset of inputs xi and outputs yi. For each xi, the network gives a predicted value y^i=f(xi,w). One then minimizes a cost function with respect to the weight parameters, for example, the sum of squares error (SSE).

Neurons and outputs are typically either real-valued (as in the preceding discussion) or complex-valued. In the next section, we will discuss interval-valued neurons and then, in the subsequent section, present an extension to convex-polygon-valued neurons. In each case, we will present results from an example. But first we define the notions of sparsity and entropy for the current context. In addition to considering the preceding two types of neural networks, the examples also explore whether performance can be improved by including sparsity and entropy criteria with appropriately small coefficients.

2 Sparsity and Entropy

Let x=[x1,⋯,xn]T be a vector in Rn, and ℓp(x)=∑i=1n|xi|p. The ℓ0 pseudonorm of x is

‖x‖0=limp→0+ℓp(x).

(1)

If we define 00=limp→0+|0|p=0, then ‖x‖0=ℓ0(x)=#{xi|xi≠0}, a count of the number of nonzero entries in x. Note that we use the word pseudonorm, since this function is not a norm: it does not satisfy a homogeneity property since, for α≠0, ℓ0(αx)=ℓ0(x)≠|α|ℓ0(x) in general. It is also called the counting norm.

The role of an ℓ0 criterion is to strongly enforce or demand sparsity, as it becomes smaller when its argument has more zero entries. The ℓ0 pseudonorm can prove difficult to work with due to its non-convex and discontinuous nature, but it can be approximated by various functions, including smooth ones. For example, let r>0 be a real number and x=[x1,⋯,xn]T be a vector in Rn. Then s(x)=1-e-rℓ2(x) is a family whose members better approximate l0(x) as r increases, with the ℓ0 pseudonorm satisfying

ℓ0(x)=limr→∞1-e-rℓ2(x).

The notion of (Shannon) entropy [6] has been modified to deal with a set of parameters that take on both positive and negative values with magnitude possibly greater than one [4]. Entropy can be defined for such a set of parameters, in our case the weights w=[w0,w1,⋯,wn]T of the neural network, by

ENT(w)=-∑i=0n|wi|Mln|wi|M

(2)

where M is a constant satisfying M≥maxi|wi|.

The role of an entropy criterion should be to maximize entropy, so we seek to minimize negative entropy, -ENT(w).

3 Interval-Valued Neurons

Interval-valued neurons are a class of neurons that process interval-valued data. The lower and upper bounds are denoted with a superscript L and U, respectively. Similar to other neural network models, an interval-valued neuron or network aims to find the relationship between its inputs, X=XL,XU, and outputs, Y=YL,YU.

The first model for interval-valued data, called the centre method (CM), was proposed in [1]. This model has a strict assumption that the lower and upper bounds share linear coefficients, which is estimated by modelling the centres,

Xc=12XL+XUandYc=12YL+YU.

Although the CM model can be easily implemented, the linear coefficients assumption restricts the model’s ability to accurately predict outputs for complex real-world data. Two methods that are based on linear regression are better at predicting interval-valued data than the CM. They are the MinMax method proposed in [2] and the centre and range method (CRM) proposed in [5]. The CRM is a similar method to the CM that splits the predicted interval-valued output into centre and range regression.

Let Xk,Yk,k=1,…,n, be an interval-valued dataset where Xk=XkL,XkU and Yk=YkL,YkU. The CRM modifies the data into centres and ranges

Xkc=12XkL+XkUandXkr=12XkL-XkU,Ykc=12YkL+YkUandYkr=12YkL-YkU,

where Xkc and Ykc are the centres and Xkr and Ykr are the ranges. This is then built into a regression model as follows:

Ykc=β0c+β1cX1c+⋯+βncXk,nc+ϵkcYkr=β0r+β1rX1r+⋯+βnrXk,nr+ϵkr.

An approach to interval-valued neurons based upon adding an additional criterion to the cost function that penalizes lower and upper bounds reversing their order was introduced in [8]. The authors called their construction a regularized artificial neural network or RANN. The RANN model is made up of 3 layers including a single hidden layer and all of the inputs and outputs are interval-valued with an upper and lower bound.

There are 2n input neurons, J hidden neurons, and 2 output neurons, where the outputs of the RANN can be written as

Y^L=f∑j=0Jzjwj,L(o)+bL(o)andY^U=f∑j=0Jzjwj,U(o)+bU(o)

where wL(o)=w0,L(o),…,wJ,L(o) and wU(o)=w0,U(o),…,wJ,U(o) are the weight vectors connecting the hidden layer to the output layer, bL(o) and bU(o) are the biases, and zj is the output from the jth neuron in the hidden layer and is written as

zj=g∑i=02nwi,j(h)Xi+bj(h)

with g as the activation function, wi,j(h) are the weights connecting the ith neuron from the input layer to the jth neuron in the hidden layer, and bj(h) is the bias. The loss function for this model combines the mean square error for both the lower and upper bounds of the intervals and an additional non-crossing term that penalizes the model whenever the bounds flip such that Y^L>Y^U. The loss function can be written as

C=∑k=1nYkL-Y^kL2+∑k=1nYkU-Y^kU2+λ∑k=1nmax0,Y^kL-Y^kU2

(3)

where n is the number of inputs and λ>0 is the regularization coefficient. Note that the cost function C is scaled by 12n in [8].

When dealing with interval-valued data, there are a few methods to evaluate the performance of the network and the ability to predict the output intervals. As usual, one such method is to calculate the root mean square error (RMSE), this time for the lower and upper bounds of intervals, written as follows

RMSEL=∑k=1nYkL-Y^kL2nandRMSEU=∑k=1nYkU-Y^kU2n

Smaller RMSE values indicates better performance by the model.

Example 1

We use an interval-valued neural network model, the mushroom dataset [7], and a cost function Cr=C+γ0ℓ0(w¯)-γentENT(w¯), where C=C(w¯) is the cost function presented in (3),

ℓ0(w¯)=∑i=02n1-e-0.5ℓ2w¯(h)+1-e-0.5ℓ2w¯L(o)+1-e-0.5ℓ2w¯U(o)

is a sparsity criterion,

ENT(w¯)=-∑i=02n∑j=1Jwi,j(h)Mlnwi,j(h)M-∑j=0Jwj,L(o)Mlnwj,L(o)M-∑j=0Jwj,U(o)Mlnwj,U(o)M

is an entropy criterion, and γ0 and γent are two chosen nonnegative constants.

The mushroom dataset typically uses the upper and lower bounds of the pileus cap width and the stipe length as the inputs and the network predicts the upper and lower bounds of the stipe thickness. Following [8], the neural network is constructed using an input consisting of a lower and upper value, a hidden layer containing 5 neurons, and two output neurons for the upper and lower predicted value. The neurons use the sigmoid activation function, the learning rate is 0.001, and the non-crossing coefficient is 1000, as suggested in [8].

Figure 1a and b illustrates some choices of γ0 in the interval-valued model. Figure 1 shows for both the lower (a) and upper (b) bound that choices of γ0≤ 1.0×10-5 provide improvement to the model performance compared to the base case. The improvement to the model is easier to notice in the graph for the lower bound (a) than the upper bound (b). For the upper bound, the base case error value is 1.2503 and the values when γ0≤1.0×10-5 are in [1.2314, 1.2425], showing some improvement. For larger choices of γ0 displayed in Fig. 1, there is no benefit to the model.

Fig. 1

Value of cost function C and choices of (a) γ0 for lower bound predictions, (b) γ0 for upper bound predictions, (c) γent for lower bound predictions, and (d) γent for upper bound predictions. Values along the x-axis are choices for 0 and s in 1.0×10-s,s=1,…,15. Choices of γ0 (ℓ0 proxy criterion) and γent (ENT criterion) for (e) lower bound and (f) upper bound. Values along each axis are choices for 0 and s in 1.0×10-s,s=1,…,15. The shades of blue represent the value of C where darker indicates a higher value

Figure 1c and d illustrates some choices of γent in the interval-valued model. Figure 1 shows for the lower bound (c) that all choices of γent provide benefit and for the upper bound (d) choices of γent≤1.0×10-6 provide improvement to the model performance compared to the base case. Once again, the improvement to the model is easier to notice in the graph for the lower bound (c) than the upper bound (d). For the upper bound the base case error value is 1.2503 and the values when γent≤1.0×10-6 are in [1.2301, 1.2408], showing some improvement. For larger choices of γent displayed in Fig. 1(d), there is no benefit to the upper bound of the model.

Figure 1e and f illustrates some choices of γ0 and γent in the interval-valued model. Figure 1e and f shows that there are many choices of γ0 and γent that provide improvement to the model performance compared to the base case. This benefit occurs at the smaller choices of both γ0 and γent with a base case error values of [0.9464, 1.2503] and the improved model error values are in [0.8146, 0.9445] and [1.2272, 1.2494], for the lower and upper bound respectively. The darker shades of blue occur when larger choices of γ0=0.01,0.001 are combined with smaller choices of γent causing a decrease in model performance.

4 Convex-Polygon-Valued Neurons

In [3], the idea of an interval-valued neuron was expanded to a network that treats convex-polygon-valued data in R2. Consider a convex q-gon. Consistent and generalized with respect to input neurons of a RANN, there are t=q·n input neurons, J hidden neurons, and q output neurons where the outputs of the convex-set-valued neuron can be written as

Y^l=f∑j=0Jzjwj,l(o)+bl(o),

where l=1,⋯,q,w¯l(o)=w1,l(o),⋯,wJ,l(o) are the weight vectors connecting the hidden layer to the output layer, bl(o) are the biases, and zj is the output from the jth neuron in the hidden layer and is written as,

zj=g∑i=0twi,j(h)Xi+bj(h)

with g as the activation function, wi,j(h) are the weights connecting the ith neuron from the input layer to the jth neuron in the hidden layer, and bj(h) is the bias.

Now, similar to the RANN network, we wish to introduce a term that penalizes non-convexity. For the following discussion, we will use the notation xi and yi to represent the coordinates of a point in the plane. We hope this causes no confusion with the dominant use of these letters to represent inputs and outputs, respectively. To develop the result, we let q be the number of vertices of the convex q-gon, given by a¯i=xiyi∈R2,i=1,…,q. To ensure that the a¯i are vertices of a convex q-gon in the plane we check that for any choice of three points a¯i,a¯j,a¯k, i,j,k∈{1,…,q}, i≠j≠k, each remaining point a¯m, m∈{1,…,n}\{i,j,k}, is not contained in the triangle with vertices a¯i,a¯j,a¯k. If a¯m does lie inside the triangle, then for some choice of λi,λj,λk∈(0,1), λi+λj+λk=1, we have

a¯m=λia¯i+λja¯j+λka¯k⇒xm=λixi+λjxj+λkxkandym=λiyi+λjyj+λkyk.

We get the linear system

xi-xkxj-xkyi-ykyj-ykλiλj=xm-xkym-yk

and solve for each λ:

λia¯i,a¯j,a¯k,a¯m:=λi=xm-xkyj-yk-xj-xkym-ykxi-xkyj-yk-xj-xkyi-ykλja¯i,a¯j,a¯k,a¯m:=λj=xi-xkym-yk-xm-xkyi-ykxi-xkyj-yk-xj-xkyi-yk.

For a¯m to be in the interior of triangle a¯i,a¯j,a¯k we need λi,λj>0 and λi+λj<1. If these conditions are true, we want to penalize our model. Next, we consider the colinear case. Let a¯i,a¯j,a¯k,i,j,k∈ {1,…,q},i≠j≠k be three distinct colinear points. Then the vectors connecting the vertices are multiples of each other such that for α>0 we have

a¯j-a¯k=αa¯i-a¯k⇒xj-xk=αxi-xkandyj-yk=αyi-yk.

(4)

Eliminating α, we obtain

xj-xk·yi-yk=yj-yk·xi-xk.

(5)

In order for non-colinearity to be guaranteed, we need (5) not to hold for distinct choices of a¯i,a¯j,a¯k,i,j,k∈{1,…,q}, i≠j≠k. Now, we consider the degenerate case, when two or three of the points are the same. Using the same a¯i, a¯j, and a¯k as above, assume α=0. Then a¯j-a¯k=αa¯i-a¯k gives a¯j=a¯k, which is a contradiction since the points are assumed to be distinct.

We use the following cost function for our model:

C=∑l=1q∑p=1tYpl-Y^pl2+β∑i,j,k,m=1tmax0,λia¯i,a¯j,a¯k,a¯mmax0,λja¯i,a¯j,a¯k,a¯mmax0,1-λia¯i,a¯j,a¯k,a¯m-λja¯i,a¯j,a¯k,a¯m2,

(6)

where the Yi’s and Y^i’s are as before, t=q·n is the number of inputs, β>0 is the regularization coefficient, and the summation is shorthand for four nested summations and the prime means that we also require that i, j, k, and m are distinct and a¯i, a¯j, and a¯k are neither colinear nor degenerate.

Example 2

We consider two trained convex-set-valued networks that forecast the movement of a 12-gon set. The first convex-set-valued network does not include the non-convexity penalty term in the cost function and the second does, with both excluding sparsity and entropy criteria. The weights of both networks are trained using the same data set and the trained networks forecast the movement of the set depicted in Fig. 2a. Figure 2b shows that when no penalty is included in the cost function during the training of the weights, the forecasted set becomes non-convex. However, in Fig. 2c, after the same number of steps, the network that included the non-convexity penalty term maintains convexity. In both cases, the forecasted set moved in the direction of the 3rd quadrant.

Fig. 2

The starting set (a) and the points propagated through the convex-set-valued network trained (b) without the non-convexity penalty term (c) with the non-convexity penalty term

Example 3

As a simple test problem, we define a convex set in the plane, with twelve vertices. We define a velocity for each vertex and create snapshots of the evolving convex set over time. The convex-set-valued data forms the training dataset for a convex-set-valued network with which we hope to forecast the future evolution of the set. We use a convex-set-valued neural network model, a dataset consisting of q-gons, and a cost function Cr=C+γ0ℓ0(w¯)-γentENT(w¯), where C=C(w¯) is the cost function presented in (6),

ℓ0(w¯)=∑i=0t1-e-0.5ℓ2w¯(h)+∑l=1q∑j=1J1-e-0.5ℓ2w¯l(o)

is a sparsity criterion and

ENT(w¯)=-∑i=0t∑j=1Jwi,j(h)Mlnwi,j(h)M+∑l=1q∑j=0Jwj,l(o)Mlnwj,l(o)M

is an entropy criterion, and γ0 and γent are two chosen nonnegative constants. The neural network is constructed using an input consisting of x’s and y’s, a hidden layer containing 5 neurons, and 24 (12 for x’s, 12 for y’s) output neurons for each predicted point (x, y). This network will demonstrate forecasting capabilities by using the previous 5 data points to forecast the subsequent data point. The network will include 121 input neurons (60 for the x coordinates from the 5 previous data points, 60 for the y coordinates from the previous 5 data points, 1 for the bias). We use a sigmoid activation function and the learning rate is 0.001.

Figure 3a and b illustrates some choices of γ0 in the convex-set-valued model. Figure 3 shows for both the x-coordinates (a) and y-coordinates (b) of the q-gon vertices that choices of 1.0×10-9≤γ0≤1.0×10-2 provide some improvement to the model performance compared to the base case. The value of the cost function for the x-coordinate and y-coordinate are in [0.1678, 0.1708] and [0.1553, 0.1582], respectively, whereas in the base case the cost value is 0.1708 and 0.1582 , respectively. We notice that for γ0=0.1 there is a decrease in model performance with a much greater cost value compared to the base case.

Fig. 3

Maximum value of cost function C across the 12 vertices of the q-gon and choices of a γ0 for the x-coordinate predictions, b γ0 for y-coordinate predictions, c γent for the x-coordinate predictions, and d γent for the y-coordinate predictions. Values along the x-axis are choices for 0 and s in 1.0×10-s,s= 1,…,15. Choices of γ0 (ℓ0 proxy criterion) and γent (ENT criterion) for the e x-coordinate and f y-coordinate predictions. Values along each axis are choices for 0 and s in 1.0×10-s,s=1,…,15. The shades of blue represent the maximum value of C across the 12 vertices of the q-gon where darker indicates a higher value

Figure 3c and d illustrates some choices of γent in the convex-set-valued model. Figure 3 shows for both the x-coordinates (c) and y-coordinates (d) of the q-gon vertices that there is no choice of γent that provides improved performance in the model. The value of the cost function for the x-coordinate and y-coordinate in the base case are 0.1708 and 0.1582, respectively.

Figure 3e and f illustrates some choices of γ0 and γent in the convex-set-valued model. Figure 3 shows for both the x-coordinates (e) and y-coordinates (f) of the q-gon vertices that there are many choices of γ0 and γent that provide improvement to the model performance compared to the base case. The improved model performance occurs when 1.0×10-9≤γ0≤1.0×10-2 combined with 1.0×10-15≤γent≤1.0×10-5, with the highest improvement occurring when γ0=0.01 and γent=0. Although the highest improvement does not occur when γent=0, there are many choices with both γ0>0 and γent>0 that provide an improvement.

5 Conclusion

We extended the notion of an interval-valued neural network to a convex-polygon-valued neural network, demonstrating that the penalty term we introduced in the latter case did indeed ensure that convexity was preserved. We explored model problems for these two types of networks, including, in addition, a sparsity criterion and an entropy criterion in the cost function. The experiments presented lead us to postulate that including both a sparsity and an entropy criteria with appropriately small coefficients generates an improvement in the performance of the network.

References

1.

Billard, L., Diday, E.: Regression analysis for interval-valued data. In: Data Analysis, Classification, and Related Methods. Springer (2000)

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Billard, L., Diday, E.: Symbolic regression analysis. In: Classification, Clustering, and Data Analysis. Springer (2002)

3.

Boreland, B.: A Mathematical Exploration of Neural Networks Using Sparsity, Entropy, and Multi-Valued Neurons. Doctoral Thesis, University of Guelph (Apr. 2023)

4.

Kunze, H., La Torre, D., Vrscay, E.R.: Solving inverse problems for DEs using the collage theorem and entropy maximization. Appl. Math. Lett. 25(12) (2012)

5.

Lima Neto, E.D.A., de Carvalho, F.D.A.: Centre and range method for fitting a linear regression model to symbolic interval data. Comput. Stat. Data Anal. 52(3) (2008)

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Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3) (1948)

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Wood, M., Stevens, F.: The fungi of California (1996). http://​www.​mykoweb.​com/​CAF

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Yang, Z., Lin, D., Zhang, A.: Interval-valued data prediction via regularized artificial neural network. Neurocomputing 331 (2019)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_8

Excursions of Solvable Scalar Diffusions

Giuseppe Campolieti1   and Yaode Sui1  

(1)

Mathematics Department, Wilfrid Laurier University, Waterloo, Canada

Giuseppe Campolieti (Corresponding author)

Email: gcampoli@wlu.ca

Yaode Sui

Email: suix3850@mylaurier.ca

Abstract

In this brief exposition we develop novel general formulae for distributions that are central to excursions of a diffusion process. We consider any scalar time-homogeneous solvable diffusion straddling two arbitrary levels within a finite time T and allow for general endpoint boundary types. In particular, we derive general formulae for both the marginal distribution of the first hitting time (post-T) to a given level and its joint distribution with the last hitting time to another level within time T. Our formulae involve only univariate integral expressions that are readily implementable for any solvable diffusion. We demonstrate their applicability by easily reproducing known formulae for standard Brownian motion. We also present general formulae for any solvable model arising from a Doob-transform.

Keywords

Solvable diffusionsStraddling excursionsLast hitting (passage) timeDoob-transformed diffusions

1 Introduction

The last passage (hitting) time is a fundamental concept in the study of stochastic processes. Some early studies in the context of standard Markov processes can be attributed to Getoor and Sharpe [1]. This groundwork was further enriched to include Bessel processes, where scholars like Pitman and Yor [2] made important strides. However, these studies focused particularly on the last passage time over an infinite time horizon. Very recent advancements were spearheaded by Campolieti and Sui [3, 4], where various general formulae were derived for the distribution of the last hitting time to a level, as well as its joint distribution with the value, realized maximum and/or minimum of the process on any finite time horizon and for any solvable scalar time-homogeneous diffusion.

In this current study, we focus on excursions of solvable diffusions. We derive new general comprehensive formulae for both marginal and joint distributions for a diffusion process to straddle two given levels within any fixed time T. This endeavor extends the research boundaries set by prior studies, which primarily revolved around Brownian Motion and for only one fixed level.

2 Main Result

Consider any regular time-homogeneous scalar diffusion process X≡{Xt}t≥0 on the state space I=(l,r) with endpoints l and r, -∞≤l<r≤∞, and with infinitesimal generator

Gf(x):=12σ2(x)f″(x)+α(x)f′(x)=1m(x)f′(x)s(x)′,

(1)

where α(x) and σ(x) are the respective infinitesimal drift and diffusion coefficient functions.¹ The domain of G consists of all bounded functions on I with Gf bounded on I and satisfying appropriate boundary conditions at the endpoints, e.g., see [5]. The diffusion X is generally assumed to have smooth scale and speed density functions, s(x):=exp-∫x2α(z)σ2(z)dz and m(x):=2σ2(x)s(x), where m(x) and s′(x) are assumed continuous on I. Fixing a probability measure P, we have the conditional probability Px(Xt∈dy)=p(t;x,y)dy where p(t; x, y) denotes the time-t transition probability density function (PDF) (w.r.t. the Lebesgue measure) of the diffusion X started at X0=x, x,y∈I,t>0. In what follows, we allow for either singular (exit-not-entrance, or entrance-not-exit or natural) endpoints with unique specification of the density or non-singular (regular) endpoints with boundaries that can be specified as either reflecting or killing. Our formulation is hence not restricted to conservative processes, i.e., we can have Px(Xt=∂†)>0 where ∂† denotes the cemetery state.

We consider excursions of X straddling a given time T>0 for two (generally different) levels, prior and post T. Hence, of interest is the last passage (hitting) time of X to level k∈I within a finite time interval [0, T],

gk(T):=sup{0≤u≤T:Xu=k},

(2)

with gk(T)≡0 if {0≤u≤T:Xu=k}=∅, as well as the earliest time instance after (“post”) T when the process reaches a level ℓ∈I:

Dℓ(T):=inf{u>T:Xu=ℓ}.

(3)

2.1 Marginal Distribution of Dℓ(T)

By conditioning on XT, where the process starts at X0=x, the cumulative distribution function (CDF) of Dℓ(T), for v∈(T,∞), takes the form:

Px(Dℓ(T)≤v)=∫lrPx(Dℓ(T)≤v|XT=y)p(T;x,y)dy=∫lrPy(Tℓ≤v-T)p(T;x,y)dy,

(4)

where p(T; x, y) is the time-T transition PDF and Py(Tℓ≤v-T) is the CDF (at v-T) of the first hitting time, Tℓ:=inf{t≥0:Xt=ℓ}, to level ℓ starting from y. As in [3], we exploit the total law of probabilities where Px(Dℓ(T)≤v|XT=∂†)=0 since the process hits ℓ past time T. Moreover, the last line is due to the Markov property in conjunction with the time-homogeneity of the process.

The PDF fDℓ(T)(v;x):=∂∂vPx(Dℓ(T)≤v) now follows directly from (4):

fDℓ(T)(v;x)=∫lℓf+(v-T;y,ℓ)p(T;x,y)dy+∫ℓrf-(v-T;y,ℓ)p(T;x,y)dy,

(5)

v∈(T,∞). Throughout, f(t;y,ℓ):=∂∂tPy(Tℓ≤t) and f±(t;y,ℓ):=∂∂tPy(Tℓ±≤t) denote the first hitting time densities, where Tℓ+ is the first hitting time up to ℓ starting from y∈(l,ℓ) and Tℓ- is the first hitting time down to ℓ from y∈(ℓ,r). The integrals in (5) encompass the contributions for hitting ℓ from both directions.

The random variable Dℓ(T) is generally not purely continuous. To complete the specification of its distribution we now also derive a formula for its discrete (“defective”) portion corresponding to Px(Dℓ(T)=∞). In what follows, we define El:={lisattractingnatural,orexit-not-entranceorregularkilling} and Elc:={l is non-attracting natural, or entrance-not-exit or regular reflecting} with identical definitions for Er and Erc pertaining to the right endpoint r. From standard diffusion theory we have

Py(Tℓ+=∞)=S[y,ℓ]S(l,ℓ]·IElandPy(Tℓ-=∞)=S[ℓ,y]S[ℓ,r)·IEr,

(6)

where IEl=1 if El holds and IEl=0 if Elc holds and similarly for IEr, and S[a,b]:=∫abs(x)dx is the scale function. Since Px(Dℓ(T)=∞)=∫lrPy(Tℓ=∞)p(T;x,y)dy,

Px(Dℓ(T)=∞)=IElS(l,ℓ]∫lℓS[y,ℓ]p(T;x,y)dy+IErS[ℓ,r)∫ℓrS[ℓ,y]p(T;x,y)dy.

(7)

Note: Py(Tℓ=∞)=Py(Tℓ+=∞), for y∈(l,ℓ], and Py(Tℓ=∞)=Py(Tℓ-=∞) for y∈[ℓ,r).

As a simple illustrative application of (5), consider standard Brownian Motion.

Example 1

Standard Brownian Motion Xt=Wt∈R: By setting the hitting level ℓ=0, with x=X0=W0=0, we recover the known PDF of D0(T) (e.g., see [5]):

P0(D0(T)∈dv)=1πvTv-Tdv,v∈(T,∞).

Indeed, the transition PDF is p(T;0,y)=e-y2/2T2πT and the first hitting time densities are f+(v-T;y,0)=|y|e-y2/2(v-T)(v-T)2π(v-T)I{y<0} and f-(v-T;y,0)=ye-y2/2(v-T)(v-T)2π(v-T)I{y>0}. Substituting these quantities into (5) gives the density of D0(T):

fD0(T)(v;0)=∫-∞0f+(v-T;y,0)p(T;0,y)dy+∫0∞f-(v-T;y,0)p(T;0,y)dy=1π(v-T)T(v-T)∫0∞ye-(v/(2T(v-T))y2dy=1πvTv-T.

(8)

Here we used the integral property: ∫0∞ye-ay2dy=12a. The CDF is then given by P0(D0(T)≤v)=1-2πsin-1Tv, v∈(T,∞), and zero otherwise, which also follows by directly employing the first hitting time CDF of standard BM into (4). Note that P0(D0(T)=∞)=0, which also follows directly from (7) since l=-∞ and r=∞ are non-attracting natural boundaries.

By using (4)–(7), the class of explicit formulae is readily extended to any solvable scalar time-homogeneous diffusion process.

2.2 Joint Distribution of (gk(T),Dℓ(T))

The last passage time gk(T) and Dℓ(T) are intrinsically intertwined. For any time-homogeneous diffusion, useful general formulae for the density of gk(T), and the joint density of the last passage time and process value, (gk(T),XT), as well as the joint distributions of gk(T),XT,MT,mT, where MT:=sup0≤t≤TXt,mT:=inf0≤t≤TXt, i.e., with imposed killing at arbitrary levels, are derived in [3], wherein novel analytically closed-form spectral expansions are also presented for several explicit families of solvable diffusions. Useful analytical formulae for the joint distribution of gk(T) and Dℓ(T) have remained largely confined to more elementary processes such as standard BM. Moreover, the levels k and ℓ have been constrained to be identical. Thus, in this section, building upon our previous works, we aim to derive a comprehensive formula that is versatile, applicable for any solvable diffusion and devoid of any stringent restrictions on the hitting levels.

Upon conditioning on Xt, while using the Markov property and time-homogeneity, the joint CDF, for all t∈[0,T],v∈(T,∞), can be expressed equivalently as:

Px(gk(T)≤t,Dℓ(T)≤v)=∫lrPx(gk(T)≤t,Dℓ(T)≤v|Xt=y)p(t;x,y)dy=∫lr[∫lrPz(Tℓ≤v-T)Py(Tk>T-t,XT-t∈dz)]p(t;x,y)dy=∫lk[∫lkp(t;x,y)pk-(T-t;y,z)dy]Pz(Tℓ≤v-T)dz+∫kr[∫krp(t;x,y)pk+(T-t;y,z)dy]Pz(Tℓ≤v-T)dz

(9)

where

Py(Tk>T-t,XT-t∈dz)=Py(mT-t>k,XT-t∈dz)≡pk+(T-t;y,z)dz,Py(MT-t<k,XT-t∈dz)≡pk-(T-t;y,z)dz.

(10)

Note that pk+ and pk- denote the time-(T-t) transition PDFs with imposed killing at level k and these are, respectively, nonzero only for y,z∈(k,r) and y,z∈(l,k).

We remark that the derivation of (9) employs a two-stage conditioning. That is, the probability of paths (starting at X0=x) having last hit level k by time t∈(0,T], and later hitting level ℓ for the first time within the time interval (T, v] is equivalent to the probability of paths (starting at X0=x) that take on any value y at time t, and then avoid hitting level k up to time T while ending up with any value z (i.e., the first hitting time to k, for the process restarted at y, is greater than T-t and takes on value z after a time T-t) and subsequently hit the second level ℓ within the final time lapse v-T.

Since p(0+;x,y)=δ(y-x), where δ() is the Dirac delta function, Eq. (9) gives the partly discrete portion of the distribution in terms of a univariate integral:

Px(gk(T)=0,Dℓ(T)≤v)=∫lkPz(Tℓ≤v-T)pk-(T;x,z)dz,x∈(l,k],∫krPz(Tℓ≤v-T)pk+(T;x,z)dz,x∈[k,r).

(11)

This equation also follows directly from the equivalence of the joint probabilities: Px(gk(T)=0,Dℓ(T)≤v)=Px(Tk>T,Dℓ(T)≤v). We can also expand out one of the integrals in (11) for the respective two cases, ℓ∈(l,k] and ℓ∈[k,r), while also using Tℓ=Tℓ+ or Tℓ=Tℓ- accordingly. Note also that the partial density ∂∂vPx(gk(T)=0,Dℓ(T)≤v) is also represented as a single integral as in (11) where Pz(Tℓ≤v-T) is replaced by the density f(v-T;z,ℓ) upon differentiating w.r.t. v.

For the jointly discrete portion of the distribution we have:

Px(gk(T)=0,Dℓ(T)=∞)=∫lkPz(Tℓ=∞)pk-(T;x,z)dz,x∈(l,k],∫krPz(Tℓ=∞)pk+(T;x,z)dz,x∈[k,r).

(12)

These integrals can also be written more explicitly using (6) for the cases ℓ∈(l,k] and ℓ∈[k,r).

By simply replacing Pz(Tℓ≤v-T) with Pz(Tℓ=∞) in (9), we have the other partly discrete portion of the distribution:

Px(gk(T)≤t,Dℓ(T)=∞)=∫lk[∫lkp(t;x,y)pk-(T-t;y,z)dy]Pz(Tℓ=∞)dz+∫kr[∫krp(t;x,y)pk+(T-t;y,z)dy]Pz(Tℓ=∞)dz.

(13)

The following theorem now gives us a general formula for the joint PDF in terms of only univariate integrals.

Theorem 1

Joint PDF of (gk(T),Dℓ(T))

The joint PDF of (gk(T),Dℓ(T)) for any regular diffusion on (l, r), started at X0=x∈(l,r), with generator in (1), is given equivalently by:

fgk(T),Dℓ(T)(t,v;x)=q(t;x,k)[∫lkp^-(τ;k,z)f(v-T;z,ℓ)dz+∫krp^+(τ;k,z)f(v-T;z,ℓ)dz]

(14)

≡q(t;x,k)[∫lℓp^-(τ;k,z)f+(v-T;z,ℓ)dz+∫ℓkp^-(τ;k,z)f-(v-T;z,ℓ)dz+∫krp^+(τ;k,z)f-(v-T;z,ℓ)dz],ℓ∈(l,k],q(t;x,k)[∫lkp^-(τ;k,z)f+(v-T;z,ℓ)dz+∫kℓp^+(τ;k,z)f+(v-T;z,ℓ)dz+∫ℓrp^+(τ;k,z)f-(v-T;z,ℓ)dz],ℓ∈[k,r),

(15)

for t∈(0,T), v∈(T,∞), and zero otherwise, where q(t;x,k):=p(t;x,k)m(k), and pk± are defined in (10) with p^±(τ;k,z):=±1s(k)∂∂ypk±(τ;y,z)|y=k± and τ:=T-t.

   □

We remark that p^±(τ;k,z) is strictly positive in both (±) cases. Moreover, the two expressions in (15), which are valid when k≠ℓ, reduce to the same single expression in the special case when k=ℓ. This is also clearly seen by setting ℓ=k in (14) where f(v-T;z,ℓ) becomes f+(v-T;z,k) in the first integral and f(v-T;z,ℓ) becomes f-(v-T;z,k) in the second integral.

Proof

The joint PDF of (gk(T),Dℓ(T)) is given by

fgk(T),Dℓ(T)(t,v;x)=∂∂t∂∂vPx(gk(T)≤t,Dℓ(T)≤v).

(16)

Hence, by differentiating within the two double integrals in (9), where by definition f(v-T;z,ℓ)=∂∂vPz(Tℓ≤v-T), we obtain

fgk(T),Dℓ(T)(t,v;x)=∫lk[∫lk∂∂t(p(t;x,y)pk-(T-t;y,z))dy]f(v-T;z,ℓ)dz+∫kr[∫kr∂∂t(p(t;x,y)pk+(T-t;y,z))dy]f(v-T;z,ℓ)dz.

The y-integrals in both of the above double integrals are evaluated as follows. Let us consider the first y-integral. By combining the backward Kolmogorov PDE (in t, y) for pk-(T-t;y,z), with generator in (1), and the forward Kolmogorov PDE (in t, y) for p(t; x, y) with adjoint generator, we have

∂∂t(p(t;x,y)pk-(T-t;y,z))=pk-(τ;y,z)∂∂y(1s(y)∂∂yq(t;x,y))-q(t;x,y)∂∂y(1s(y)∂∂ypk-(τ;y,z)),

(17)

where q(t;x,k):=p(t;x,k)m(k), τ:=T-t.

Now, using this expression within the above first y-integral and subsequently applying integration by parts, with exact cancelation of two intermediate integral terms, gives

∫lk∂∂t(p(t;x,y)pk-(T-t;y,z))dy=pk-(τ;y,z)1s(y)∂∂yq(t;x,y)|y=l+y=k--q(t;x,y)1s(y)∂∂ypk-(τ;y,z)|y=l+y=k-=-1s(y)q(t;x,y)∂∂ypk-(τ;y,z)|y=k-≡q(t;x,k)p^-(τ;k,z),

(18)

where p^-(τ;k,z):=-1s(k)∂∂ypk-(τ;y,z)|y=k-.

The final expression in (18) arises by taking into account all types of lower boundary l in evaluating the limits within the second equation line. The first term with y=k- vanishes from the imposed killing boundary condition pk-(τ;k-,z)=pk-(τ;k,z)=0 and where 1s(k-)∂∂yq(t;x,y=k-)=1s(k)∂∂yq(t;x,k) is bounded for any interior point k. Moreover, both terms with limits y=l+ also vanish. Indeed, if l is regular killing, natural, or exit-not-entrance, then both 1s(l+)∂∂ypk-(τ;l+,z) and 1s(l+)∂∂yq(t;x,l+) are bounded (being zero in the natural case) and pk-(τ;l+,z)=q(t;x,l+)=0. Otherwise, if l is regular reflecting or entrance-not-exit, then we have 1s(l+)∂∂ypk-(τ;l+,z)=1s(l+)∂∂yq(t;x,l+)=0, with both pk-(τ;l+,z) and q(t;x,l+) being bounded.

The second y-integral above is evaluated in exactly the same fashion as given in (17)-(18), with interval (k, r) replacing (l, k), pk+(τ;y,z) replacing pk-(τ;y,z) and upper boundary r replacing lower boundary l. For all types of upper boundary r, we have

∫kr∂∂t(p(t;x,y)pk-(T-t;y,z))dy=q(t;x,k)p^+(τ;k,z),

(19)

where p^+(τ;k,z):=1s(k)∂∂ypk+(τ;y,z)|y=k+. Combining both expressions in (18) and (19) gives (14).

Upon differentiating (13) w.r.t. t while employing (18) and (19) also gives us a formula for the partial density involving only univariate integrals:

∂∂tPx(gk(T)≤t,Dℓ(T)=∞)=q(t;x,k)[∫lkPz(Tℓ=∞)p^-(τ;k,z)dz+∫krPz(Tℓ=∞)p^+(τ;k,z)dz].

(20)

This expression can also be expanded and recast in exactly the same form as the formula in (15) where the densities f+(v-T;z,ℓ) and f-(v-T;z,ℓ) are now replaced by Pz(Tℓ+=∞) and Pz(Tℓ-=∞), respectively.

The complete distribution of the pair (gk(T),Dℓ(T)) is therefore given by (11), (12), (14) and (20). These all involve only univarate integrals and lead to newly closed-form analytical formulas for any one-dimensional time-homogeneous solvable diffusion. Most theoretical discussions in the literature have largely revolved around the standard Brownian motion and where the hitting levels k and ℓ were constrained to be of the same value. By employing spectral expansions for the transition densities and the transition densities with imposed killing, our formulas also give rise to spectral expansions for the joint distribution of (gk(T),Dℓ(T)). These spectral expansions are similar to the expressions already exhaustively treated in Campolieti and Sui [3], where closely related spectral expansions were derived and applied across several solvable diffusion models. We do not discuss these in this brief paper. Rather, as a simple example of the applicability of our new main formulas, we now simply apply them to standard Brownian motion and juxtapose our findings with known results.

Example 2

Standard Brownian Motion Consider standard Brownian motion Xt=Wt, X0=W0=0≡x, where we simply take both hitting levels as zero, i.e., k=ℓ=0. The analytical expression for the joint PDF of (g0(T),D0(T)) is well-known (e.g., see [5]):

P0(g0(T)∈dt,D0(T)∈dv)=dtdv2πt(v-t)3,t∈(0,T),v∈(T,∞).

(21)

The transition PDF for y,z∈(0,∞) with killing at lower level k=0 and starting at the point y>0 is given by (τ≡T-t):

p0+(τ;y,z)=12πτ(e-(z-y)22τ-e-(z+y)22τ).

(22)

Taking the derivative with respect to y and evaluating at y=0+ yields:

∂∂yp0+(τ;y,z)|y=0+=2τ2πτze-z22τ≡2τ2πτ|z|e-z22τ.

For y,z∈(-∞,0), p0-(τ;y,z) is given by the same expression in (22) where

-∂∂yp0-(τ;y,z)|y=0-=-2τ2πτze-z22τ≡2τ2πτ|z|e-z22τ.

Since s(k)=1, p^±(τ;0,z)=±2τ2πτze-z22τ≡2τ2πτ|z|e-z22τ.

Substituting p^±(τ;0,z) into (14), while using the first hitting time density, i.e., f(v-T;z,0)=|z|e-z2/2(v-T)(v-T)2π(v-T), and q(t;0,0)=122πt, and the fact that the product p^±(τ;0,z)f(v-T;z,0) is an even function about z=0 gives the joint PDF in (21):

fg0(T),D0(T)(t,v;0)=12πt∫0∞p^+(τ;0,z)f(v-T;z,0)dz=(τ(v-T))-3/22π2πt∫-∞∞z2e-z22τ(v-T)/(v-t)dz=12πt(v-t)3.

(23)

Here, we used ∫-∞∞z2e-z2/2σ2dz=2πσ3, with σ≡τ(v-T)/(v-t).

The joint distribution is purely continuous in this case, where P0(g0(T)=0)=0 and P0(D0(T)=∞)=0. We recover the known purely continuous marginal distribution, fg0(T)(t)=∫T∞fg0(T),Dℓ(T)(t,v;0)dv=1πt(T-t), giving the arcsine law: P0(g0(T)≤t)=2πsin-1t/T, 0≤t≤T.

2.3 Doob-Transformed Solvable Diffusions

The general formulas for the marginal and joint distributions in the previous section are readily extended to more families of solvable diffusions via a simple application of a Doob-h transform. The Doob-h transform can also be combined with a smooth mapping to construct several alternative families of solvable diffusions that have seen applications in finance. This transformation was originally coined “diffusion canonical transformation”. For more details see, e.g., [6] and references therein. Here we only give the briefest account and present two main formulas that are useful for excursions.

Given an underlying diffusion X∈(l,r) with generator in (1), the Doob-transformed diffusion, denoted as X(ρ)≡{Xt(ρ)}t≥0∈(l,r), has infinitesimal generator

G(ρ)f(x):=12σ2(x)f″(x)+α(x)+σ2(x)u^ρ′(x)u^ρ(x)f′(x),

(24)

where u^ρ(x) is the strictly positive generating function that solves Gu^ρ(x)=ρu^ρ(x) with an arbitrary parameter ρ>0. The speed and scale densities are respectively given by mρ(x)=m(x)u^ρ2(x) and sρ(x)=s(x)/u^ρ2(x). As in (2)-(3), the last passage time and first hitting time post-T are gk(ρ)(T):=sup{0≤u≤T:Xu(ρ)=k} and Dℓ(ρ)(T):=inf{u>T:Xu(ρ)=ℓ}, respectively.

The fundamental quantities for X(ρ) are simply related to those of X. In particular, Px(Xt(ρ)∈dy)=p(ρ)(t;x,y)dy, where p(ρ)(t;x,y)=e-ρtu^ρ(y)u^ρ(x)p(t;x,y), and the transition PDFs for X(ρ) killed at k are given by pk(ρ)±(τ;y,z)=e-ρτu^ρ(z)u^ρ(y)pk±(τ;y,z). The first hitting time density of X(ρ) is related to that of X, i.e., f(ρ)(v-T;y,ℓ)=e-ρ(v-T)u^ρ(ℓ)u^ρ(y)f(v-T;y,ℓ). From these relations it readily follows that the PDF of Dℓ(ρ)(T) and joint PDF of gk(ρ)(T),Dℓ(ρ)(T) are simply related to the respective PDFs for the X diffusion:

fDℓ(ρ)(T)(v;x)=e-ρvu^ρ(ℓ)u^ρ(x)fDℓ(T)(v;x),

(25)

fgk(ρ)(T),Dℓ(ρ)(T)(t,v;x)=e-ρvu^ρ(ℓ)u^ρ(x)fgk(T),Dℓ(T)(t,v;x).

(26)

The discrete portions of the distribution, i.e., analogues of (7), (11), (12) and (20), also readily follow. Hence, these formulae further extend the realm of analytical expressions for various families of solvable diffusions.

Acknowledgements

This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grant number 2018-06176.

References

1.

Getoor, R.K., Sharpe, M.J.: Last exit times and additive functionals. Ann. Probab., 550–569 (1973)

2.

Pitman, J., Yor, M.: Bessel processes and infinitely divisible laws. Stoch. Integr., 285–370 (1981)

3.

Campolieti, G., Yaode, S.: Last Hitting Time Distributions for Solvable Diffusions. Working paper

4.

Yaode, S.: Last Passage Time and Excursion Theory for Solvable Diffusions with Applications in Mathematical Finance, Doctoral dissertation, Wilfrid Laurier University (2024)

5.

Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae, Series: Probability and its Applications, 2nd ed. Birkhäuser Basel (2002)

6.

Campolieti, G., Makarov, R.N.: Solvable diffusion models with linear and mean-reverting nonlinear drifts. SIAM J. Financ. Math. 8(1), 146–170 (2017)MathSciNet

Footnotes

1

Throughout we consider diffusions with zero instantaneous killing measure, although the framework is readily extended to include a nonzero killing measure.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_9

Hybrid Impulsive Formation Control of Vehicle Platoons Using Neural Networks

Zhanlue Liang1   and Xinzhi Liu1  

(1)

University of Waterloo, Waterloo, Ontario, Canada

Zhanlue Liang

Email: z22liang@uwaterloo.ca

Xinzhi Liu (Corresponding author)

Email: xinzhi.liu@uwaterloo.ca

Abstract

In this paper, we study formation control problems for vehicle platoons via a neural network-based hybrid impulsive strategy. Specifically, intrinsic nonlinear vehicle dynamics are approximated using the radial basis function neural networks technique (RBFNNs), while overall system stabilization depends on stabilizing control impulses. Based on the Lyapunov-Razumikhin technique, graph theory and LMI method, sufficient stabilization criteria are derived so that tracking errors can be restrained under a compact error bound and vehicle platoon formation control objectives can be further achieved. Lastly, a numerical example is provided to validate the performance and effectiveness of our proposed control strategy.

Keywords

Vehicle platoonFormation trackingImpulsive controlNeural networks

1 Introduction

In recent years, formation control of vehicle platoons has attracted considerable research interests due to its extensive applications associated with multi-agent systems and automated highway vehicle systems. In practice, it aims to improve traffic efficiency while reducing transportation congestion [1]. These objectives can be accomplished by continuously modifying the vehicle’s speed in the platoon and inter-vehicle distance through adaptive cruise control [2], vehicle-to-vehicle (V2V) communication [3] etc. Thus, in view of predefined platoon configuration and limited sensing range, decentralized displacement-based formation control subject to relative information exchange [4, 5] is considered to be one of efficient approaches to maintain overall stability of vehicle platoons.

In order to increase fault tolerance, enhance transmission security and reduce control costs, we adapt the hybrid impulsive framework to the design of the formation control protocol. In this manner, continuous control signals are partially generated, and stabilizing control impulses are transmitted only at predefined time instants to change system states abruptly. Many pioneering works such as [6, 7] have provided a comprehensive view regarding the fundamentals of impulsive and hybrid systems, but there are few results that incorporate both types of systems in the stabilization of formations. Moreover, we use radial basis function neural networks (RBFNNs) for adaptive estimation of intrinsic dynamics, since they are robust to noisy data, model nonlinearity, and perform interpolation. During the past decade, continuous control results for nonlinear multi-agent systems using neural networks in addition to [8, 9] have been addressed, but we are integrating RBFNNs into a hybrid impulsive formation control framework.

Inspired by the above discussion, we develop a novel hybrid impulsive vehicle platoon control protocol by incorporating neural network-based adaptive estimation using RBFNNs. We provide an adaptive updating law and some sufficient conditions for string stabilization of vehicle platoons under compact tracking errors.

We organize the remaining paper as follows. In Sect. 2, we provide preliminaries and problem formulation. In Sect. 3, neural network-based hybrid-impulsive control protocols of vehicle platoons are developed. Section 4 presents numerical examples to validate the performance and effectiveness of our proposed control scheme. Finally, the conclusion is drawn in Sect. 5.

2 Preliminaries

2.1 Notations

Denote G=(V,E) as a directed graph (or diagraph) of order n, which includes a vertex set V={ν1,ν2,ν3,...,νn}, and an edge set E={(ei,ej)}⊂E×E. An edge eij=(ei,ej) is added as long as node νj is able to receive information from node νi. Ni={j∈V|eij∈E} denotes the i-th neighbour set associated with G. The weighted adjacency matrix A=[aij]∈Rn×n satisfies aij>0 for eij∈E and aij=0 otherwise. The Laplacian matrix of the graph L=[lij]∈Rn×n is defined as L=D-A, where D is a diagonal matrix with lii=∑i≠jaij and lij=-aij ∀i≠j. A sequence of edges e12, e23,... represents a directed path in G. Let Z+, N be the set of positive integers and natural numbers, In as the n×n identity matrix, ‖.‖ as the Euclidean norm, ⊗ as the Kronecker product, while the minimum and maximum eigenvalues of matrix L are λmin(L), λmax(L) respectively. In addition, 0n×n, 0n and 1n are set to be the n×n matrix, n×1 column vector of zeros and ones.

2.2 Problem Statement

According to [10], the dynamics of each vehicle in the platoon of n vehicles can be modelled as follows:

p˙i(t)=vi(t),v˙i(t)=ai(t),a˙i(t)=-1κai(t)+fi(t)+ui(t),i=1,2,...,n,

(1)

while a virtual leader is planted to regulate the overall vehicle platoon by

p˙0(t)=v0(t),v˙0(t)=a0(t),a˙0(t)=-1κa0(t),

(2)

where pi,vi,ai∈R2 are position, velocity and acceleration states of vehicle i, p0,v0,a0∈R2 represent the corresponding states for the virtual leader, fi(t) is the nonlinear intrinsic dynamics, and ui(t)∈R2 is the hybrid impulsive control input to be designed hereafter. Meanwhile, we denote ξi=[piT,viT,aiT]T and consider the set of impulsive instants via the sequence {tl,l∈Z+} satisfying t0<t1<...<tl-1<tl<..., liml→∞tl=∞. For consistency, the right continuity with ξi(t)=ξi(t+) and Δξi(t)=ξi(t+)-ξi(t-) are considered throughout the paper. Moreover, we denote p^i(t)=pi(t)-p0(t)-pdi where pdi∈R2 prescribes the desired displacement between the i-th vehicle and the virtual leader, v^i(t)=vi(t)-v0(t), and a^i(t)=ai(t)-a0(t). Then set p¯=[p^1T,...,p^nT]T, v¯=[v^1T,...,v^nT]T, a¯=[a^1T,...,a^nT]T, f¯=[f1T,...,fnT]T, ξ^i=[p^iT,v^iT,a^iT]T and ξ¯=[p¯T,v¯T,a¯T]T. Thus, the error dynamics of the i-th vehicle can be described as follows:

ξ^˙i(t)=Eξ^i(t)+0202ui(t)+fi(t),i=1,...,n,ξ^i(t0)=ξ^i0,

(3)

where ξ^i0 is the initial state and E=02×2I202×202×202×2I202×202×2-(1κ)I2 for i=0,1,...,n.  

The following assumptions, lemmas and definitions are given for the rest of paper.

Assumption 1

The overall topology G contains a spanning tree, where its root node is the virtual leader.

Assumption 2

Assume that fi(t) can be linearly parameterized on a prescribed compact set using radial basis function neural networks (RBFNNs) as:

fi(t)=WiTsi(t)+ϵi,

(4)

where Wi is the ideal RBFNN weight matrix with compatible dimension, si(t) is the basis function vector, while ϵi represents the arbitrarily small residual error satisfying ‖ϵi‖≤ϵm.

Remark 1

In practice, the nonlinear intrinsic dynamics fi(t) remains undetected. Therefore, in order to compensate for such uncertainties, we employ RBFNNs for adaptive approximation. Here we denote W^i as the current RBFNN weight and W~i=Wi-W^i.

Lemma 1

[6] For any symmetric matrix M>0 and vectors u,v∈Rn, we have 2uTv≤uTMu+vTM-1v.

Definition 1

The global practical stabilization of the vehicle platoon is achieved if for any given bounded initial states such that

limt→∞‖ξ^i(t)‖≤ζ,

(5)

where ζ is a positive constant.

Definition 2

[11] The average impulsive interval (AII) of the impulsive sequence F[Tα,ϱ]={tl}l∈Z+ is equal to Tα>0 if there exists ϱ>0 so that

T-tTα-ϱ≤N(t,T)≤T-tTα+ϱ,∀0≤t≤T,

(6)

where N(t, T) denotes the total number of impulse instants of the impulsive sequence F[Tα,ϱ] in the time interval (t,T).

Fig. 1

Hybrid topological structure of the vehicle platoon subject to predecessor-following string

Remark 2

Fig. 1 illustrates the overall communication topology of the vehicle platoon via the hybrid impulsive control framework satisfying Assumption 1 with virtual leader 0. The topological structure in Fig. 1a for vehicle 1–6 is defined as the predecessor-following string.

3 Main Results

We design our hybrid impulsive cooperative control protocol subject to stabilizing control impulses as follows:

ui(t)=-W^iTsi(t)+k1PC(p^i(t),v^i(t),a^i(t))+k2PD(p^i(t),v^i(t),a^i(t))+k3PL(p^i(t),v^i(t),a^i(t)),

(7)

with

PC(.,.,.)=-∑j∈Ni1bij(p^i(t)-p^j(t))-∑j∈Ni1bij(v^i(t)-v^j(t))-∑j∈Ni1bij(a^i(t)-a^j(t)),PD(.,.,.)=-∑l=1∞∑j∈Nibij(p^i(t)-p^j(t))δ(t-tl)-∑l=1∞∑j∈Nibij(v^i(t)-v^j(t))δ(t-tl)-∑l=1∞∑j∈Nibij(a^i(t)-a^j(t))δ(t-tl),PL(.,.,.)=-∑l=1∞cip^i(t)δ(t-tl)-∑l=1∞civ^i(t)δ(t-tl)-∑l=1∞cia^i(t)δ(t-tl),

(8)

where k1, k2, k3 are positive gains, Ni1⊂Ni is the neighbor set associated with the hybrid topology within non-impulse interval, δ(.) is the Dirac function, bij is the coupling strength, and ci represents the impulsive tracking gain. Then, since Eξdi=06×6 with ξdi=[pdiT,02T,02T]T, it gives the compact error dynamics of (3) as:

ξ¯˙(t)=(E¯+E¯1)ξ¯(t)+02n02n(W~iTsi(t)+ϵi)⊗1n,t≠tl,Δa¯(t)=-k3Cp¯(t)-k3Cv¯(t)-k3Ca¯(t)-k2Lp¯(t)-k2Lv¯(t)-k2La¯(t),t=tl,ξ¯(t0)=ξ¯0,

(9)

where ξ¯0=ξ^i0⊗1n, E¯=E⊗In, E¯1=E1⊗In=02n×2n02n×2n02n×2n02n×2n02n×2n02n×2nk1L′k1L′k1L′, C=cI2n, and L′ represents the Laplacian matrix of topological structure during the non-impulse interval. Denote Σ=-k2L-k3C, E¯2=(1-k3)I2n02n×2n02n×2n02n×2n(1-k3)I2n02n×2nΣΣI2n-Σ, P=P11P12P13P21P22P23P31P32P33∈R6×6 and ϵ¯=02n02nϵi⊗1n for later use. The following theorem gives sufficient conditions of global practical stabilization of vehicle platoon.

Theorem 1

Suppose that Assumptions 1 and 2 are satisfied, P¯=P⊗In>0, β>0, q1>0, ϱ>0, θ(t)=1 for t≠tl and θ(t)=ρ1 for t=tl,l∈Z+. Meanwhile, set W¯~=W~i⊗In and W¯^=W^i⊗In, W¯=Wi⊗In so the adaptive updating law of W¯ is designed as:

W¯^˙=(βsia^iTP33+βsia^iTP33T)⊗In.

(10)

If for ε1>0 the following criterion can be fulfilled:

η1+lnρ1Tα<-q1<0,

(11)

with η1=λmax((P¯(E¯+E¯1)+(E¯T+E¯1T)P¯+ε1‖P¯‖2)P¯-1), ρ1=λmax(E¯2TP¯E¯2P¯-1), Then the global practical stabilization of vehicle platoon with respect to system (1) and (2) is achieved, and the compact tracking error is bounded by

‖ξ¯(t)‖≤2ρ1ϱε0q1λmin(P¯),ε0=ϵm22ε1.

(12)

Proof

Construct the Lyapunov function H(t) as:

H(t)=12ξ¯T(t)P¯ξ¯(t)+θ(t)2βtr(W¯~TW¯~).

(13)

Then, based on (9), (10) and Lemma 1, it gives

H˙(t)=12ξ¯T(t)(P¯E¯+E¯TP¯)ξ¯(t)+12ξ¯T(t)(P¯E¯1+E¯1TP¯)ξ¯(t)+12(ξ¯T(t)P¯ϵ¯+ϵ¯TPξ¯(t))+∑i=1n(a^iTP33W~iTsi+siTW~iP33a^i)-1βtr(W¯~TW¯^˙)≤12ξ¯T(t)(P¯(E¯+E¯1)+(E¯T+E¯1T)P¯)ξ¯(t)+∑i=1n(1βtr(βsia^iTP33W~iT)+1βtr(βsia^iTP33TW~iT)-1βtr((βsia^iTP33+βsia^iTP33T)W~iT))+ε1‖P¯‖22ξ¯T(t)ξ¯(t)+12ε1ϵ¯Tϵ¯,

(14)

such that

H˙(t)≤12ξ¯T(t)(P¯(E¯+E¯1)+(E¯T+E¯1T)P¯+ε1‖P¯‖2)ξ¯(t)+εm22ε1≤η1H(t)+ε0.

(15)

At each impulsive instant t=tl, we have

12ξ¯T(tl)P¯ξ¯(tl)=12ξ¯T(tl-)E¯2TP¯E¯2ξ¯T(tl-)≤12λmax(E¯2TP¯E¯2P¯-1)ξ¯T(tl-)Pξ¯(tl-)=12ρ1ξ¯T(tl-)P¯ξ¯(tl-).

(16)

Then, (16) leads to

H(tl)≤12ρ1ξ¯T(tl-)P¯ξ¯(tl-)+ρ12βtr(W¯~T(tl-)W¯~(tl-))≤ρ1H(tl-).

(17)

Based on (15) and (17), H(t) is now upper bounded by

H˙max(t)=η1Hmax(t)+ε0,t≠tl,Hmax(tl)=ρ1Hmax(tl-),t=tl,

(18)

where Hmax(0)=H(0)=H0, H(t)≤Hmax(t) for t>t0. Then one has

Hmax(t)=H0eη1(t-t0)+∫t0teη1(t-z)ε0dz,t∈[t0,t1),

(19)

and

Hmax(t1)=ρ1Hmax(t1-)=ρ1H0eη1(t1-t0)+ρ1∫t0t1eη1(t1-z)ε0dz.

(20)

Recursively for t∈[t1,t2), we have

Hmax(t)=ρ1H0eη1(t-t1+t1-t0)+ρ1∫t0t1eη1(t-t1+t1-z)ε0dz+∫t1teη1(t-z)ε0dz,

(21)

and

Hmax(t2)=ρ1Hmax(t2-)=ρ12H0eη1(t2-t0)+ρ12∫t0t1eη1(t2-z)ε0dz+ρ1∫t1t2eη1(t2-z)ε0dz.

(22)

Therefore, based on Definition 2 and (11), one has the following for t∈[tl-1,tl):

Hmax(t)=ρ1l-1H0eη1(t-t0)+∑m=1l-1∫tm-1tmρ1l-meη1(t-z)ε0dz+∫tl-1teη1(t-z)ε0dz,

(23)

and

H(t)≤Hmax(t)=ρ1N(t0,t)H0eη1(t-t0)+∫t0tρ1N(z,t)eη1(t-z)ε0dz≤ρ1ϱH0e(η1+lnρ1Tα)(t-t0)+∫t0tρ1ϱe(η1+lnρ1Tα)(t-z)ε0dz≤ρ1ϱH0e-q1(t-t0)+∫t0tρ1ϱe-q1(t-z)ε0dz≤ρ1ϱH0e-q1(t-t0)+1q1ρ1ϱε0,

(24)

so that Definition 1 is satisfied with the compact error bound

‖ξ¯(t)‖2≤2ρ1ϱε0q1λmin(P¯).

(25)

As a result, the global practical stabilization of vehicle platoon with respect to system (1) and (2) is achieved. The proof is complete.

Remark 3

Condition (11) indicates that the stabilization process depends on the strength of stabilizing impulses ρ1, average impulsive interval Tα and instability tolerance of continuous dynamics η1.

4 Numerical Simulation

The total simulation time is 10 s and we simulate a platoon of n=6 predecessor-following vehicles. The initial positions, velocities and accelerations of followers are arbitrarily generated from the range of [0,15]×[-2,2], [0,2.5]2 and [-5,5]2 respectively; the initial states of virtual leader are set to be p0=[15,0]T, v0=[2.5,0]T and a0=[1,1]T. We consider the impulsive sequence {tl}l∈Z+={t2n-1=0.05n-0.03,t2n=0.05n}n∈Z+ so that Tα=0.025s and ϱ=2. We also set our control parameters bij=1.25, ci=0.5, k1=k2=1, and k3=2.5. Moreover, suppose the RBFNNs contain 36 neurons, the centre of corresponding Gaussian radial basis function is evenly distributed on [-3,3]×[-3,3], the variance of RBFNNs is chosen to be 1.2 and each entry of weight matrix is initialized in the range [0, 1]. Lastly, our desired configuration is set to be pdi=[-5-3(i-1),2]T for i=1,...,6.

Example 1 We obtain η1=2.09 and ρ1=0.847 by using LMI toolbox so that condition (11) is satisfied with q1=4.3. Based on Fig. 2, the practical stability of tracking mismatches with respect to position, velocity and acceleration are successfully obtained with ‖ξ¯(t)‖≤0.158 due to inevitable RBFNN residual errors. Meanwhile, the string formation trajectory evolution in Fig. 3 also suggests that the global practical stabilization of vehicle platoon can be achieved via our hybrid impulsive control protocol and adaptive updating law, thus Theorem 1 is valid.

Fig. 2

Convergence of a position b velocity c acceleration d total mismatch of vehicle platoon

Fig. 3

Trajectory evolution of vehicle platoon

5 Conclusions

In this paper, we have developed a hybrid impulsive formation control protocol for a vehicle platoon using neural networks. Global practical stabilization has been achieved by using Lyapunov Razumikhin technique, graph theory and LMI approach. We have carried out the adaptive estimation of ideal RBFNN weights and specified the compact error bound subject to impulse strength. Finally, we present a numerical example to validate our theoretical results in terms of control effectiveness and stabilization performance.

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada which is gratefully acknowledged.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_10

Pinning Impulsive Control for Synchronization of Complex-Valued Delayed Multi-Layer Networks

Yuan Shen1   and Xinzhi Liu1  

(1)

University of Waterloo, Waterloo, ON, Canada

Yuan Shen

Email: y57shen@uwaterloo.ca

Xinzhi Liu (Corresponding author)

Email: xinzhi.liu@uwaterloo.ca

Abstract

This paper studies the synchronization problem of complex-valued multi-layer networks (CVMLNs) via delay-dependent pinning impulsive control. At first, a multi-layer directed network model is constructed in complex domains, and the effects of intra-layer and inter-layer coupling delays are both considered. In order to achieve synchronization of CVMLNs, a delay-dependent pinning impulsive control strategy is designed in the complex field. The pinning impulsive controller depends on not only the network states at each impulsive instant but also the states at history time. By taking advantage of the Lyapunov function method in the complex field and algebraic graph theory, sufficient conditions for synchronization of CVMLNs with time-varying coupling delays are established. Finally, a numerical example is presented to illustrate the effectiveness and validity of the theoretical result.

Keywords

Complex-valued multi-layer networksSynchronizationPinning algorithmImpulsive controlTime delay

1 Introduction

Nowadays, complex networks (CNs) have received considerable attention in science and engineering. In reality, many natural and man-made networks can be described by CNs. Synchronization is a significant and pervasive collective behavior of CNs, which refers to the coordination and alignment of dynamics among interconnected entities within networks. Over the past decades, many significant achievements have been carried out in the study of synchronization of CNs [1, 3, 4].

In the previous literature on CNs, most studies have focused on single-layer networks (SLNs). However, with the development of network science, it has been found that SLNs may not be able to fully capture the diverse and nuanced relationships that exist within networked systems. In many practical scenarios, networks do not exist independently, and there are also interdependencies and interactive relationships between networks. For example, in social networks, individuals interact in various ways, such as friendships, family ties, and professional collaborations, and each type of interaction forms a distinct layer within the network. In view of the wide range of demand for practical applications, multi-layer networks (MLNs) have attracted increasing research attention. Compared with conventional SLNs, MLNs enable the representation of various relationships and interactions between different entities across multiple layers, thus allowing for more effective characterization and processing of complex relationships in real-world networks. Meanwhile, in the process of information exchange, time delays commonly exist at interactions among nodes due to the limitation of transmission speed. Therefore, the effect of time delays cannot be neglected. In recent years, there have been several studies focusing on the modeling and analysis of MLNs, and some important results have been achieved in the study of synchronization of MLNs with time delays [8, 12].

It is worth noting that the above results focus on MLNs with real variables. In fact, complex numbers can better depict some natural phenomena in real life. In signal processing and communication engineering, the amplitude and phase information of signals can be acquired and processed simultaneously by employing complex-valued networks (CVNs). Additionally, complex-valued neural networks (CVNNs) have recently become a hot research topic because of their potential applications in computer vision, imaging, remote sensing, etc. As we know, CVNNs usually consist of an input layer, one or more hidden layers, and an output layer, which can be regarded as a kind of MLN. Based on this practical application, it is natural to generalize real-valued MLNs to CVMLNs and to study dynamical behaviors of general CVMLNs. However, synchronization of CVMLNs has rarely been considered, even for the delay-free case, which motivates our current research.

In view of the complexity of node dynamics and network topology, most MLNs cannot synchronize themselves. Recently, various control schemes have been proposed to achieve synchronization of MLNs, such as intermittent feedback control [11], impulsive control [6], pinning control [7], etc. Among these strategies, impulsive control appears to be an effective method that applies control inputs to network nodes only at discrete moments, which can reduce communication or control costs of MLNs. On the other hand, a MLN is normally composed of numerous high dimensional nodes, and it is uneconomical and almost infeasible to control all of them. To tackle this issue, the pinning control scheme, which allows for controlling a small portion of nodes, has been proposed. Obviously, pinning control can greatly reduce control costs for large-scale MLNs. To further reduce the control cost, the idea of combining pinning control and impulsive control, named pinning impulsive control, was adopted to study synchronization of MLNs [5, 8]. However, to the best of our knowledge, time delay effects are seldom considered in pinning impulsive controllers to address the problem of synchronization control for CVMLNs.

Inspired by the above discussions, this paper studies synchronization of delayed CVMLNs via delay-dependent pinning impulsive control. The main contributions of this paper are summarized as follows: Firstly, a directed MLN model with both intra-layer and inter-layer couplings is constructed in the complex field, which is an extension of the existing models of real-valued MLNs. Moreover, the effects of intra-layer and inter-layer transmission delays are also considered. Secondly, a delay-dependent pinning impulsive control (PIC) scheme is directly designed in complex domains, where the pinning impulsive controller depends on not only the states of network nodes at each impulsive instant but also the states at history time. Lastly, some sufficient conditions on synchronization of CVMLNs with time-varying coupling delays are established by taking advantage of the Lyapunov function method in the complex field, the algebraic graph theory and an extended impulsive differential inequality.

The rest of this paper is organized as follows. In Sect. 2, some basic notions in graph theory and some necessary preliminaries are introduced. In Sect. 3, the synchronization result for delayed CVMLNs is presented. A numerical example is provided in Sect. 4, and conclusions are given in Sect. 5.

2 Preliminaries and Problem Formulation

2.1 Graph Theory

Consider a MLN consisting of M layers and N nodes in each layer. The network topology of layer q(q∈M:={1,2,…,M}) is denoted by a weighted digraph G(q)=(V(q),E(q),A(q)), where V(q)={vi(q)|i∈I:={1,2,…,N}} and E(q)⊆V(q)×V(q) represent the set of nodes and the set of edges in layer q, respectively, and vi(q) denotes the ith node in the qth layer network. Moreover, A(q)=[aij(q)]N×N is the weighted adjacency matrix of digraph G(q). (vi(q),vj(q)) denotes a directed edge of digraph G(q) meaning that there is a connection from node i to node j in layer q. The weighted adjacency matrix A(q) is defined as follows: aij(q)>0(i≠j) if (vj(q),vi(q))∈E(q), otherwise, aij(q)=0. Let (G1,C) denote the projection of weighted digraphs G(q),q∈M, where C=[cij]N×N is the weighted adjacency matrix of digraph G1 with cij=maxq∈M{aij(q)}. The digraph G1 is called strongly connected if there is a directed path connecting any two arbitrary nodes in G1. The Laplacian matrix L=[lij]N×N of the weighted digraph G1 is defined by L:=H1-C, where H1=diag{Ξ1,Ξ2,…,ΞN} with Ξi=∑j=1,j≠iNcij, for i∈I. Let digraph (G2,B) denote the network topology between layers, where B=[bql]M×M is defined as follows: bql>0(q≠l) if there is a one-to-one connection between the nodes from the lth layer network to the qth layer network, otherwise, bql=0. The inter-layer Laplacian matrix L~=[l~ij]M×M of the weighted digraph G2 is defined by L~:=H2-B, where H2=diag{Ω1,Ω2,…,ΩM} with Ωq=∑l=1,l≠qMbql, for q∈M.

2.2 Problem Formulation

Consider the following CVMLN consisting of M (M≥2) layers and N (N≥2) nodes in each layer. The dynamical equation of the CVMLN is described by

z˙i(q)(t)=f(zi(q)(t))+σ∑j=1,j≠iNaij(q)(zj(q)(t-r(t))-zi(q)(t-r(t)))+ϑ∑l=1,l≠qMbql(zi(l)(t-τ(t))-zi(q)(t-τ(t)))+ui(q)(t),

(1)

for i∈I:={1,2,…,N},q∈M:={1,2,…,M}, where zi(q)(t)=(zi1(q)(t),zi2(q)(t),..., zin(q)(t))T∈Cn denotes the state of the ith node in the qth layer network; f(·):Cn→Cn is a nonlinear complex-variable vector-valued function; σ>0 and ϑ>0 represent the intra-layer and the inter-layer coupling strength, respectively; r(t) and τ(t) are the intra-layer and inter-layer time-varying transmission delays satisfying 0≤r(t)≤r and 0≤τ(t)≤τ, respectively; A(q)=[aij(q)]N×N represents the intra-layer coupling weight matrix of the qth layer network, where aij(q)>0 (i≠j) when the node i receives information from the node j in the qth layer network, otherwise, aij(q)=0; B=[bql]M×M is the inter-layer coupling weight matrix, which is defined as: bql>0 (q≠l) if there is a connection between nodes from layer l to layer q, otherwise bql=0. ui(q)(t) denotes the control input of the ith node in the qth layer.

The dynamics of the target system are given by

s˙(t)=f(s(t)),

(2)

where s(t)=(s1(t),s2(t),...,sn(t))T∈Cn is the objective state.

Definition 1

The CVMLN (1) is said to achieve synchronization with the target system (2) if limt→∞‖zi(q)(t)-s(t)‖=0,i∈I,q∈M.

In order to synchronize the CVMLN (1) with the target system (2), the pinning impulsive controller (PIC) is designed as follows:

ui(q)(t)=∑k=1∞[ζi(q)ei(q)(t)+ηi(q)ei(q)(t-d)]δ(t-tk),i∈D(q)(tk),0,i∉D(q)(tk),q∈M,

(3)

where ei(q)(t):=zi(q)(t)-s(t) is the synchronization error of node i in layer q, ζi(q) and ηi(q) are impulsive control gains to be determined; d represents the impulse delay; δ(·) is the Dirac Delta function. The impulsive sequence {tk}k∈N satisfies 0<t1<t2<⋯<tk<⋯ and limk→∞tk=∞. Let hq denote the number of pinned nodes in layer q at each impulsive instant. The index set D(q)(tk) is defined as follows: At the impulsive instant tk, reordering the error states e1(q)(tk-),e2(q)(tk-),...,eN(q)(tk-) in layer q such that ‖eθ1(q)(tk-)‖≥‖eθ2(q)(tk-)‖≥⋯≥‖eθhq(q)(tk-)‖≥‖eθhq+1(q)(tk-)‖≥⋯≥‖eθN(q)(tk-)‖, then D(q)(tk):={θ1,θ2,...,θhq}, and so ♯D(q)(tk)=hq.

According to systems (1), (2) and the PIC (3), the error system can be written as where f~(ei(q)(t))=f(zi(q)(t))-f(s(t)), Δei(q)(tk)=ei(q)(tk+)-ei(q)(tk-). The initial condition is given by ei(q)(t0+s)=ψi(q)(s), for i∈I, q∈M, where ψi(q)∈PC([-ϖ,0], Cn) with ϖ=max{r,τ,d}. It is assumed that solutions of system () are right-continuous at tk, i.e., ei(q)(tk+)=ei(q)(tk) and ei(q)(tk-)=limt→tk-ei(q)(t).

Assumption 1

For any u,v∈Cn, there exists constant Lf>0 such that

(f(u)-f(v))∗(u-v)+(u-v)∗(f(u)-f(v))≤Lf(u-v)∗(u-v).

Assumption 2

The digraphs (G1,C) and (G2,B) are strongly connected.

Lemma 1

[10] Consider the following impulsive differential inequality:

D+u(t)≤au(t)+∑i=1hbi[u(t)]τi,t≠tk,t≥t0,u(tk+)≤pku(tk-)+∑j=1rqkj[u(tk-)]σj,k∈N+,u(s)=ϕ(s),s∈[t0-τ,t0],

where a,bi,pk,qkj,τi,σj are constants and bi≥0,pk≥0,qkj≥0,τi≥0,σj≥0, i=1,2,...,h,j=1,2,...,r, u∈PC([t0,∞],R+), ϕ is continuous on [t0-τ,t0], [u(t)]τi=supt-τi≤s≤tu(s) and [u(tk-)]σj=suptk-σj≤s<tku(s), k∈N+. Suppose that pk+∑j=1rqkj<1 and

a+∑i=1hbipk+∑j=1rqkj+ln(pk+∑j=1rqkj)tk-tk-1<0,k∈N+.

Then there exist constants M>1 and λ>0 such that u(t)≤M‖ϕ(t0)‖τe-λ(t-t0),t≥t0, where ‖ϕ(t0)‖τ=supt0-τ≤s≤t0‖ϕ(s)‖,τ=max{τi,σj,i=1,2,...,h,j=1,2,...,r}.

3 Main Result

Theorem 1

Let Ξi and Ωq, i∈I,q∈M be the diagonal elements of L and L~, respectively. Suppose that Assumptions 1 and 2 hold. The CVMLN (1) can achieve synchronization with the target system (2) under the PIC (3) if ζi(q)∈(-22-1,22-1) for i∈I,q∈M, and

α+β1+β2γ+ρ+ln(γ+ρ)tk-tk-1<0,k∈N+,

(4)

where α=maxi∈I,q∈M{Lf+2σΞi+2ϑΩq}, γ=maxi∈I,q∈M{1-hqN[1-2(1+ζi(q))2]}, ρ=maxi∈I,q∈M{2(ηi(q))2}, β1=2σmaxi∈I{Ξi}, β2=2ϑmaxq∈M{Ωq}.

Proof

Consider the Lyapunov function as V(t)=∑q=1M∑i=1NΩqΞiVi(q)(t), where Vi(q)(t)=(ei(q)(t))∗ei(q)(t). Then, for t∈[tk-1,tk), k∈N+, we have

D+V(t)=∑q=1M∑i=1NΩqΞi[f~∗(ei(q)(t))ei(q)(t)+(ei(q)(t))∗f~(ei(q)(t))]+2∑q=1M∑i=1NΩqΞiRe[(ei(q)(t))∗σ∑j=1Naij(q)(ej(q)(t-r(t))-ei(q)(t-r(t)))]+2∑q=1M∑i=1NΩqΞiRe[(ei(q)(t))∗ϑ∑l=1Mbql(ei(l)(t-τ(t))-ei(q)(t-τ(t)))].

(5)

According to Assumption 1, we can obtain that

(ei(q)(t))∗f~(ei(q)(t))+f~∗(ei(q)(t))ei(q)(t)≤Lf‖ei(q)(t)‖2.

(6)

By applying the Cauchy-Schwarz inequality in the complex field, we have

2Re[(ei(q)(t))∗σ∑j=1Naij(q)(ej(q)(t-r(t))-ei(q)(t-r(t)))]≤2σ∑j=1Naij(q)|(ei(q)(t))∗(ej(q)(t-r(t))-ei(q)(t-r(t)))|≤2σ∑j=1Naij(q)‖ei(q)(t)‖‖ej(q)(t-r(t))-ei(q)(t-r(t))‖≤2σ∑j=1Ncij‖ei(q)(t)‖2+2σ∑j=1Ncij‖ei(q)(t-r(t))‖2+σ∑j=1Ncij(‖ej(q)(t-r(t))‖2-‖ei(q)(t-r(t))‖2).

(7)

Similarly, we can obtain

2Re[(ei(q)(t))∗ϑ∑l=1Mbql(ei(l)(t-τ(t))-ei(q)(t-τ(t)))]≤2ϑ∑l=1Mbql‖ei(q)(t)‖2+2ϑ∑l=1Mbql‖ei(q)(t-τ(t))‖2+ϑ∑l=1Mbql(‖ei(l)(t-τ(t))‖2-‖ei(q)(t-τ(t))‖2).

(8)

Moreover, according to Theorem 2.3 of [2], we have

∑q=1M∑i=1N∑j=1NΩqΞicij(‖ej(q)(t-r(t))‖2-‖ei(q)(t-r(t))‖2)=0,

(9)

and

∑q=1M∑i=1N∑l=1MΩqΞibql(‖ei(l)(t-τ(t))‖2-‖ei(q)(t-τ(t))‖2)=0.

(10)

Then, it follows from (5)–(10) that

D+V(t)≤∑q=1M∑i=1NΩqΞi(Lf+2σΞi+2ϑΩq)‖ei(q)(t)‖2+∑q=1M∑i=1N2σΩqΞi2‖ei(q)(t-r(t))‖2+∑q=1M∑i=1N2ϑΩq2Ξi‖ei(q)(t-τ(t))‖2≤αV(t)+β1sups∈[-r,0]V(t+s)+β2sups∈[-τ,0]V(t+s).

For t=tk,k∈N+, according to (), we have

V(tk)=∑q=1M∑i∈D(q)(tk)ΩqΞi[(1+ζi(q))ei(q)(tk-)+ηi(q)ei(q)(tk-d)]∗[(1+ζi(q))ei(q)(tk-)+ηi(q)ei(q)(tk-d)]+∑q=1M∑i∉D(q)(tk)ΩqΞi(ei(q)(tk-))∗ei(q)(tk-)≤2∑q=1M∑i∈D(q)(tk)(1+ζi(q))2ΩqΞi(ei(q)(tk-))∗ei(q)(tk-)+2∑q=1M∑i∈D(q)(tk)(ηi(q))2ΩqΞi(ei(q)(tk-d))∗ei(q)(tk-d)+∑q=1M∑i∉D(q)(tk)ΩqΞi(ei(q)(tk-))∗ei(q)(tk-).

(11)

Let γi(q)=1-hqN[1-2(1+ζi(q))2]. By the pinning rule and ζi(q)∈(-22-1,22-1) for i∈I,q∈M, we have

∑q=1M∑i∉D(q)(tk)ΩqΞi(1-γi(q))(ei(q)(tk-))∗ei(q)(tk-)≤∑q=1M∑i∈D(q)(tk)N-hqhqΩqΞi(1-γi(q))(ei(q)(tk-))∗ei(q)(tk-)=∑q=1M∑i∈D(q)(tk)ΩqΞi[γi(q)-2(1+ζi(q))2](ei(q)(tk-))∗ei(q)(tk-).

Hence, we can obtain

2∑q=1M∑i∈D(q)(tk)(1+ζi(q))2ΩqΞi(ei(q)(tk-))∗ei(q)(tk-)+∑q=1M∑i∉D(q)(tk)ΩqΞi(ei(q)(tk-))∗ei(q)(tk-)≤∑q=1M∑i=1NΩqΞiγi(q)(ei(h)(tk-))∗ei(q)(tk-).

(12)

Then, it follows from (11) and (12) that

V(tk)≤maxi∈I,q∈M{γi(q)}V(tk-)+2maxi∈I,q∈M{(ηi(q))2}∑q=1M∑i=1NΩqΞi(ei(q)(tk-d))∗ei(q)(tk-d)≤γV(tk-)+ρV(tk-d).

According to condition (4), we have γ+ρ<1. Then, it follows from Lemma 1 that there exist constants M>1 and λ>0 such that V(t)≤M‖V(t0)‖ϖe-λ(t-t0), t≥0. In addition, we have V(t)≥mini∈I,q∈M{ΩqΞi}∑q=1M∑i=1N‖ei(q)(t)‖2, hence we can obtain ∑q=1M∑i=1N‖ei(q)(t)‖2→0 as t→∞, which implies that limt→∞‖ei(q)(t)‖=0 for any i∈I,q∈M. Therefore, the CVMLN (1) achieves synchronization with the target system (2) under the PIC (3).    □

4 Numerical Example

Example 1

Consider the CVMLN (1) consisting of three layers with five nodes in each layer, where σ=0.5,ϑ=1,r(t)=0.02etet+1,τ(t)=0.01sin2(t),

A(1)=0000000100000000000101000,A(2)=0100000000000100100001000,A(3)=0000000100100000100100000,B=00.200.200.200.20.

The target system (2) is chosen as a 3-D complex-valued chaotic Chen system:

s˙1=10s2-s1,s˙2=28s1-s1s3+s2,s˙3=12s¯1s2+s1s¯2-83s3,

(13)

where s1,s2∈C,s3∈R. Figure 1 shows the the chaotic attractors of system (13) in some domains with the initial state s(0)=[2-3j,-1+j,4]T. It has been shown in [9] that Assumption 1 is satisfied with Lf=49.

The topology of the 3-layer CVN (1) is depicted in Fig. 2. It can be seen from Fig. 2 that the digraph (G1,(cij)5×5) and digraph (G2,(bql)3×3) are both strongly connected, thus Assumption 2 holds. Meanwhile, we can obtain {Ξ1,Ξ2,Ξ3,Ξ4,Ξ5}={1,1,2,2,1} and {Ω1,Ω2,Ω3}={0.2,0.4,0.2}. After simple calculations, we can get α=51.8,β1=2 and β2=0.8.

Fig. 1

The chaotic attractors of system (13) (a) in Re(s1)-Im(s1)-s3 plane; (b) in Re(s2)-Im(s2)-s3 plane with the initial condition s1(0)=2-3j, s2(0)=-1+j and s3(0)=4

Fig. 2

Topology of the 3-layer network (1) with 5 nodes in each layer

Fig. 3

Trajectories of Re(ei1(1)(t)), Im(ei1(1)(t)), Re(ei2(1)(t)), Im(ei2(1)(t)) and ei3(1)(t)

Fig. 4

Trajectories of Re(ei1(2)(t)), Im(ei1(2)(t)), Re(ei2(2)(t)), Im(ei2(2)(t)) and ei3(2)(t)

Fig. 5

Trajectories of Re(ei1(3)(t)), Im(ei1(3)(t)), Re(ei2(3)(t)), Im(ei2(3)(t)) and ei3(3)(t)

Let h1=h3=3, h2=4 and d=0.005, then we have ϖ=0.02. Choose the impulsive control gains ζi(q)=-0.832+0.001(2q+i)∈(-1-12,-1+12) and ηi(q)=0.205+0.001(5q+i) for i=1,2,...,5,q=1,2,3. After simple calculations, we can get that γ=0.438 and ρ=0.101. Choose the impulsive sequence as t3k-2=0.024k-0.016, t3k-1=0.024k-0.01, t3k=0.024k, k∈N+, then we get tk-tk-1≤0.01 for k∈N+. Hence, the condition (4) of Theorem 1 holds. It follows from Theorem 1 that the three-layer CVN (1) achieves synchronization with the target system (13) under the PIC (3). The initial condition of the three-layer CVN (1) is chosen as zi(q)(s)=(0.2(q-2)+i-3)[1-3j,-2+j,4]T, s∈[-0.02,0],i=1,...,5,q=1,2,3. Figures 3, 4 and 5 show the trajectories of real and imaginary parts of 3-D synchronization errors between the nodes in the first layer, the second layer and the third layer of network (1) and the target system (13), respectively. It can be observed from Figs. 3, 4 and 5 that synchronization between the three-layer CVN (1) and the target system (13) is achieved under the PIC (3).

5 Conclusions

In this paper, the problem of synchronization of CVMLNs with intra-layer and inter-layer coupling delays has been studied. A delay-dependent pinning impulsive controller has been proposed in the complex field to synchronize delayed CVMLNs. With the help of the Lyapunov function method in the complex field, algebraic graph theory and an extended impulsive differential inequality, some synchronization criteria of delayed CVMLNs have been established. Our result shows that synchronization of CVMLNs with time-varying delays can be achieved if the pinning impulsive controller is properly designed. Lastly, a numerical example has been presented to demonstrate the feasibility of the theoretical result.

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Wang, Z., Jin, X., Pan, L., Feng, Y., Cao, J.: Quasi-synchronization of delayed stochastic multiplex networks via impulsive pinning control. IEEE Trans. Syst. Man Cybern. Syst. 52(9), 5389–5397 (2021)

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_11

Quasi-Synchronization of Fractional-Order Multiplex Networks with Parameter Mismatch via Intermittent Control

Yao Xu1  , Xinzhi Liu2   and Wenxue Li3  

(1)

Tongda College of Nanjing University of Posts and Telecommunications, Yangzhou, China

(2)

Department of Applied Mathematics, University of Waterloo, Waterloo, Canada

(3)

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, PR China

Yao Xu (Corresponding author)

Email: yaoxu0125@163.com

Xinzhi Liu

Email: xzliu@uwaterloo.ca

Wenxue Li

Email: wenxuetg@hitwh.edu.cn

Abstract

This paper addresses the quasi-synchronization issue of fractional-order multiplex networks with parameter mismatch using intermittent control. To provide a more comprehensive description of the complexity, ambiguity, and network topology in the real world, it simultaneously introduces parameter mismatch, intra-layer, and inter-layer structures into the mathematical model of fractional-order multiplex networks. Supported by graph theory and the Lyapunov method, and by employing an aperiodically intermittent control strategy, sufficient conditions are established for quasi-synchronization and estimate the allowable error bound. Finally, the theoretical results are applied to power systems and numerical simulations are presented to demonstrate the effectiveness of our findings.

Keywords

Quasi-synchronizationMultiplex networksFractional-order systems with short memoryParameter mismatchIntermittent control

This work was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (No. 24KJB120011).

1 Introduction

In recent years, synchronization has become an often studied key collective behavior in complex networks due to its potential applications in secure communication, pattern recognition, edge detection, and so on [4, 6, 7, 22]. Researchers mainly focus on different types of synchronization, which include complete synchronization, projective synchronization, cluster synchronization, bipartite synchronization, finite-time synchronization, fixed-time synchronization, quasi-synchronization, and more [10, 11, 20]. Among them, quasi-synchronization signifies that the synchronization error does not surpass a permissible level, which is a common weak cooperative behavior in the case of parameter mismatch. Lately, many researchers have discussed and developed promising results of quasi-synchronization of complex networks with parameter mismatch, see [2, 19, 21] for examples.

Fractional-order complex networks possess the memory and genetic qualities of fractional calculus, providing them with more dynamic behaviors, while also inheriting the advantages of integer-order complex networks. There are lots of studies concentrated on the synchronization of fractional-order complex networks [5, 8, 18], in which multiple nodes are arranged in a plane. Multiplex networks are arranged in a space composed of multiple complex networks. Therefore, multiplex networks are more complicated than complex networks in structure, which have become a promising network model. Now, the dynamic behaviors of fractional-order multiplex networks have been intensively investigated by some applicable control strategies such as adaptive control, impulsive control, sliding mode control, and so on [14, 15, 17]. More notably, intermittent control, as a discontinuous control strategy, has critical advantages such as flexibility of the control time, no need to change state immediately, and energy savings. Some synchronization results of fractional-order complex networks have been obtained based on intermittent control, see [1, 13, 16]. To the best of authors’ knowledge, few works have been done on the quasi-synchronization of fractional-order multiplex networks by intermittent control.

Based on the above discussion, in this paper, we study the quasi-synchronization of fractional-order multiplex networks with parameter mismatch (FMNPM) by intermittent control. Empowered by the Lyapunov method and graph theory, certain quasi-synchronization criteria are obtained, and an estimate of the permissible error bound is made. Then, the numerical simulations on power systems are elaborated to illustrate the effectiveness of the theoretical results, followed by the conclusions provided at the end.

2 Preliminaries and Problem Statement

In the following discussion, we show the definitions of Caputo’s fractional derivative, the Mittag-Leffler function, and the quasi-synchronization of an FMNPM. Then, the detailed problem statement of this paper is introduced.

Definition 1

([12]) The Caputo’s fractional derivative for x(t)∈R is defined as tlCDtαx(t)=1Γ(1-α)∫tltx′(τ)(t-τ)αdτ, where 0<α<1, Γ(1-α)=∫0+∞t-αe-tdt.

Definition 2

([12]) The function Eα,β(h)=∑k=0+∞hkΓ(αk+β) denotes the two-parameter Mittag-Leffler function, where α, β are positive real numbers, and h is a complex number. When β=1, Eα,β(h) turns into Eα(h)=∑k=0+∞hkΓ(αk+1)=Eα,1(h).

Consider the following FMNPM:

tlCDtαζi[k](t)=Xiζi[k](t)+YiH(ζi[k](t))+m∑j=1Paij[k]ζj[k](t)-ζi[k](t)+n∑s=1Qbksζi[s](t)-ζi[k](t)+Li[k](t)+ui[k](t),t∈[tl,δl);δlCDtαζi[k](t)=Xiζi[k](t)+YiH(ζi[k](t))+m∑j=1Paij[k]ζj[k](t)-ζi[k](t)+n∑s=1Qbksζi[s](t)-ζi[k](t)+Li[k](t),t∈[δl,tl+1),

(1)

where i=1,2,…,P, k=1,2,…,Q, 0<α<1 is the order of fractional derivative; ζi[k](t)∈Rσ is the state vector of FMNPM; Xi, Yi∈Rσ×σ are real matrices; H(ζi[k](t))=(H1(ζi[k](t)),H2(ζi[k](t)),…,Hσ(ζi[k](t)))T is a nonlinear vector-valued function; m>0 and n>0 are the intra-layer and inter-layer coupling strengths, respectively; aij[k]∈{0,1} and bks∈{0,1} represent the connections between nodes in intra-layers and inter-layers; Li[k](t) is the external input; and ui[k](t) is the control input, where tl and δl, respectively, denote the time instants of the start and end of the lth control interval, t1=0, l∈Z+.

The target system is described by

tlCDtαη(t)=Xη(t)+YH(η(t))+L(t),t∈[tl,δl);δlCDtαη(t)=Xη(t)+YH(η(t))+L(t),t∈[δl,tl+1),

(2)

where η(t)∈Rσ; X, Y∈Rσ×σ are real matrices different from Xi, Yi; and L(t) is the external input with L(t)≠Li[k](t). Define the error vector as ϑi[k](t)=ζi[k](t)-η(t).

Definition 3

If there exists a compact set C such that the error vector ϑi[k](t) converges into the set C={ϑ||ϑ|≤ρ} as t→+∞, then it is said that network (1) realizes the quasi-synchronization with system (2), where ϑ=(ϑ1[1],…,ϑP[1],…,ϑP[Q])T.

The objective of this paper is to realize the quasi-synchronization. To this aim, the following aperiodically intermittent control is designed:

ui[k](t)=-di[k]ϑi[k](t),t∈[tl,δl);0,t∈[δl,tl+1),

(3)

where di[k]>0 is the control gain. For intervals, we define some notations below: τl=δl-tltl+1-tl, l∈Z+, τ~=inf{τl}, T^=sup{tl+1-tl}, T~=inf{tl+1-tl}.

According to ϑi[k](t)=ζi[k](t)-η(t), the error system can be derived as follows:

tlCDtαϑi[k](t)=Xiϑi[k](t)+YiH¯(ϑi[k](t))+m∑j=1Paij[k]ϑj[k](t)-ϑi[k](t)+n∑s=1Qbksϑi[s](t)-ϑi[k](t)+Ti[k](η(t))-di[k]ϑi[k](t),t∈[tl,δl);δlCDtαϑi[k](t)=Xiϑi[k](t)+YiH¯(ϑi[k](t))+m∑j=1Paij[k]ϑj[k](t)-ϑi[k](t)+n∑s=1Qbksϑi[s](t)-ϑi[k](t)+Ti[k](η(t)),t∈[δl,tl+1),

(4)

where i=1,2,…,P, k=1,2,…,Q, Ti[k](η(t))=(Xi-X)η(t)+(Yi-Y)H(η(t))+Li[k](t)-L(t), H¯(ϑi[k](t))=H(ζi[k](t))-H(η(t)).

Some technical lemmas are given below.

Lemma 1

([3]) Consider ζ(t)∈Rσ be a vector of differentiable functions, then we have

t0CDtαζT(t)Pζ(t)≤2ζT(t)Pt0CDtαζ(t),

for any α∈(0,1] and any time instant t≥t0, where P∈Rσ×σ is a positive-definite matrix.

Lemma 2

([1]) Let μ(t) be a non-negative continuous function on [0,∞) satisfying

tlcDtαμ(t)≤-β1μ(t)+ϖ1,tl≤t<δl,δlcDtαμ(t)≤β2μ(t)+ϖ2,δl≤t<tl+1,

where l∈Z+,0<α<1,β1,β2>0, and ϖ1,ϖ2≥0. Denote τl= δl-tltl+1-tl,l∈Z+and τ˘=infl∈Z+τl, where τl denotes the control rate of the lth control, called the lth control rate. If τ˘>0,Ψ=Eα-β1τ˘T~αEαβ21-τ˘T^α<1, then

μ(t)≤μ(0)ΨelnΨT^t+ϑ01-Ψ+ρ1,t≥0,

where ρ1=ϖ1/β1,ρ2=ϖ2/β2, and ϑ0=ρ1+ρ2Eαβ2((1-τ˘)T^)α-ρ2.

3 Main Results

We make the following assumptions.

Assumption 1

The target vector η(t) is bounded. That is, for any initial values η(t0), there is a positive constant B1 such that |η(t)|≤B1 holds.

Assumption 2

The function fi satisfies the Lipschitz condition. That is, for any a,  b∈R, there is a positive constant l such that |fi(a)-fi(b)|≤l|a-b|.

Assumption 3

There is a positive constant B2 such that |Li[k](t)-L(t)|≤B2.

Assumption 4

The digraphs (M,(Aij)P×P) and (N,(bks)Q×Q) are strongly connected.

Some notations are listed as follows:

X~i=Xi-X, Y~i=Yi-Y, wi=3λmax(X~iTX~i)+3λmax(Y~iTY~i)l2;

ι=mini,k{μiγk}, Aij=maxk{aij[k]}, β1=mini,k{β~i[k]}, β2=maxi,k{β^i[k]};

β^i[k]=β~i[k]+2di[k], β~i[k]=λmax(Xi+XiT)-2di[k]+λmax(YiYiT)+l2+4m∑j=1PAij+4n∑s=1Qbks+1, ϖ1=∑i=1P∑k=1Qμiγk(wiB12+3B22);

Ψ=Eα-β1τ˘T~αEαβ21-τ˘T^α, ρ1=ϖ1/β1,ρ2=ϖ1/β2, ϑ0=ρ1+ρ2Eαβ2((1-τ˘)T^)α-ρ2.

Theorem 1

Presume that Assumptions 1–4 are satisfied. Under aperiodically intermittent control (3), if

β^i[k]>0,β~i[k]>0,τ˘>0,Ψ<1,

then network (1) can realize the quasi-synchronization with system (2), and the allowable error bound is estimated as ϑ0ι(1-Ψ)+ρ1ι.

Proof

Consider the following Lyapunov function:

V(t)=∑i=1P∑k=1Qμiγk|ϑi[k](t)|2,

where μi>0, γk>0 are the cofactors of the ith and kth diagonal elements of the Laplacian matrices of the digraphs (M,(Aij)P×P) and (N,(bks)Q×Q), which are a consequence of Assumption 4 based on [9]. When t∈[tl,δl), under Lemma 1, we calculate the fractional derivative of V(t) below:

tlCDtαV(t)≤2∑i=1P∑k=1Qμiγk(ϑi[k](t))TtlCDtαϑi[k](t)=2∑i=1P∑k=1Qμiγk(ϑi[k](t))T[Xiϑi[k](t)-di[k]ϑi[k](t)+m∑j=1Paij[k]ϑj[k](t)-ϑi[k](t)+YiH¯(ϑi[k](t))+n∑s=1Qbksϑi[s](t)-ϑi[k](t)+Ti[k](η(t))].

(5)

In terms of Assumption 2, one gets

2(ϑi[k](t))TYiH¯(ϑi[k](t))≤(ϑi[k](t))TYiYiTϑi[k](t)+(H¯(ϑi[k](t)))TH¯(ϑi[k](t))≤(λmax(YiYiT)+l2)|ϑi[k](t)|2.

(6)

By Assumption 1, one arrives at

|Ti[k](η(t))|2≤3|(Xi-X)η(t)|2+3|(Yi-Y)H(η(t))|2+3|Li[k](t)-L(t)|2≤3|Xi-X|2|η(t)|2+3|Yi-Y|2H(η(t))|2+3|Li[k](t)-L(t)|2≤wiB12+3B22,

(7)

which implies that

2(ϑi[k](t))TTi[k](η(t))≤|ϑi[k](t)|2+|Ti[k](η(t))|2≤|ϑi[k](t)|2+wiB12+3B22.

(8)

Based on the basic inequality, we have

2m(ϑi[k](t))T∑j=1Paij[k]ϑj[k](t)-ϑi[k](t)≤2m∑j=1Paij[k]|ϑi[k](t)||ϑj[k](t)-ϑi[k](t)|≤m∑j=1Paij[k]|ϑi[k](t)|2+m∑j=1Paij[k]|ϑj[k](t)-ϑi[k](t)|2≤4m∑j=1PAij|ϑi[k](t)|2+m∑j=1PAij(|ϑj[k](t)|2-|ϑi[k](t)|2),

(9)

and similarly, we derive

2n(ϑi[k](t))T∑s=1Qbksϑi[s](t)-ϑi[k](t)≤4n∑s=1Qbks|ϑi[k](t)|2+n∑s=1Qbks(|ϑi[s](t)|2-|ϑi[k](t)|2).

(10)

On the basis of (6)–(10), inequality (5) becomes

tlCDtαV(t)≤∑i=1P∑k=1Qμiγk(λmax(Xi+XiT)-2di[k]+λmax(YiYiT)+l2+4m∑j=1PAij+4n∑s=1Qbks+1)|ϑi[k](t)|2+m∑i=1P∑k=1Q∑j=1PμiγkAij(|ϑj[k](t)|2-|ϑi[k](t)|2)+n∑i=1P∑k=1Q∑s=1Qμiγkbks(|ϑi[s](t)|2-|ϑi[k](t)|2)+∑i=1P∑k=1Qμiγk(wiB12+3B22)≤-β1V(t)+ϖ1,t∈[tl,δl).

(11)

When t∈[δl,tl+1), one has

δlCDtαV(t)≤2∑i=1P∑k=1Qμiγk(ϑi[k](t))T[Xiϑi[k](t)+m∑j=1Paij[k]ϑj[k](t)-ϑi[k](t)+YiH¯(ϑi[k](t))+n∑s=1Qbksϑi[s](t)-ϑi[k](t)+Ti[k](η(t))].

(12)

Then based on (6)–(10), inequality (12) becomes

δlCDtαV(t)≤∑i=1P∑k=1Qμiγk(λmax(Xi+XiT)+λmax(YiYiT)+l2+4m∑j=1PAij+4n∑s=1Qbks+1)|ϑi[k](t)|2+m∑i=1P∑k=1Q∑j=1PμiγkAij(|ϑj[k](t)|2-|ϑi[k](t)|2)+n∑i=1P∑k=1Q∑s=1Qμiγkbks(|ϑi[s](t)|2-|ϑi[k](t)|2)+∑i=1P∑k=1Qμiγk(wiB12+3B22)≤β2V(t)+ϖ1,t∈[δl,tl+1).

(13)

By combining (11) and (13), we further obtain

tlCDtαV(t)≤-β1V(t)+ϖ1,t∈[tl,δl);δlCDtαV(t)≤β2V(t)+ϖ1,t∈[δl,tl+1).

(14)

Therefore, by using Lemma 2, one has

V(t)≤V(0)ΨelnΨT^t+ϑ01-Ψ+ρ1,t≥0,

Then, based on the expression of V(t), we have

ϑ(t)≤V(0)ιΨelnΨ2T^t+ϑ0ι(1-Ψ)+ρ1ι,t≥0.

It indicates that network (1) can realize the quasi-synchronization with system (2). The proof is completed.

4 Numerical Simulations of Power Systems

In this section, a numerical example of power systems built on the FMNPM is elaborated to illustrate the effectiveness of the obtained results.

Fig. 1

An FMNPM with two layers, which has three nodes on each layer

Consider a fractional-order power system that acts as the target system in the form of system (2): X=010-0.15, Y=000-0.18, η(t)=(η1(t),η2(t))T, h(η(t))=(0,sinη1(t))T, L(t)=(0,0.04+0.056cos(0.027t))T. In addition, consider an FMNPM with two layers, which has three nodes on each layer (see Fig. 1). The parameters in FMNPM (1) are set as follows: Xi=010-0.25-0.005i, Yi=000-0.25-0.015i, Li[k](t)=(0,(0.025+0.075i)+(0.065i)cos(0.027t))T, m=0.1, n=0.1, (aij[1])3×3=001100010, (aij[2])3×3=011001000, (bks)2×2=0110.

Fig. 2

a The trajectories of FMNPM (1) without control and system (2); b The trajectories of error system (4) without control

Fig. 3

a The trajectories of FMNPM (1) with control (3) and system (2); b The trajectories of error system (4) with control (3)

By conducting simulations, we have: the left one in Fig. 2 describes the trajectories of FMNPM (1) without control and system (2), and the right one shows the trajectories of error system (4) without control. From Fig. 2, it is apparent that FMNPM (1) does not achieve the quasi-synchronization.

In order to make the FMNPM (1) quasi-synchronized, the intermittent control is introduced, in which the control gain di[k]=20, τ˘=0.8, T^=0.3, T~=0.1. By verifying the conditions in Theorem 1 and making some simulations, we know that FMNPM (1) can realize the quasi-synchronization with system (2) from Fig. 3.

5 Conclusions

This paper has studied the quasi-synchronization issue of fractional-order multiplex networks with parameter mismatch using intermittent control. Utilizing graph theory and the Lyapunov method, we have derived sufficient conditions for quasi-synchronization under an aperiodically intermittent control strategy and determine the allowable error bounds. The effectiveness of our results is demonstrated through numerical simulations in the context of power systems.

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Part IIIMathematical Modelling in Engineering, Physical and Chemical Sciences

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_12

A Gradient-Based Optimization Approach to Image Fusion Using Structural Similarity

Fatemeh Hirbodvash1   and Mehran Ebrahimi1  

(1)

Faculty of Science, Ontario Tech University, Oshawa, ON, Canada

Fatemeh Hirbodvash

Email: fatemeh.hirbodvash@ontariotechu.net

Mehran Ebrahimi (Corresponding author)

Email: mehran.ebrahimi@ontariotechu.ca

Abstract

Multiple imaging sensors are being used in surveillance, medical imaging, and machine vision. To effectively combine the information from multiple imaging sensors, image fusion techniques have emerged to produce an improved fused single image. This paper introduces a general framework for image fusion, incorporating the structural similarity index measure (SSIM), a highly effective measure of visual proximity aligned with the human visual system. To formulate image fusion, the SSIM-Mean is used for data fidelity of an optimization problem. To solve it numerically by gradient descent method an alternate expression of SSIM is proposed, utilizing Mean values and vector norms, alongside its derivative calculation. We apply the proposed method to the resolution enhancement of a set of given degraded images, where SSIM-Mean is used as data fidelity. The results will be evaluated both qualitatively and quantitatively. A significant advantage of the proposed formulation lies in its flexibility to accommodate various degradation operators and regularization terms, which marks a notable strength in this approach. Our code is available at https://​github.​com/​fatemehhirbodvas​h/​SSIM-Mean- .​

Keywords

Structural similarity indexImage fusionImage quality measuresImage enhancementNumerical optimization

1 Introduction

Image fusion is the process of combining multiple images into a single composite image, thereby providing the most relevant information from several image observations to the viewer in a single image [1]. Image fusion is vital for many applications in computer vision, robotics, remote sensing, and biomedical imaging [2–4]. Traditionally, mean squared error (MSE) has been the standard measure for evaluating signal fidelity and quality, as well as for designing and optimizing signal processing algorithms and systems, due to its ease of computation and desirable mathematical properties. However, MSE (and other similar metrics) is a poor predictor of perceived signal fidelity or quality. The Structural Similarity Index Measure (SSIM), originally proposed by Wang et al. [8, 9] can be considered a suitable data fidelity measure consistent with Human Visual System (HVS). The SSIM has been widely accepted as a suitable measure of visual quality and perceptual proximity between images based on subjective quality assessments involving large databases. For imaging applications, the Mean SSIM (MSSIM) as the arithmetic average of all local SSIM scores for an entire image has been defined in [10].

The SSIM-Mean (not to be confused with MSSIM), defined by Brunet in his thesis [11], was used as a data fidelity term in the inverse problem formulation and would be suitable for image fusion, since the aim is to integrate complementary information from multiple images to produce a higher-quality image.

In this paper, we formulate image fusion as an optimization problem using SSIM-Mean and provide a flexible framework that can apply various degradation operators and regularization terms, which is its advantage compared to Brunet’s work in [11]. The novelty of this paper lies in (1) the formulation of the image fusion problem in terms of SSIM using the so-called SSIM-Mean, and (2) the use of gradient-based optimization approaches to obtain image fusion results.

First, we review the definitions of SSIM, MSSIM, SSIM-Mean, and the general formulation of inverse problems using SSIM-Mean. Next, we solve the problem numerically with gradient-based methods. Finally, we present numerical results using total variation as a regularization term.

2 Background Material

2.1 Regularization in Inverse Problems

Many image processing problems can be modelled as an energy minimization approach to estimate a minimizer of an objective function f(x). Such a problem is called ill-posed if the solution to this minimization is not unique or cannot be estimated stably [12]. To solve ill-posed problems, theoretical strategies and numerical tools have been developed, and regularization theory has become mainstream. In regularization, the ill-posed problem pointing to a non-unique and/or unstable solution is replaced with a well-posed problem pointing to an approximate solution that continuously depends on given data [16]. The purpose of the regularization is to prevent over-fitting by introducing prior information about problems to penalize the model. The prior information is usually a penalty for complexity, such as restrictions for smoothness in total variation regularization [13] and bounds on the norm of the space vector in Tikhonov regularization [14, 15]. This term is usually known as a penalty and the general regularized model can be formulated as the minimization of

f(x)+λg(x).

(1)

2.2 Structural Similarity Index Measure (SSIM)

SSIM provides a measure of visual proximity between an ideal image and a distorted version of it. Due to the assumption that the ideal image is always available, SSIM can be considered as a full-reference measure of image quality assessment [8]. By comparing the luminance, contrast, and structure of two images, this SSIM measurement can be defined. In particular, given two images x∈Rn×m and y∈Rn×m (i.e., x and y are n×m dimensional images), μx and μy represent the mean values of vectors x and y, σx and σy denote the standard deviations of the images. Their cross-correlation coefficient is denoted as σxy. The SSIM is defined as

SSIM(x,y)=2μxμy+C1μx2+μy2+C1α2σxσy+C2σx2+σy2+C2βσxy+C3σxσy+C3γ

(2)

=l(x,y)αc(x,y)βs(x,y)γ,

(3)

where l(x, y) quantifies changes in luminance between two images, c(x, y) measures the contrast and s(x, y) measures the structure comparison. Here α, β, and γ are weights on the luminance, contrast, and structure terms respectively and C1, C2, and C3 are constants. In [8] the authors simplify Eq. (2) by setting α=β=γ=1 and C3=C2/2 which leads to

SSIM(x,y)=2μxμy+C1μx2+μy2+C1(2σxy+C2σx2+σy2+C2).

(4)

2.3 Mean SSIM (MSSIM)

Local SSIM methods are used rather than global SSIM methods, using a sliding Gaussian window of fixed size to determine the SSIM for each point of an image. The result of this analysis is a map that indicates how image quality varies spatially. Calculating the average of SSIMs can yield a global SSIM. It is in this case that mean SSIM is introduced as

MSSIM(x,y)=1N∑i=1NSSIM(xi,yi).

(5)

The number of local windows in the image is N, and xi and yi represent the ith sample sliding square windows, also known as patches in images x and y.

2.4 SSIM-Mean

SSIM-Mean is defined as a weighted average scheme that aims to find an ideal image/vector that maximizes the sum of the SSIM values of a set of input images/vectors [11]. Given {y1,y2,⋯,yn} a set of n vectors and {p1,p2,⋯,pn} a set of weights, find x that maximizes

∑i=1npiSSIM(x,yi).

(6)

2.5 SSIM-Based Optimization Problems

The SSIM-based optimization problem is defined in [11, 18–20] as

argminxT(x,y)+λg(x),

(7)

where T(x,y)=1-SSIM(x,y) and g(x) is the regularization term. Considering that SSIM is the measure of similarity, hence T(x,y) is considered a measure of dissimilarity between x and y.

3 Generalization of the SSIM-Mean Involving Arbitrary Linear Operators

We propose to use SSIM as a data fidelity term in an image fusion formulation as follows. Given {y1,⋯,yn} a set of n images in vector format of size n×1, {p1,⋯,pn}, a set of associated weights, {H1,H2,⋯,Hn} a set of linear degradation operators of size n×m , g(x) a regularization term, and λ its corresponding regularization parameter, find x of size m×1 that maximizes

∑i=1npiSSIM(Hix,yi)+λg(x).

(8)

Generally, when λ is strictly greater than zero, the regularization term is employed to force the optimal solution to reside within a predefined “target” space, allowing for a priori information regarding the solution to be used. Regularization can be particularly useful in the scenario where HTH is weakly conditioned or rank-deficient, in which case it can contribute to well-defined and stably computable solutions. In any case, since the first term in Eq. (8) is not convex, the whole cost function is not either convex or concave. Therefore, there is no guarantee that a unique global minimizer exists for Eq. (8). For the special case where Hi is the identity for every i, and λ=0, this formulation yields the SSIM-Mean Eq. (6) proposed by Brunet [11]. The methodology proposed by [11] is analytical and cannot be trivially extended to our proposed generalization. However, it is possible to develop efficient numerical methods to estimate an optimal solution to the problem, as demonstrated in [19] for the case where i=1. We aim to solve the above problem numerically using gradient-based schemes. As gradient-based methods are employed to solve this problem, the derivatives of expressions are required. Due to the challenge of directly differentiating the SSIM expression, an alternative expression of SSIM in terms of mean and vector norm can be introduced in the next Section.

3.1 Another Expression for SSIM

The following expression of SSIM applies the mean and vector norm instead of mean, variance, and covariance. This new expression would simplify calculations.

SSIM(x,y)=2x¯y¯+ε1x¯2+y¯2+ε1||x-x¯+y-y¯||22-||x-x¯||22-||y-y¯||22+ε2||x-x¯||22+||y-y¯||22+ε2.

(9)

where x¯ and y¯ are mean of x, y, ε1=C1, and ε2=(n-1)C2.

3.2 Derivative of Alternative Expression of SSIM

Let {y1,y2,⋯,yn} be a set of n vectors of size n×1 and , {H1,H2,⋯,Hn} a set of degradation operators of size n×m.

SSIM(Hix,yi)=S1(Hix,yi)S2(Hix,yi)

(10)

where

S1(Hix,yi)=2Hix¯yi¯+ε1Hix¯2+yi¯2+ε1,

(11)

S2(Hix,yi)=||Hix-1THix¯+yi-1Tyi¯||22-||Hix-1THix¯||22-||yi-1Tyi¯||22+ε2||Hix-1THix¯||22+||yi-1Tyi¯||22+ε2,

(12)

and 1=1⋯1 is a 1×m vector. Using the quotient rule

∂S1(Hix,yi)∂x=2nHiT1T(yi¯-Hix¯S1(Hix,yi))(yi¯2+Hix¯2+ε1),

(13)

and

∂S2(Hix,yi)∂x=2HiT(I-1nJn×n)[(yi-1Tyi¯)-(I-1nJn×n)HixS2(Hix,yi)]||yi-1Tyi¯||22+||Hix-1THix¯||22+ε2,

(14)

where Jn×n represents an n×n matrix with every element equal to 1, and I is the identity matrix of size n×n.

Fig. 1

CheckDerivative result for implementation of SSIM derivatives: linear decay (solid line) for ||f(x+hv)-f(x)|| and quadratic decay (dashed line) for ||f(x+hv)-f(x)-hdf(x)v|| up to machine precision is shown on a logarithmic scale

An easy way to check the implementation of a derivative is to use the Derivative Test. Let f be a multivariate function f:Rn⟶R and let v∈Rn be an arbitrary vector in the Taylor expansion f(x+hv)=f(x)+hdf(x)v+O(h2). The Matrix A is the derivative of f if and only if the difference between ||f(x+hv)-f(x)-hAv|| is quadratic in h. That difference is calculated by the function check derivative (see, e.g., [22]). The result of applying the check-Derivative function to the SSIM derivative is shown in Fig. 1. In this case, the result is consistent since the difference is quadratic with respect to h.

4 Experiment Result

4.1 One Dimensional Case

We can pass the calculated derivative to an efficient gradient-based optimization scheme, e.g., l-BFGS. In [11], Brunet addresses the problem of maximizing SSIM Mean for a set of points and presents a solution. To evaluate our proposed method, we replicate the example provided on page 97 of his thesis considering one-dimensional data, where we take degradation operators equal to the identity. The problem of finding the maximum SSIM Mean between two given vectors y1 and y2 in his thesis [11], i.e. the problem is finding x such that

argmaxxSSIM(x,y1)+SSIM(x,y2).

(15)

The problem is solved numerically using scipy.optimize, which offers various optimization methods. The L-BFGS-B method in Scipy-optimization is used when ϵ1=0 in this case. Figure 2 shows plot of the function x for y1=0.1 and y2=0.4 when ϵ1=0. A local maximum of 0.2 is achieved, which is identical to Brunet’s result using the Matlab code provided in his thesis, except that our formulation can now handle more generalized scenarios.

Fig. 2

Function f(x)=SSIM(x,0.1)+SSIM(x,0.4). The blue point represents the local maximum obtained by the L-BFGS-B method in the Scipy-optimization package when the starting point 0.1

4.2 Using MSSIM for Multi-input Fusion Aimed at Image Zooming

In this Section, we address the following image fusion problem which assumes the form given in Eq. 7 above. Let y1,y2,⋯,yn be a set of n degraded images and {p1,p2,⋯,pn} a set of associated weights across patches, {H1,H2,⋯,Hn} a set of degradation operators, find x such that

argminx∑i=1npiTM(Hix,yi)+λ‖∇x‖TV.

(16)

Here, TM(Hix,yi)=1-MSSIM(Hix,yi). The regularization term g(x) employed in this study is the isotropic total variation (TV) regularization operator, ‖∇x‖TV=∑x‖∇x(x)‖22. (TV is commonly used in image processing.) It is important to note that any degradation operator can, in principle, be employed in the proposed method. When images are used as inputs instead of vectors, local SSIM (MSSIM) is applied rather than SSIM. Therefore, for gradient-based optimization approaches, the derivative of MSSIM, as presented in [20], is required. We skip detailed derivative calculations of MSSIM here and refer the reader to [20] for further information. In the following experiment, five distortions of the test image Cameraman were produced using the following degradation operators: (1) mean contrast stretching (multiplication of each greyscale value by 1.2), (2) mean shift (by adding 10 to each greyscale value), (3) blur (using a 5-by-5 Gaussian filter), (4) salt-and-pepper noise and (5) additive Gaussian white noise (variance 0.005). These distortions then undergo local averaging followed by downsampling by a factor of 2, all associated with the degradation operators Hi in Eq. 16. The resized degraded images, obtained by bicubic interpolation, correspond to y1, y2, y3, y4, and y5 in Eq. 16, and pi=1 for simplicity. When λ=0, only the SSIM-Mean term is employed in Eq. 16. The optimal solution is found by gradually increasing λ until achieving a balance between the fidelity term and the regularization term. However, excessively large values of λ can neglect the fidelity term. Other factors, such as the learning rate, number of iterations, and fudge parameter, also influence gradient-based methods. The numerical results in Table 1 show that in the selected range of λ the superior fused image is obtained when λ=4.5 in terms of Mean Squared Error (MSE), Peak signal-to-noise ratio (PSNR), and SSIM, and λ=7 in terms of SSIM. It can be observed that the values of PSNR and SSIM are higher than all of the degraded image inputs and MSE is lower than all of those as expected. The visual results are provided in Fig. 3. The fused image corresponding to λ=7, obtained by combining a set of degraded images, is visually superior to each individual degraded image and the initial starting point. The plots of MSE, PSNR and SSIM for λ are displayed in Fig. 4. A more comprehensive set of experiments can be reviewed in the first author’s thesis [17].

Table 1

Numerical results for the fusion of a set of distortion images

MSE

PSNR

SSIM

Resized version of mean contrast stretched image

1043.53

17.94

0.45

Resized version of mean shift image

459.57

21.50

0.42

Resized version of Blurred image

695.54

19.70

0.56

Resized version of salt and pepper noisy image

745.85

19.40

0.55

Resized version of Gaussian noisy image

348.40

22.70

0.44

Fused image-λ=0

459.57

21.50

0.42

Fused image-λ=4.5

341.61

22.79

0.69

Fused image-λ=5.5

343.93

22.76

0.71

Fused image-λ=7

352.43

22.60

0.72

Fig. 3

Fusion results of a set of distortions of the test image Cameraman. a Original image. b The resized version of noisy mean shift (starting point). c Fused image (λ=0). d Fused image (λ=4.5). e Fused image (λ=5.5). f Fused image (λ=7)

Fig. 4

Plots of changes of MSE, PSNR, and SSIM value with respect to λ

5 Conclusion

We formulated new expressions to facilitate the calculation of SSIM derivatives in the proposed image fusion approach. A main advantage of the proposed formulation compared to previous work [11, 19] is its flexibility, i.e. the incorporation of additional terms as well as the use of different types of regularization terms. Furthermore, the proposed inverse problem offers flexibility in accommodating arbitrary linear degradation operators. Notably, based on our knowledge, previous works exclusively consider the identity matrix as a degradation operator [11], setting them apart from our proposed formulation, or only taking a single input rather than a set [19]. It is important to acknowledge that the choice of a starting guess in the proposed gradient-based iterative method and its parameters could significantly influence the numerical results, as a local optimal solution is being estimated.

Acknowledgements

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank the anonymous reviewers for their careful reading of our manuscript and their valuable and insightful comments.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_13

Hamiltonian Engineering with Time-Ordered Evolution for Unitary Control of Electron Spins in Semiconductor Quantum Dots

Bohdan Khromets1  , Zach D. Merino1   and Jonathan Baugh1  

(1)

Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

Bohdan Khromets (Corresponding author)

Email: bohdan.khromets@uwaterloo.ca

Zach D. Merino

Email: zmerino@uwaterloo.ca

Jonathan Baugh

Email: baugh@uwaterloo.ca

Abstract

We present a unitary control pulse design method for a scalable quantum computer architecture based on electron spins in lateral quantum dots. We employ simultaneous control of spin interactions and derive the functional forms of spin Hamiltonian parameter pulses for a universal set of 1- and 2-qubit logic gates. This includes selective spin rotations with weak local g-factor variations in the presence of the global oscillating field and a Control-Phase operation with the simultaneous control of g-factors and exchange couplings. We outline how to generalize the control scheme to multiqubit gate operations and the case of constrained or imperfect control of the Hamiltonian parameters.

Keywords

Spin qubitsSemiconductor quantum dotsQuantum optimal controlHamiltonian engineeringConstrained shape pulse design

1 Introduction

Among the paths toward large-scale quantum computing, spin-based quantum information processing in semiconductor quantum dots is particularly enticing. This is due to large coherence times in isotopically purified materials like Si or Ge, the possibility of operation at relatively high temperatures [13] (1–4 K), and the highly developed MOS/CMOS fabrication technology for semiconductor electronics [9]. The recent demonstrations of fidelity exceeding the fault tolerance threshold for one-[15] and two-qubit quantum logic gates [10] boost hope for efficient, universal, and scalable computers in the foreseeable future. Still, high fault-tolerance thresholds and susceptibility to pulse mis-settings and charge noise are significant bottlenecks on the path toward scalability. Thus, developing reliable and precise optimal control algorithms for Hamiltonian engineering is an important direction for tackling these problems. In this work, we establish an analytical pulse design method for unitary evolution, applicable to a specific architecture of semiconductor quantum dot spin qubit processors [2] and compatible with the simultaneous manipulation of different spin interactions. Certain parts of the proposed method are an extension of quantum circuit physical optimization proposals from [3] to larger functional spaces. The analytical nature of our approach bears similarity with the one by Barnes [1] for the exact evolution operator design with a partial reverse-engineering of the Hamiltonian. The freedom to choose the pulse functional profiles offered by our method makes possible the incorporation of the ideas from other proposals such as SMART protocol [7]. We expect our results to be instrumental in both the control of experimental devices and the numerical engineering of pulses with improved properties (for example, noise robustness) [11, 14].

2 Spin Interactions in a Semiconductor Quantum Processor

We focus on the realization of a universal, scalable semiconductor spin-qubit architecture [2] based on electron spins in Si/SiO2 lateral quantum dots. It consists of a planar grid of few-qubit computational nodes, where electron states are controlled locally with voltages on electrodes and globally with two perpendicular (static and oscillatory) magnetic fields. Entangled states can be distributed between neighboring nodes through electron shuttling. The entire rectangular network of nodes undergoes error correction according to the surface code protocol [5].

A minimal computational node (Fig. 1) contains a linear array of four quantum dots formed near the semiconductor (Si)—insulator (SiO2) interface: one for the data qubit (leftmost), two for ancilla qubits (middle), and one for the transfer of electrons between the nodes. Plunger gate voltages Vi accumulate quantum dots and host electrons in them, whereas the tunneling gate voltages Wi control the potential barriers between the dots (and thus the spin exchange coupling strengths). All spins are identically coupled to two global fields: an in-plane static Zeeman field Bz and a perpendicular oscillating field BRF(t) for electron spin resonance (ESR).

Fig. 1

Schematic of a computational node. Electrons are accumulated in the quantum dots at the Si/SiO2 interface and coupled to two (static and oscillatory) magnetic fields. The two-electron interactions and Larmor frequencies are voltage-controlled

Our goal is to establish a procedure for coherently controlling electron spins in such interacting quantum dot systems. The spin Hamiltonian of a linear array of N spins in the laboratory frame reads:

HL(t)=∑j=1NgjV(t),W(t)2μBBzZj⏟Zeeman+∑j=1N-1Jj,j+1V(t),W(t)4σj·σj+1⏟exchange+μBBRF(t)∑j=1Ncos(∫0tωRF(t′)dt′+ϕ(t))Xj⏟ESRwithglobaloscillatingfield.

(1)

Here, Xi,Yi,Zi denote Pauli operators of the ith spin, σi·σj=XiXj+YiYj+ZiZj, and μB is Bohr magneton. V(t) and W(t) represent the vectors of plunger and tunneling gate voltages, respectively. The Hamiltonian terms describe different spin couplings:

1.

Voltage-induced g-factor shifts introduce variation of the qubit Larmor frequencies gjμBBz/2ħ of order 10-4. This is a consequence of a weak but nonzero spin-orbit coupling in Si and is sometimes referred to as a Stark shift.

2.

Direct exchange interaction Ji,j between ith and jth electron arises from the Pauli exclusion principle together with Coulomb repulsion. This coupling is strongly (exponentially) dependent on the tunnel barrier height and electron separation.

3.

The global ESR field is characterized by its time-dependent envelope BRF(t), angular frequency ωRF(t), and phase ϕ(t).

We now move to the rotating frame synchronized with the global ESR field. Then, the Hamiltonian (1) and evolution operator will transform as follows:

UR=ULR,HR=RHLR†+iħR˙R†,R=expi2∫0tωRF(t′)dt′∑j=1NZj.

(2)

The substitution of R gives the rotating-frame Hamiltonian in frequency units:

H=HRħ=12∑j=1NμBBzħgj(t)-ωRF(t)Zj+∑j=1N-1Jj,j+1(t)4ħσj·σj+1+μBBRF(t)2ħ∑j=1Ncosϕ(t)Xj+sinϕ(t)Yj,

(3)

Note that the rotating-wave approximation (ignoring the fast-oscillating terms) has been applied. Clearly, the evolution under the laboratory and rotating frame Hamiltonians is equivalent when the operator R performs an integer number of Z-rotations during a pulse of length T: ∫0TωRF(t)dt=2πn,n∈Z⇒UR(T)≡UL(T). For the typical values of n∼103-104, this condition can always be satisfied experimentally by properly choosing the pulse length T.

3 Hamiltonian Engineering with the Universal Control Shape

Given the Hamiltonian (3), we aim to identify functions gj(t), Jj,j+1(t), BRF(t), ωRF(t), and ϕ(t) that yield deterministic evolution operators constituting a universal set of quantum logic gates. First, it is important to define the initial and final state of the system, in which no qubit evolution in the rotating frame will take place (we will refer to it as the idling state). Keeping the ESR field turned off and making exchange couplings negligibly small by increasing the tunneling barriers before and after each pulse can readily be achieved experimentally. The only additional requirement to define an idling state is to tune the quantum dots into the regime where the g-factors of all electrons are the same. Then, by choosing the rotating frame appropriately, we set all Z-terms in the Hamiltonian to zero at the beginning and the end of a control pulse, in addition to all other terms:

∀i:gi(Vidle,Widle)-gidle=0,ωRF(0)=ωRF(T)=μBBzħgidle.

(4)

We note that the SMART protocol [7] implies the same choice of idling state.

3.1 Spin Interactions as Basic Generators of Evolution

The most basic quantum operations are easiest to realize by manipulating one part of the Hamiltonian (3) at a time while keeping others equal to zero. The requirements on the profiles of active pulses then become particularly simple:

1.

Z-rotations of individual qubits by varying their Larmor frequencies:

ROTj(z^,θi)=e-i2θjZj:∫0TdtμBBzħgj(t)-ωRF(t)=θjmod2π.

(5)

2.

SWAPk gates on separated qubit pairs by varying their exchange couplings. The following holds (up to a global phase) due to σj·σj+1=12(1+SWAPj,j+1):

SWAPj,j+1kj≡e-i2πkjSWAPj,j+1:∫0TdtJj,j+1(t)ħ=πkjmod2π.

(6)

3.

Rotation of all qubits around an axis within x-y plane n=[cosφ,sinφ,0] by the same angle θ using an on-resonance ESR pulse with a fixed phase ϕ(t)≡φ:

∏j=1NROTj(n,θ)=∏j=1Ne-i2θn·σj:∫0TdtμBBRF(t)ħ=θmod2π.

(7)

Formulas (5), (6), and (7) are merely integral constraints on the control functions gj(t),Jj,j+1(t),BRF(t),ωRF(t). That is, one can choose any functional profiles of the control pulse combinations under the integrals as long as they integrate to the specified values. This observation will help us in designing arbitrary single-qubit rotations – an extra building block needed for universal control – and in extending the method to more complex quantum gates, including multi-qubit operations.

3.2 Addressing Individual Qubits with a Global ESR Field

The fundamental feature of material systems like Si/SiO2 is the weakness of spin-orbit coupling. This makes the range of g-factor variations very small for realistic voltage controls. For typical Larmor frequencies ∼10 GHz, their tunability range due to g-factor deviations is ∼0.1 to 1 MHz [8]. Thus, qubits cannot be shifted far from resonance to implement selective spin rotations in a traditional way (this is known as frequency crowding [12]). We notice, however, that the control ranges of the Larmor and Rabi frequencies μBBRF/ħ are comparable, and identify that:

The combination of global ESR control and weak local g-factor variations makes possible the effective local control of individual spins. Specifically, combining the resonant Rabi drive with the simultaneous Larmor frequency manipulation to achieve a 2π rotation at the end of the pulse restores the initial states of the qubits that should be effectively detuned from resonance.

Controlling both X- and Z-components of the Hamiltonian shifts the axis of rotation, and the deterministic rotation by a desired angle can be achieved if the axis of rotation remains fixed during the pulse. This holds when the control pulses are proportional at all times:

∀t∈[0,T]:μBBzħgj(t)-ωRF(t)∝μBBRF(t)ħ.

(8)

This constraint is not very limiting, as the control pulses can be chosen arbitrarily so long as their shapes are the same.

With these considerations in mind, we now explicitly derive the pulses for a general ROT+(n,θ) operation on resonant qubits (labeled as “+”), and simultaneously, a 2π rotation around some shifted axis ROT-(2π) on the non-resonant qubits (labeled as “-”). We rewrite the proportionality condition (8) by expressing the control parameters in terms of a generalized control shape S(t) and coefficients A±,ß:

μBBzħg±,j(t)-ωRF(t)=A±,jS(t)T,μBħBRF(t)=ßS(t)T.

(9)

For S(t), one can choose any continuous function that starts and ends at zero (as in realistic experimental conditions) with its average value normalized to 1:

S(0)=S(T)=0,S=1T∫0TS(t)dt=1.

(10)

Each single-qubit part of the Hamiltonian then reads:

H±,j=12S(t)T[A±Zj+ß(cosϕXj+sinϕYj)]=12S(t)TA±2+ß2σj·n±,j,

(11)

n±,j=ßcosϕ,ßsinϕ,A±T/A±2+ß2,‖n±,j‖=1.

(12)

We align the axis of rotation of resonant qubits with the desired axis: n+=n=nx,ny,nzT, and impose requirements on rotation angles of the qubits of two kinds:

A+2+ß2=θ,A-2+ß2=2π.

(13)

From these two conditions, we obtain exact expressions for the amplitudes A±,ß and thus the pulses implementing a ROT(n,θ) operation on selected qubits:

μBBzħg+,j(t)-ωRF(t)μBBzħg-,j(t)-ωRF(t)μBBRF(t)ħ=S(t)Tnzθ±(2π)2-θ2nx2+ny2θnx2+ny2signS(t)ϕ(t)=atan2(nysignS(t)signθ,nxsignS(t)signθ),

(14)

while all exchange couplings are kept negligibly small. Possible sign changes of S(t) are taken into account. The ESR phase then becomes a piecewise constant function instead of a constant to keep the ESR field envelope BRF(t) always positive (BRF(t) is proportional to |S(t)| in formula (9) in this case).

3.3 General Shape Function Formalism for Time-Ordered Evolution

In the spirit of the constraint on control pulse shapes (9), we generalize our method by making all control parameters of an N-qubit Hamiltonian (3) proportional to each other (except the ESR phase which is kept fixed or at most piecewise constant):

H(t)=S(t)TH0,H0=12∑j=1NAjZj+ßcosϕXj+sinϕYj+∑j=1N-1Cjσj·σj+1,

(15)

Here, H0 is a constant operator with the coefficients defined by expressions (9) combined with the following formula for exchange pulses:

Jj,j+1(t)=ħCjS(t)T.

(16)

The key feature of such a Hamiltonian is that it commutes with itself at all times: H(t′),H(t′′)≡0∀t′,t′′∈[0,T]. The evolution operator is thus time-ordered and is given by an ordinary matrix exponential:

U(t)=exp(-i∫0tH(t)dt)=exp(-iH0∫0tS(t′)Tdt′),U(T)=exp(-iH0).

Consequently, complex multi-qubit unitary operations U(T) can be engineered by properly choosing the coefficients of H0. A notable example is the Control-Phase two-qubit gate that can be implemented by simultaneously varying exchange couplings and g-factors. Burkard and Loss [3] noticed that a 4×4 Hamiltonian of type H0=12(A1Z1+A2Z2)+C4σ1·σ2 is block diagonal in the computational basis, and thus its matrix exponential can be found analytically. The coefficients A1,A2,C for a CPHASE(α) gate, derived in [3] as the amplitudes of constant pulses:

CPHASE(α):1····1····1····eiαC=2π-α,A1,2=12α±α4π-α,α∈[0,π]-π+α±α4π-α2,α∈[π,2π),

(17)

can now define smooth pulse profiles with shapes S(t) chosen arbitrarily.

Fig. 2

3-pulse sequence for an exact CNOT operation. Each pulse is 1μs long, and their shapes are Gaussians with σ=0.1,0.15 and 0.3, respectively. The system parameters are Bz=1 T, gidle=2, and ωRF/2π=27.99248987 GHz is fixed (not modulated). The value of ωRF is calculated from relation (4)

To demonstrate the control scheme, we find a pulse sequence that realizes a CNOT gate with one control-Z and two Hadamard gates (Fig. 2). This quantum circuit exemplifies both the selective qubit rotations (π-rotation around (101) axis for a Hadamard operation, and a 2π-rotation of the non-resonant qubit) and simultaneous control of g-factors and exchange to realize CPHASE(π)≡CZ. For pulse shapes, we choose normalized Gaussian functions offset vertically to start and end at zero:

Sσ(τ)=e-τ-1/222σ2-e-18σ2∫01dτ′[e-τ′-1/222σ2-e-18σ2],τ=tT∈[0,1].

(18)

We intentionally choose different widths σ for each of the 3 pulses in the sequence to emphasize that any normalized function S(t) for the pulse shape profile will yield an exact unitary operation (i.e., with theoretical fidelity 1).

The pulse shapes and coefficients of the Hamiltonian (3) for all quantum logic gates derived in this section are summarized in Table 1. Note two groups of gates can run in parallel in quantum circuits, and the shape functions of certain simultaneous controls need not be the same. Both of these scenarios are related to the Hamiltonian splitting into a sum of commuting operators.

4 Discussion and Conclusions

The idea of engineering a Hamilonian in the form (15) (a product of a scalar function and a constant operator) to make evolution time-ordered is clearly applicable to systems of any number of qubits N. Furthermore, it is straightforward to genenarize it (possibly, with different restrictions on the experimentally controlled parameters) to other architectures involving, for example, hole spin qubits or electric dipole spin resonance (EDSR). As the number of controls available in our architecture gj(t),Jj,j+1(t) grows linearly with N while the Hilbert space size grows as 2N, unitaries generated by a single N-qubit Hamiltonian of type (15) become progressively more restricted relative to the full unitary group U(2N) when N grows. Then, one can design a sequence of M smooth pulses that generates a desired multiqubit unitary with a certain fidelity by numerically optimizing the coefficients Aj,ß,Cj of each pulse (M does not have to be large). The optimization procedure is mathematically equivalent to a GRAPE method [6, 11] with M steps, but does not affect the freedom to choose any smooth S(t) for each of the pulses.

Additionally, we note that the design of pulses with improved properties can be treated as an optimization problem for a function S(t) within the proposed control scheme. For example, robustness to imperfections of controls due to miscalibrations or noise represented as an “undesired” part of the Hamiltonian H′(t) is achieved when the directional derivative of U(t) along H′(t) [6] is minimized. Then, the optimal S(t) can be found by minimizing ‖DUDH′‖ variationally with respect to S(t). The geometric formalism for dynamically corrected operations [4, 16] is an alternative way to optimize the shapes of control pulses to improve their robustness properties.

In conclusion, we devised a pulse design technique for the unitary control of a semiconductor spin qubit processor architecture [2]. Thanks to the freedom to choose a functional profile of pulses, our method is capable of realizing experimentally useful control pulse sequences. Within the limitations on the control of spin couplings in the device architecture, we engineered pulses for arbitrary spin rotations SWAPk, and Control-Phase operations with theoretically perfect gate fidelities. Notably, this gate set is suitable for universal quantum computation. The general, functional approach to pulse design could be adapted to other qubit architectures with different Hamiltonian controls and extended with the variational optimization of pulse shape in the presence of noisy controls.

Table 1

Two groups of single- and two-qubit gates that can be run in parallel on semiconductor spin quantum processors proposed in [2]

a due to the fact that n·(σ1+σ2) commutes with σ1·σ2 for any n, a SWAPk gate can run simultaneously with any pair of synchronous qubit rotations (i.e., same axis and angle of rotation)

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_14

Exploring the Use of Gradients in the Structural Similarity Image Quality Measure

Amelia Kunze1, 2   and Edward R. Vrscay1  

(1)

University of Waterloo, Waterloo, Canada

(2)

Università Degli Studi Dell’Insubria, Varese, Italy

Amelia Kunze (Corresponding author)

Email: agkunze@uwaterloo.ca

Email: akunze@uninsubria.it

Edward R. Vrscay

Email: ervrscay@uwaterloo.ca

Abstract

In this paper, we investigate if the well-known Structural Similarity image quality measure (SSIM) can be improved by incorporating gradient information. We propose a simple gradient similarity measure which yields results similar to the canonical correlation method. Using the LIVE image database, we show that our proposed gradient-based SSIM exhibits improved performance for degraded images in the mid-to-low quality range.

Keywords

Image quality measuresPerceptual image qualityStructural similarityGradients in imagingGradient-based similarity measureHuman visual system

1 Introduction: Image Quality Assessment and the MSE

The L2-based mean squared error (MSE) and its variations continue to be the most widely employed metrics in image processing. This is most probably due to the fact that (1) the MSE is simple to compute and (2) it possesses a number of convenient mathematical properties, including differentiability and convexity. It is well known, however, that these L2-based measures perform poorly in terms of measuring the visual quality of images. Their failure is partially due to the fact that the L2 metric does not capture spatial relationships between pixels. This was a motivation for the introduction of the so-called Structural Similarity (SSIM) image quality measure [1] which, along with its variations, continues to be one of the most effective measures of visual quality. The SSIM index measures the similarity between two images by combining three components of the human visual system—luminance, contrast, and structure. In particular, for two corresponding image patches x,y∈RM×M, the SSIM index is usually defined as the product,

SSIM(x,y)=S1(x,y)·S2(x,y)·S3(x,y)=2x¯y¯+C1x¯2+y¯2+C12σxσy+C2σx2+σy2+C2σxy+C3σxσy+C3,

(1)

where C1, C2, and C3 are stability constants. Local SSIM values are computed in patches tiling the entire image, then averaged to yield a so-called “mean SSIM” (MSSIM) value. The reader is directed to [1] for a more detailed discussion of the SSIM. For our interests, it is important to note that S3(x,y) computes the correlation between the patches x and y, and it is our belief, which we thoroughly investigated in [2], that the S3 term is the most important component of the SSIM.

To illustrate these concepts, we present a particularly compelling failure of the MSE. Consider the so-called “Einstein images”—a set of six 256×256 pixel, 8 bits-per-pixel grayscale images—depicted in Fig. 1.¹ The images blur, contrast, impulse, jpg, and meanshift are all perturbations of the reference Einstein image original. (A more detailed discussion and analysis of the Einstein images can be found in [2]). For each of these five degradations, the degree of distortion was adjusted to yield nearly equal MSE relative to original. Because the degraded images differ significantly and obviously in perceptual quality, they provide a striking example of the failure of the MSE to measure perceptual quality. On the other hand, the MSSIM scores are more indicative of the apparent perceptual quality of the images. The root mean squared error (RMSE) and MSSIM of each of the degraded images relative to original are reported alongside the Einstein images in Fig. 1.

Fig. 1

The reference Einstein image original and its perturbations

Perhaps more pertinent than the actual MSSIM values are their relative values. The MSSIM scores reported in Fig. 1 imply the following ordering of the distorted images in decreasing order of quality:

meanshift>contrast>impulse>blur>jpg.

(2)

Ideally the ranking in Eq. (2) is consistent with the reader’s own subjective preferences of the distorted images in Fig. 1.

In this paper, we seek to develop a gradient-based image quality measure. This work is taken from a larger study [2] exploring the use of gradients in imaging, which involves not only image quality measures but also best approximation problems. In the following discussion, many valuable details were necessarily omitted due to space restrictions. Indeed, the interested reader is directed to [2] for a more complete development of our approach. Underlying our study is the question of how sensitive the human visual system could be with respect to changes in the gradient vectors of an image, in terms of either direction or magnitude, or perhaps both. Naturally, we are interested to explore if an image quality measure employing gradients can assess visual quality as well as, or perhaps even better, than current image quality measures, such as the MSSIM, which do not use gradient information.

To begin, we first need to decide on the manner in which to compute the image gradients. There are numerous reasonable choices: Indeed, we acknowledge that in the image processing literature, many different formulas—which can be expressed in terms of “filters” operating on the matrices representing the images—are employed. Furthermore, different gradient filters will most probably yield different computational results. That being said, our primary purpose in this paper is only to introduce the idea of using gradients in image quality measures. As such, we will define the gradient using simple forward differences, i.e., the gradient of the image x∈RN×M at the (i,j)th pixel will be defined by,

∇xij=(x(i+1,j)-x(i,j),x(i,j+1)-x(i,j)).

(3)

We assume an even extension of the image, so that at the edges of the image we have x(N+1,j)=x(N,j) and x(i,M+1)=x(i,M). Note that this produces a zero value for the appropriate component of the gradient vector.

The following simple experiment motivates the application of gradient information for image quality assessment. For those Einstein images in Fig. 1, we compute the usual L2 distance between gradients. Specifically, for the original image x and each of its five perturbations y, we compute the following distances,

‖∇x-∇y‖2=1N∑i=1N∑j=1N‖∇xij-∇yij‖221/2,

(4)

where N=256 and the ∇x and ∇y are computed using the forward differences previously described. These results are presented in Table 1. (Because the meanshift image is produced from original by merely adding a constant to the greyscale values of the latter, the gradients of the two images are identical. Hence, one would expect the meanshift RMSE in Table 1 to be exactly 0. The observed deviation is due to a few pixels whole meanshift values are restricted by limits on the greyscale range.)

Table 1

RMSE between the gradients of the original Einstein image and its perturbations

blur

contrast

impulse

jpg

meanshift

17.0349

5.5426

23.9110

17.6135

0.1309

We can immediately observe a significant deviation in these values. In other words, the structural information encoded in our simplistic gradient allows the MSE, which does not otherwise consider spatial relationships between pixels, to differentiate between the Einstein images. It is clear that computing the MSE between gradients already offers a marked improvement over the MSE.

Letting the gradient distances define an ordering of image quality, we obtain,

meanshift>contrast>blur>jpg>impulse.

(5)

Unfortunately, this is not in agreement with the ordering dictated by the MSSIM as reported in Eq. (2). However, observe that the only difference between Eq. (2) and Eq. (5) is the placement of the impulse image. Indeed, it is easy to understand why impulse noise is particularly bothersome to the gradient distance: Impulse noise strongly affects not only the gradient at each contaminated pixel, but also the gradients of its adjacent neighbours. It is possible that some preprocessing methods, such as blurring or downsampling of the images, could mitigate this effect, perhaps even to the extent that one could retrieve a ranking of gradient distance in agreement with the ordering according to the MSSIM. That being said, we will instead proceed by pursuing other, and hopefully better, ways of computing gradient similarity.

2 Computing Gradient Similarity

Our foremost source of inspiration on which to model a gradient similarity measure is the correlation. Indeed, in the same way that the SSIM index computes the correlation between two M×M patches x and y, one might be tempted to compute the correlation between their gradients ∇x and ∇y. However, ∇x and ∇y will each be an M×M block of vectors, i.e., each component of the M×M matrix will be a 2-vector as written in Eq. (3). There is a proper way to compute the correlation between vectors of M×M blocks, namely, the “canonical correlation” method introduced by Hotelling in 1936 [5]. However, this method is somewhat expensive computationally, which led us to examine the effectiveness of simpler methods to compute the correlation. A number of rather simple methods were explored in [2]. Here we shall simply report on the “best” such method, which generally produced results quite comparable to the canonical correlation method.

Our “best” measure of gradient similarity is computed as follows: For two corresponding image patches, let a denote the correlation between the x-components of the gradients and b denote the correlation of the y-component of the gradients. Then compute the “normalized magnitude” of the vector (a, b), i.e., the quantity

S4(a,b)=12(a2+b2)1/2,notingthatS4(a,b)∈[0,1].

(6)

Table 2 reports the values of the normalized magnitude S4(a,b) for each perturbed Einstein image relative to original. In each case, these values correspond to using non-overlapping image patches of size 32×32 pixels.

Table 2

Average S4(a,b) values of the original Einstein image and its degradations

blur

contrast

impulse

jpg

meanshift

0.4391

0.9964

0.5573

0.3571

1.0000

If we let these normalized magnitude values dictate the relative quality of the images, we retrieve the ranking imposed by the MSSIM, i.e.,

meanshift>contrast>impulse>blur>jpg.

(7)

(In [2], we show that the ordering in Eq. (7) is preserved for various patch sizes.) A comparison of the MSSIM values and those S4 values in Table 2 reveals that the gradient correlation is much more punitive than the usual MSSIM. However, using this small set of Einstein images alone we cannot conclude whether our punitive S4 is performing “better” than the MSSIM. In the next section, we will use a larger set of images, i.e., the LIVE image database [6], to attempt to answer this question. (In the process, we will need to define what we mean by “better” performance.)

3 Our Quest for a Gradient-Based Image Quality Measure

The LIVE image database contains a total of 982 reference and distorted images. Each image is associated with a unique “difference mean opinion score” (DMOS) indicating its perceptual quality. (Refer to [2] or [7] for a detailed discussion of the database and the method used to obtain the scores.) DMOS=0 indicates perfect perceptual quality, i.e., the associated image has no visible distortions, while DMOS increases as the presence of distortions becomes more visible and bothersome.

In Fig. 2a, we plot MSSIM against DMOS for all images in the LIVE database. Here, we have adopted the convention in the literature which puts DMOS along the vertical axis. The scatterplot includes a curve of best fit according to the expected nonlinear relationship provided in [7] and which was computed using the Matlab function “fitnlm”. To provide an indication of how well a given image quality measure, e.g., the MSSIM, is performing, we measure the scatter of the points relative to the fitted curve in the following manner: For all images j in the LIVE database, we compute the distance,

dist=∑j=1982(DMOS(j)-Quality(MSSIM(j)))21/2,

(8)

where Quality(x) denotes the value of the fitted curve at a point x. Because the curve is obtained through least squares regression, our “dist” measure can be thought of as the variance between the fitted curve and the DMOS values as a function of the algorithm score. This “dist” value is provided in the title of Fig. 2a.

The data points in Fig. 2a are quite concentrated about the curve at the bottom right region of the plot, which may be considered as the “low DMOS” region (i.e., DMOS under 20) or “high MSSIM” region (i.e., MSSIM near 1). As we move leftward and upward, however, the data points are distributed more diffusely about the regression curve. This leads us to think of one possible criterion for “improving” the MSSIM, namely, decreasing the diffusiveness of the data points in the lower DMOS region. With this goal in mind, we propose the following,

gradSSIM(x,y)=SSIM(x,y)·S4(x,y)=S1(x,y)·S2(x,y)·S3(x,y)·S4(x,y),

(9)

where, in keeping with the spirit of the SSIM, our “normalized magnitude” S4 has been added as a fourth multiplicative term. Recall that our S4 term is defined in terms of the correlations (i.e., quotients) a and b. In both a and b, we would like to include a stability constant C4 to keep the denominator away from 0. This raises the question of which value should be taken by the stability constant C4.

In the usual SSIM, the stability constants C1, C2, and C3 are included in both the numerator and denominator of the S1, S2, and S3 terms, respectively. The suggested values for these constants, provided by Wang et al. in [8], are C1=6.5, C2=58.5 and C3=29.2. In this spirit, one possibility is to include C4 in both the numerator and denominator of the correlations a and b. Because we think of these correlations as gradient-based analogues of the S3 component, we set C4 (practically) equal to C3, i.e., we let C4=30. This reasoning is justified by our observation that the gradient correlations are similar in magnitude to the regular correlations for the Einstein images. This S4 formulation is plotted against DMOS in Fig. 2b.

However, [8] describes C1, C2 and C3 as “small” constants to protect against numerical instabilities in the denominator; Given this understanding, their suggested values are significantly larger than we would expect. Moreover, we are unable to find in the literature any clear description of how or why these suggested values were obtained. Our instinct is to include a constant C4≪1 only in the denominators of a and b. The result of using this formulation is shown in Fig. 2c.

Fig. 2

For all LIVE images, we plot a SSIM and our S4 measure with b C4=30 in the numerator and denominator and c C4=10-5 in the denominator only. The legend denotes distortion type: “jp2k” and “jpeg” for JPEG 2000 and JPEG compression, respectively, “wn” for white noise, “gblur” for Gaussian blur, and “ff” for transmission errors in the JPEG 2000 bit-stream

The algorithm scores are generally significantly increased from Fig. 2a and b, which indicates that having the relatively high value C4=30 in both the numerator and denominator artificially pushes the quotients a and b towards 1. On the other hand, the points in Fig. 2c are reasonably close to the fitted curve along its entire length. We are inclined to use the S4 with C4≪1 in the denominator only for our “gradSSIM” measure. This choice, although a departure from conventions set by the SSIM, is further supported by Fig. 3 which shows that this S4 is highly correlated with the aforementioned “rigorous” canonical correlation. Indeed, results virtually identical to those presented in Figs. 4 and 5 are obtained if one replaces our simple S4 term with the canonical correlation.

Fig. 3

Our preferred S4 is highly correlated with the canonical correlation

The result of computing the corresponding gradSSIM, as defined in Eq. (9), for the LIVE images is shown in Fig. 4b. In Fig. 4a we have included the usual MSSIM for comparison. Here we also flip the orientation of the axes, i.e., we now consider DMOS to be the independent variable. To the best of our knowledge, this understanding has not been adopted elsewhere in the literature. However, because the DMOS scores are obtained from subjective experiments, they are, or at least could be considered to be, the independent values. Given this understanding, it is actually the scatter in the algorithm scores relative to the curve that characterizes goodness of fit. We use a simple quadratic function to fit the “flipped” data and denote the “flipped” distance by “distDMOS” to distinguish it from the “dist” value computed previously.

Fig. 4

The “flipped” SSIM and the gradSSIM using our preferred S4

The gradSSIM in Fig. 4b has a much improved fit in the high DMOS region compared to the MSSIM in Fig. 4a. Unfortunately, this success is balanced by a loss in the low DMOS region where the gradSSIM is exhibiting its diffuseness. Still, as the “distDMOS” values indicate, there appears to be slightly less overall spread in Fig. 4b than in Fig. 4a. Compared to the MSSIM, the gradSSIM offers a better choice for applications dealing with heavily degraded images. However, according to a subjective evaluation performed in [2], the low-to-mid DMOS region likely captures reasonable distortion levels for most practical applications. In this light, a gain in the high DMOS region may not be worthwhile if it comes at the expense of the good fit for low DMOS.

Looking at Fig. 4, it would be most desireable to construct an image quality measure which behaves like the MSSIM for low DMOS and the gradSSIM for high DMOS. In [2], we explored some possible “blended” measures towards that end. Ultimately, we arrive at the following proposal involving SSIM-based exponents: For two image patches x and y, we define the following “blended” local similarity function,

gradSSIM1(x,y)=SSIM(x,y)·S4(x,y)1-SSIM(x,y).

(10)

For SSIM near 1, S4(x,y)1-SSIM(x,y)≈1. For SSIM near 0, S4(x,y)1-SSIM(x,y)≈S4. In this way, we reduce the effect of S4 in the low DMOS region, where it exhibits a great deal of scatter, and increase the effect of S4 in the high DMOS region, where it is more successful than the MSSIM. If the effect of the S4 term is not sufficiently strong for larger DMOS, we may prefer the following definition,

gradSSIM1(x,y)=SSIM(x,y)·S4(x,y)1-SSIM(x,y)2.

(11)

Fig. 5

We plot both versions our the gradSSIM1, as defined in a Eq. (10) and b Eq. (11)

The results of computing the gradSSIM1 as written in Eqs. (10) and (11) are shown in Fig. 5. Both formulations appear to be working well, with little difference between the two plots. According to the “distDMOS” value, Eq. (11) is performing slightly better. In general, the gradSSIM1 improves the scatter in the mid-to-high DMOS without sacrificing the good fit for low DMOS.

For the heavily distorted images with high DMOS, the attempt to cluster the data points may be considered as more of an academic exercise than one of practical value. However, it may still be of practical concern to improve the fit in the mid-DMOS range, i.e., 40–80. Indeed, the great majority of mid-DMOS points lie above the fitted curve in the MSSIM plot shown in Fig. 4a. In [2], we showed that the placement of these points is largely invariant to changes in the stability constants C1, C2, and C3. On the other hand, our gradSSIM1 can produce smaller values for both the mid-DMOS and high DMOS range which, on the basis of our subjective evaluation performed in [2], are warranted.

4 Conclusion

The goal of this paper was to investigate if the MSSIM could be improved by incorporating gradient information. By adding a simple gradient similarity measure, we demonstrated that the resulting “gradSSIM1” improves the fit in the mid-to-high DMOS range without sacrificing much in terms of fit in the low DMOS range. In future studies, we suggest that DMOS be considered as the independent variable.

Other gradient-based similarity measures have been proposed in the literature. In [3], an SSIM-like gradient similarity measure seeks to rate the presence of edges using gradient operators. Edge maps are obtained by Haar wavelet filters and subsequently compared in [4]. However, the rather complicated mathematical machinery employed in both [3, 4] makes it difficult to understand exactly how the edge data informs the resulting similarity index. It is unclear to what degree these complicated methods of collecting and aggregating gradient information reflect an honest matching of the gradient vectors. Our approach based on first principles is mathematically tractable and clearly seeks to match the image gradient vectors.

Another novelty in our approach lies in its ability to seamlessly blend the behaviour of different measures using SSIM-based exponents. Our study led us to question the demand that a single mathematical formula should accommodate all types of distortions across the entire quality spectrum. Throughout our work, we speculated that the human visual system may access a variety of processes when judging different aspects of an image. This thought led us to consider a new way of thinking about image quality measures as “blended” formulas which might be more aligned with how the human visual system aggregates information. In this way, our approach introduces a new way of considering image quality measures. It is our hope that this novel “blended” framework will stimulate future research in this area.

Acknowledgements

This research was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Discovery Grant (ERV). AK would also like to thank the Faculty of Mathematics, University of Waterloo for partial financial support.

References

1.

Wang, Z., Bovik, A.C., Sheikh, H.R.: Digital Video Image Quality and Perceptual Coding, Chapter 7: Structural Similarity Based Image Quality Assessment. CRC Press (2005)

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Kunze, A.: An investigation of the use gradients in imaging, including best approximition and the Structural Similarity image quality measure. Master’s Thesis, Department of Applied Mathematics, University of Waterloo (2023)

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Liu, A., Lin, W., Narwaria, M.: Image quality assessment based on gradient similarity. IEEE Trans. Image Process. 21(4), 1500–1512 (2012)MathSciNet

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Reisenhofer, R., Bosse, S., Kutyniok, G., Wiegand, T.: A Haar wavelet-based perceptual similarity index for image quality assessment. Signal Process.: Image Commun. 61, 33–43 (2018)

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Hotelling, H.: Relations between two sets of variates. Biometrika 28(3–4), 321–277 (1936)

6.

Sheikh, H.R., Wang, Z., Cormack, L., Bovik, A.C.: LIVE image quality assessment database release 2. http://​live.​ece.​utexas.​edu/​research/​quality

7.

Sheikh, H.R.: Image Quality Assesment Using Natural Scene Statistics. Ph.D. thesis, The University of Texas as Austin (2004)

8.

Wang, Z., Bovik, A.C., Sheikh, H.R.: Digital Video Image Quality and Perceptual Coding, Chapter 7: Structural Similarity-Based Image Quality Assessment. CRC Press (2005)

Footnotes

1

We are grateful to have obtained these Einstein images from Prof. Z. Wang, Department of Electrical and Computer Engineering, University of Waterloo. To the best of our knowledge, these images were first presented by Wang, Bovik, and Sheikh in [1].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_15

Simulated Charge Stability in a MOSFET Linear Quantum Dot Array

Zach D. Merino1  , Bohdan Khromets1   and Jonathan Baugh2  

(1)

University of Waterloo, Waterloo, ON, Canada

(2)

Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada

Zach D. Merino (Corresponding author)

Email: zmerino@uwaterloo.ca

Bohdan Khromets

Email: bohdan.khromets@uwaterloo.ca

Jonathan Baugh

Email: baugh@uwaterloo.ca

Abstract

In this study, we address challenges in designing quantum information processors based on electron spin qubits in electrostatically-defined quantum dots (QDs). Numerical calculations of charge stability diagrams are presented for a realistic double QD device geometry. These methods generalize to linear QD arrays, and are based on determining the effective parameters of a Hubbard model Hamiltonian that is then diagonalized to find the many-electron ground state energy. These calculations enable the identification of gate voltage ranges that maintain desired charge states during qubit manipulation, and also account for electrical cross-talk between QDs. As a result, the methods presented here promise to be a valuable tool for developing scalable spin qubit quantum processors.

Keywords

Quantum computingSemiconductor quantum dots2-dimensional electron gasCMOSMOSFETQuantum dot Hubbard modelExact diagonalizationCharge stability

1 Introduction

In recent years, certain Quantum Computing (QC) implementations have demonstrated a quantum advantage over classical processors [15]. While further demonstrations of quantum supremacy may be realized, quantum processors (QPs) consisting of thousands to millions of physical qubits will be required to perform large-scale fault-tolerant QC.

The scaling requirements for fault tolerant QC make electrostatically-defined quantum dot (QD) spin qubits a promising approach. In particular, spin qubits in silicon (Si) have attracted much recent interest due to their compatibility with industrial complementary metal-oxide semiconductor (CMOS) fabrication techniques, which could yield high quality, densely packed qubits [9, 11]. A schematic Si double QD device is shown in Fig. 1. A 2-dimensional electron gas accumulates at the Si/SiO2 interface due to positive voltage applied to a gate electrode. Mesoscopic QDs are formed beneath the plunger gate fingers, localized by the presence of screening gates. The QDs are tuned into the few-electron regime via the plunger gates, where tunnel coupling to a reservoir (not shown) allows for charge transfer. The spin of an unpaired electron in a QD with an odd number of charges constitutes the qubit. Spin qubits in Si and SiGe devices have exhibited long coherence compared to quantum gate operation time scales [13], high-fidelity gate operations required for fault-tolerant QC [7], qubit control via applied electric gate signals [11], and qubit operation at relatively high temperatures (1–4 K) [8]. Furthermore, scalable spin qubit architectures suitable for implementing surface code quantum error correction have been proposed [2, 12].

Fig. 1

A schematic MOSFET double QD device in Si: (a) Side cross-sectional view showing a 2-dimensional electron gas (2DEG) formed by a plunger gate electrode. (b) Top-view cross-section showing the double well confining potential formed by the plunger (V1,V2) and tunnel (W1) electrodes. The electrostatic potential energy at the SiO₂/Si interface is calculated by a finite–element Poisson solver based on a 3D device model. (c) Contour plot of the potential energy landscape in (b)

Scaling QPs generally increases the complexity in initializing, controlling, and measuring quantum computational states. Each additional QD adds independent control parameters (plunger and tunnel gates) and is a source of cross-talk between neighboring QDs [4, 11]. A critical requirement is the ability to predict the charge state of a linear QD array for a given vector of gate voltages. For spin qubits in singly-occupied QDs, it is necessary to determine the control boundaries within which single charge occupations are maintained. Thus, accurately modeling the charge state of a realistic linear QD array as a function of gate voltages is a necessary first step, prior to the design of spin qubit control pulses.

Previous studies have employed Hubbard models as approximate descriptions of linear arrays of tunnel-coupled QDs [5, 10]. Given certain assumptions about the underlying electrostatic potential, others have shown that effective parameters of the Hamiltonian can be calculated [3, 14] and a resulting charge configuration determined [6]. In this work, electrostatic potential landscapes for realistic QD device geometries found by a finite–element Poisson solver [1] are used as inputs for calculations that determine stable charge states as a function of gate electrode voltages.

2 Hubbard Model for Quantum Dots

2.1 Hamiltonian

A generalized Hubbard Hamiltonian developed by [3, 14] for electrostatically defined QD systems is composed of

H=Hμ+Ht+HU+HJ,

(1)

where terms describing the chemical potential, Hμ, tunnel coupling, Ht, Coulomb repulsion, HU, and spin exchange, HJ, are included. The first term, Hμ, represents the occupation of spin states in an array of N quantum dots:

Hμ=-∑σ=↑,↓∑i=1Nμin^i,σ

(2)

Here, i is the index of each QD in a linear array of N QDs, μi are their chemical potentials, σ∈{↑,↓} is the spin state, and n^i,σ is the electron number operator for the spin state σ of the ith QD. The number operator is defined as n^i,σ=c^i,σ†c^i,σ, where c^i,σ† (c^i,σ) are the usual creation (annihilation) operators.

The second term, Ht, represents the tunnel couplings between QDs where an electron with spin state σ is allowed to tunnel to an adjacent QD.

Ht=-∑σ=↑,↓∑j=1N-1tj,j+1c^j,σ†c^j+1,σ+c^j+1,σ†c^j,σ

(3)

Here, tj,j+1 are the tunnel coupling, or inter-site hopping terms for adjacent QDs. The operators c^j,σ†c^j+1,σ, c^j+1,σ†c^j,σ, account for j→(j+1), j←(j+1), tunneling events.

The third term, HU, is the Coulomb repulsion between all electron pairs.

HU=∑i=1NUin^i,↑n^i,↓+∑σ1,σ2∈{↑,↓}∑j=1N-1Uj,j+1n^j,σ1n^j+1,σ2

The intra-Coulomb, or onsite, energy, Ui, accounts for the Coulomb repulsion energy of electrons on the same QD, while the inter-Coulomb energy, Uj,j+1, accounts for the Coulomb energy between pairs of electrons in adjacent QDs. In general, the inter-Coulomb energy accounts for all combinations of electron pairs in different QDs, however, in this work only nearest neighbor contributions are considered.

The last term, HJ, accounts for energy contributions from spin-exchange, pair-hopping, and occupation-modulated hopping. However, this term is much smaller compared to the other three components and will be neglected for the purposes of this work.

2.2 Effective Parameters

The inter-Coulomb repulsion is calculated using the expression given in [14],

Uij=∫-∞∞dr1∫-∞∞dr2φi(r1)2φj(r2)2v(r1,r2)

(4)

where φi is the single-particle ground state wavefunction of the ith QD and,

v(r1,r2)≡e24πϵ0ϵr1r1-r2

is the Coulomb repulsion term. The onsite repulsion term for the ith QD is defined likewise but contains only the single-particle wavefunction φi of the QD in consideration:

Ui=Uii

(5)

The wavefunctions φi are determined by solving the Schrödinger equation

p^i22m∗+Virφir=εiφir,

(6)

where m∗ is the electron effective mass, for all electrostatic potentials Vir representing localized individual QDs. An example of the procedure that allows us to determine Vir for each QD in a double-dot potential is shown in Fig. 2. Here, each of the regions near the local minima (describing individual QDs) is fitted to a 2D Gaussian function while the other ones are masked with a high potential value. The procedure preserves the effects of gate voltage cross-talk on QDs even after the isolation of their electrostatic potentials.

Fig. 2

Electrostatic potential, Φ, for: (a) left single QD and masked right QD (b) right single QD and masked left QD. Regions are in circled where there is a ≈99%, ≈95%, and ≈65% probability of finding an electron

While the Eqs. (4) and (5) are well defined, the integral expressions appear to be singular due to the Coulomb potential term v(r1,r2) when r1=r2. However, the apparent singularity can be avoided by performing the change of variable

r=r1-r2

and integrating r in polar coordinates. This yields the following integral that can be performed numerically:

Uij=e24πϵ0ϵr∫0∞dr∫02πdθ∫-∞∞dr2φi(r+r2)2φj(r2)2

(7)

Next, to determine the chemical potential from equation (2), Vir is defined by masking all but the ith QD’s electrostatic potential, and the single-electron ground state energy is found by solving equation (6).

μi=εi(g)

Lastly, the tunnel coupling term from Eq. (3) is derived from a two-level Hamiltonian that includes interdot tunneling in a double QD, and by masking the electrostatic potential for all but the jth QD pair,

tj,j+1=12(Ej,j+1(e)-Ej,j+1(g))2+(εj(g)-εj+1(g))2

(8)

where Ej,j+1e/g are the first excited/ground state energies for the double QD potential of the jth QD pair and εj/j+1(g) are the ground state energies corresponding to the single QD potentials.

2.3 Charge State via Exact Diagonalization

To determine the charge state of a linear QD array at a given voltage configuration, the components of the Hamiltonian defined by Eqs.  (2)–(4) are calculated for a chosen charge configuration basis. The Hubbard model (1) accounts for all charge and spin states with up to two electrons per QD following the Pauli exclusion principle. For example, a choice of basis for a double QD with total electron number 2 is the following:

α1≡c1,↑†c2,↑†0,0=↑,↑α2≡c2,↓†c2,↑†0,0=0,↑↓α3≡c1,↓†c2,↑†0,0=↓,↑α4≡c2,↓†c1,↑†0,0=↑,↓α5≡c1,↓†c1,↑†0,0=↑↓,0α6≡c1,↓†c2,↓†0,0=↓,↓

The ordering of the basis is arbitrary, however, the above order was chosen for consistency with [6] so that the Hamiltonian has only diagonal contributions from Hμ and HU and non-zero off diagonal element contributions from Ht. In general, the elements of the Hamiltonian are given by Hnm=αnHαm. After determining the matrix elements Hnm for a given gate voltage configuration, the Hamiltonian is diagonalized to find the charge state associated with the ground state energy. This allows us to identify the charge state of a linear QD array as a function of the voltages applied to the gate electrodes.

3 Results and Discussion

The following Subsects. (1) compare analytic and simulated results for onsite and inter-Coulomb energy calculated for a theoretical double QD electrostatic potential, and (2) present a charge stability diagram (CSD) calculated from realistic double QD electrostatic potentials. An electrostatic potential of two QDs separated by d is approximated by a quartic function with a confinement strength parameter ω:

U(x,y)=m∗ω22(x2-d24)2d2+y2

(9)

and 2D Gaussian functions are used to approximate the single-particle ground state wave functions of each QD as described in [14]. For the CSD, the electrostatic potential as a function of plunger gate voltages, V=[V1,V2], and tunnel gate voltage, W=[W1], is calculated via finite-element Poisson solver nextnano [1], and then used as input to calculate the Hubbard model parameters and determine the charge state for a given voltage configuration.

3.1 Numerical Accuracy and Efficiency

Figure 3a plots the onsite energy as a function of confinement strength (in angular frequency units), ω1, and compares the analytic solution to numerically obtained results with a varying number of mesh grid points. Note that the confinement strength ω1 is directly related to the QD radius, r0=ħ/(m∗ω1). For weakly confined electrons (large QD), accurate results can be obtained using fewer mesh grid points, however, as the QD size decreases the number of mesh grid points must increase.

Fig. 3

Comparison of numerical integration results to analytic results based on a quartic electrostatic potential. (a) Onsite Coulomb energy U1 versus QD confinement energy, ω1. (b) Interdot Coulomb energy U12 versus dot separation, d. (c) Comparison of two integration methods, showing the average CPU time versus number of mesh grid points evaluated to obtain equivalent U1 results. The data points labelled ‘fast’ refer to an integration algorithm that evaluates mesh grid points only where the electron probability density is non-negligible

Electrostatically-defined QDs typically have radii on the order of tens of nanometers, giving ω≈1011 to 1012 rad/sec, which reduces the required number of mesh grid points and therefore the computation time compared to smaller QDs. Similarly, Fig. 3b shows the inter-Coulomb energy as a function of QD separation parameter, d, for two QDs with confinement ω1=ω2=5·1012 rad/sec. A greater number of mesh grid points is required for numerical accuracy as interdot separation d is increased, which is a feature of the mesh grid formalism used in the numerical polar integration.

The numerical integration in (7) involves a quadruple integral, with each variable being integrated over n points within the coordinate boundaries defining the double QD electrostatic potential. Therefore, the number of mesh grid points to be evaluated during the numerical integration grows as 4n. Figure 3c shows how the CPU time scales with the number of mesh grid points for the brute force numerical integration as well as an improved integration algorithm. The improved algorithm only evaluates mesh grid points in regions where the electron is likely to be located, which significantly reduces the overall mesh grid size and therefore the required CPU time.

3.2 Charge Stability Diagram

The effective parameter calculations in Sect. 3.1 were performed for a quartic electrostatic potential that was a function of two control parameters, i.e., the confinement strength ω and dot separation d. In realistic devices, besides the chosen 3D geometry, the control parameters are the plunger, V, and tunnel, W, gate voltage vectors that govern the potential landscape.

Fig. 4

Charge stability diagrams (CSD), where [n, m] labels the electron occupation numbers of the two dots. (a) Theoretical CSD based on a capacitive model in which μi(V1,V2) accounts for cross talk between QDs, while t12, U1, U2, and U12 are held constant. (b) CSD resulting from the numerical Hubbard model approach. The gate voltage-dependent Hubbard parameters are calculated based on realistic QD potentials from a finite–element Poisson solver. Two pairs of triple points (in red) are indicated; the left pair are in a regime of stronger tunnel coupling than the pair on the right

We determine the CSD for a realistic double QD device shown in Fig. 1c by first constructing a charge basis consisting of 16 charge states with total charge ranging from 0 to 4 electrons, with the Hubbard model restriction that no more than 2 electrons can occupy each QD. The Hubbard Hamiltonian is calculated and diagonalized to find the charge state with lowest energy. This calculation is repeated for all simulated voltage configurations.

In Fig. 4a, a theoretical CSD is constructed assuming a capacitive model [3] and quartic potential. In the model, μi(V1,V2) are functions of the plunger gate voltages for a double QD device, and t12, U1, and U12 are assumed to be constant and voltage-independent. Figure 4b shows the CSD resulting from the numerical Hubbard model approach outlined in the present work applied to realistic double QD potentials, such as the one shown in Fig. 1. Although the voltage ranges of the plunger gates are different, the general features shown in Fig. 4a and b are comparable. Cross-capacitance effects are evident as the sloped charge transition lines. There are two primary advantages in using our Hubbard model approach with realistic QD potentials to determine the CSD in Fig. 4b:

1.

Subtle voltage dependencies in the onsite and inter-Coulomb energy are captured, which can be seen in Fig. 4b by the varying charge transition line slopes and triple point separations.

2.

Strong voltage dependencies on the interdot tunnel coupling are captured, evident as variable smoothing of the charge transitions about the triple points for different regions of the CSD.

4 Conclusion

In this work, an algorithmic approach has been presented for calculating the effective parameters of a Hubbard Hamiltonian that models electrons in realistic linear QD arrays. This was used to identify regions of charge stability as a function of control voltages which captures features missed by other simplified models. Although the results presented are for a QD array of length two, the method generalizes to arbitrary-sized arrays. Further development is underway to perform larger QD array calculations for application to a node network surface code. These methods are a valuable tool within the larger toolbox of simulation methods applicable to the development of scalable spin qubit quantum processors.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_16

Vreman Stabilization for Nonlinear Greenshield’s Model for Traffic Flow

Monika Neda1  , Jorge Reyes2   and Jiajia Waters3  

(1)

University of Nevada Las Vegas, Las Vegas, NV, USA

(2)

Virginia Tech, Blacksburg, VA, USA

(3)

Los Alamos National Lab, Los Alamos, NM, USA

Monika Neda

Email: monika.neda@unlv.edu

Jorge Reyes (Corresponding author)

Email: reyesj@vt.edu

Jiajia Waters

Email: jwaters@lanl.gov

Abstract

This report investigates a stabilization method for the Lighthill-Whitham-Richards (LWR) model which utilizes the continuity equation to model the behavior of traffic. The LWR equations are combined with the Greenshield’s model for velocity resulting in a first order nonlinear PDE. It is known that the usual unstabilized finite element method (FEM) contains spurious oscillations for non-smooth solutions for this nonlinear PDE. Herein, Vreman stabilization, that involves filtering and deconvolution, is implemented for treatment of the shocks and the affects of higher order deconvolution are investigated.

Keywords

Lighthill-Whitham-Richards modelGreenshield’s velocity modelVreman stabilizationDeconvolutionFinite element method

1 Introduction

Traffic flow refers to the examination of the movement of vehicles from their starting points to their destinations, as well as the interactions among individual drivers. Traffic analysis can be consideblack at different scales: microscopic (individual vehicle interactions), macroscopic (fluid dynamics-like models), and mesoscopic (probability functions for vehicle distributions) [8, 18, 27]. The LWR model is a macroscopic model developed in the 1950’s [25] with variants still being the subject of current research [5, 9, 10]. Herein, traffic flow is consideblack to be an inviscid but compressible fluid (fluid-dynamic model) and the traffic flow variables: density ρ, velocity v, and flow f, were defined as continuous variables in time and space. According to this model, traffic flow f was defined as a function of density ρ and velocity v defined by

∂∂tρ(t,x)+∂∂xf(t,x)=0,

(1)

where ρ represents the traffic density of the vehicles which is related to the flux f and velocity v, according to the relation f=ρv. Assuming constant velocity, v¯, results in the linear form of this PDE for x∈Ω⊂R and t>0 is given by

∂∂tρ+v¯∂∂xρ=0.

(2)

Herein, we instead consider the Greenshield’s velocity model which assumes that velocity depends on density with the following linear relationship

v(ρ)=vf1-ρρm,

(3)

where vf is the free flow speed and ρm is the maximum jam density. The free flow speed vf represents the speed of the traffic when the density ρ is zero. The maximum density ρm is the traffic density at which speed of the traffic v is equal to zero. Due to the relation from Equation (3), the graph between the flux f and the density ρ assumes a concave shape, since ∂2f∂ρ2<0. This relationship between density, flow, and velocity is shown in Fig. 1. 

Fig. 1

Experimental relationship between density, flow and velocity based on LWR and Greenshield’s model

The LWR (1) with the Greenshield’s velocity model is given as

∂ρ∂t+vf-2vfρmρ∂ρ∂x=0.

(4)

The first-order hyperbolic system poses several challenges for the continuous Galerkin method. While optimal convergence is established on specific cases of structublack grids, in general, a decrease in accuracy by one power of the mesh width can be anticipated [13, 29]. On general unstructublack grids, in addition to the expected decrease in accuracy, the unstabilized FEM may generate spurious oscillations for non-smooth solutions [6]. Various stabilization methods have been proposed to address this issue, including the SUPG and SOLD methods [4, 20], subgrid stabilization methods [16, 17], and filter-based stabilization methods [7, 12, 14]. It is recognized that when solving the hyperbolic traffic flow PDE as described in Equation (4), oscillations manifest around the shock solutions. Here, we employ an FEM stabilization method introduced in [12], based on Vreman filtering [30].

This article is organized as follows. In Sect. 2 we cover preliminaries about the finite element method and introduce the algorithms. Section 3 gives the stability results of the algorithm’s of interest. Then we present a numerical simulation that supports the theoretical results in Sect. 4. We conclude our work in the last Sect. 5.

2 Notation, Preliminaries and Algorithm

Here we introduce notation, preliminaries results that are applied in the numerical analysis, and the algorithm.

Let Ω denote the domain, while the L2(Ω) norm and the inner product are denoted by . and (., .) respectively. The function spaces used are the closure of C∞, periodic functions on Ω in the H1 norm denoted by X:=H#1(Ω), which have been used in a similar fashion in [7, 10, 12]. Let Xh⊂X be the finite element subspace.

The continuous mean u¯∈X of u∈X is the solution of the PDE [15],

-δ2Δu¯+u¯=u.

Moreover, the discrete mean u¯h∈Xh of u∈X [26] is defined as the unique solution of

δ2(∇u¯h,∇vh)+(u¯h,vh)=(u,vh),∀vh∈Xh,

where the mean quantities are associated with the filtering length scale δ. We also define the averaging operator G:X↦X and the discrete averaging operator Gh:Xh↦Xh, respectively, Gu=u¯ and Ghuh=uh¯h.

There are numerous procedures for deconvolution, [3]. Our study is built upon the Nth van Cittert deconvolution operator that is computed by repeated filtering and it can be compactly defined as follows.

Definition 1

The Nth order van Cittert continuous and discrete deconvolution operators as DN:X↦X and DNh:X-h↦Xh are, respectively,

DN:=∑n=0N(I-G)n,andDNh:=∑n=0N(I-Gh)n.

(5)

The mathematical stability and accuracy properties of continuous and discrete averaging and deconvolution have been studied in [2, 11, 22–24, 26]. We present a lemma proved in [7] that is used in our analysis below.

Lemma 1

(Smoothing Property [7]) For any ρh∈Xh,

δ2▵hDNhρh¯h+δ∇DNhρh¯h+DNhρh¯h≤C(N)ρh,

where ▵h is the discrete Laplacian operator.

We now present the variational formulations for the LWR model with Greenshield’s linear velocity model given by 4 and we incorporate the Vreman stabilizing term into the formulation. This leads to a stabilized finite element variational representation of Eq. (4), which can be expressed as follows: Find ρh∈Xh such that ∀vh∈Xh

∂∂tρh,vh+vf∂∂xρh,vh-2vfρmb(ρh,ρh,vh)+χδ2∂∂xρh∗,∂∂xvh∗=0.

(6)

The new stabilization term, χδ2∂∂xρ∗,∂∂xvh∗, is inspiblack by Vreman [30]. Here ρ∗=ρ-ρ¯h with primary purpose of this term is to enhance accuracy while mitigating undesirable spurious oscillations, all while maintaining computational efficiency.This also introduces the dimensionless stabilization parameter χ, which is traditionally manually tuned to increase or decrease the affects of the added stabilization term.

From [21] we also define the trilinear term b(·,·,·):X×X×X↦R as

b(u,v,w)=13∫Ω((uv)′+uv′)wdx.

(7)

In [21, 28] it is proven that b is skew-symmetric, i.e. b(u,v,w)+b(u,w,v)=0.

For all numerical simulations presented in Section 4 we employ backward-Euler temporal discretization. The discretized finite element algorithm for time interval (0, T] is stated below

Algorithm 1

(Fully Discrete Method for Greenshield’s model) For n=1,2…,M find ρhn∈Xh such that ∀vh∈Xh

ρhn-ρhn-1Δt,vh+vf∂∂xρhn,vh-2vfρmb(ρhn,ρhn,vh)+χδ2∂∂xρhn∗,∂∂xvh∗=0.

(8)

3 Numerical Analysis

In this section, we delve into the stability analysis of the algorithms under consideration. The proof of stability is given by the following lemma.

Theorem 1

For a sufficiently small Δt solutions to the fully discrete Algorithm () are stable and satisfy

ρhM2+2χδ2Δt∑n=1M∂∂xρhn∗2≤Cρ02.

(9)

Proof

Choosing vh=ρhn in (8) and rearranging terms yields

1Δt(ρhn-ρhn-1,ρhn)-2vfρmb(ρhn,ρhn,ρhn)+χδ2∂∂xρhn∗2=-vf∂∂xρhn,ρhn.

(10)

Using Cauchy-Schwarz and Young’s inequalities for the first term results in

(ρhn-ρhn-1,ρhn)≤12ρhn-ρhn-1.

(11)

Since b is skew-symmetric the nonlinear term vanishes. The right hand side term also vanishes due to the following argument using the periodic boundary conditions and fundamental theorem of Calculus

-vf∂∂xρh,ρh=-vf∫Ω∂ρh∂xρhdx=-vf∫Ω12∂∂x|ρh|2dx=0.

(12)

Plugging in Eqs. (11) and (12) into (10) yields

12Δtρhn2-ρhn-12+χδ22∂∂xρhn∗2≤0.

Multiplying by 2Δt and summing from n=1⋯M gives

ρhM2+χδ2Δt∑n=1M∂∂xρhn∗2≤ρh02,

where ρh0 is taken to be the L2 projection of ρ0 which yield the result.    □

4 Numerical Experiments

Section 4 covers numerical simulations that support the theoretical result. We discuss how the algorithm was implemented and showcase simulation on a moving shock benchmark problem.

4.1 Algorithm Implementation

Our algorithm given by (8) can be written as: For n=1,2…,NT find ρhn∈Xh such that ∀vh∈Xh

1Δt(ρhn-ρhn-1,vh)+vf∂∂xρhn,vh-2vfρmb(ρhn,ρhn,vh)+χδ2∂∂xρhn∗∗,∂∂xvh=0.

(13)

Herein, ρhn∗∗ cannot be treated explicitly, so we used a fixed point iteration on the implicit terms. In the case N=0, ρhn∗∗=ρhn-2ρ¯hn+ρ¯¯hn, when ρhn is computed, and the pseudo-code is:

In the case N=1, ρhn∗∗=ρhn∗-D1ρ¯hn∗ where D1ρ¯hn∗=2ρ¯hn∗-ρ¯¯hn∗. Thus, ρhn∗∗=ρhn∗-2ρ¯hn∗+ρ¯¯hn∗. Continuing this process (recall that now ρhn∗=ρhn-D1ρhn), results in ρhn∗∗=ρhn-4ρ¯hn+6ρ¯¯hn-4ρ¯¯¯hn+ρ¯¯¯¯hn, where ρhn is computed. Hence, the pseudo-code will be identical except for lines 4 and 5, i.e.

The tolerance used in subsequent simulations was set to 10-8.

4.2 Shock Moving to the Right

We now investigate problem (14) where the parameters of the problem are chosen to insure a positive shock velocity. For the computational results, we use P2 elements in space for the simulations and a final time T=3. We assign the following parameters the values of ρm=0.5, vf=25, ρl=0.01 and ρr=0.03. We used h=200500=0.4 with Δt=0.005. All of the experiments are implemented in FreeFem++ version 4.4.3 [19] and data visualizations are done using Paraview version 5.7.0 [1].

ρt+vfρx-2vfρmρρx=0,-100≤x≤100,0<t≤3ρ(0,t)=ρl,t>0,ρ(x,0)=ρl,-100≤x<0,ρr,0<x≤100.

(14)

For the chosen ρl and ρr, the shock speed can be computed as

λ=f(ρr)-f(ρl)ρr-ρl=ρlvf(1-ρlρm)-ρrvf(1-ρrρm)ρl-ρr

where f(ρ)=ρv=ρvf1-ρρm. Plugging in the prescribed parameters, we have a shock speed of λ=5 resulting in the shock that moves to the right 5 units for every 1 unit of time t.

As it can be observed in Fig. 2, the unstabilized FEM (χ=0) is unable to numerically simulate the stationary shock and produces an undesirable amount of oscillations.

Fig. 2

Numerical results of Algorithm 1 with χ=0 at time t=1,3 with Δt=0.005, h=0.4, δ=h

In Fig. 3 we see the addition of the Vreman stabilization term results in a blackuction of the oscillations. Additionally we note that in the case when χ=0.25 the higher order deconvolution N=1 performs better than the N=0. This notable improvement diminishes as we increase to χ=1 with both N=0 and N=1 nearly removing all oscillations.

Fig. 3

Numerical results of Algorithm 1 (χ=0.25, Left and χ=1, Right) at times t=1,2,3 (Top to Bottom) with Δt=0.005, h=0.4, δ=h

5 Conclusion

We investigated numerical simulation studies for the LWR with Greenshield’s velocity model. We investigate the affects of adding a Vreman stabilization term and present a fully discrete finite element algorithm. Stability is proven for the fully discrete case and we showcase some successful numerical simulations. This is a promising result that warrants future study for cases where a continuous initial condition may lead to a shock developing in time.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_17

Computational Modelling of Heat Transport in Kinked Nanowires

Alexander N. Robillard1  , Graham W. Gibson1   and Ralf Meyer1  

(1)

Laurentian University, Sudbury, ON, Canada

Alexander N. Robillard (Corresponding author)

Email: an_robillard@laurentian.ca

Graham W. Gibson

Email: gw_gibson@laurentian.ca

Ralf Meyer

Email: rmeyer@cs.laurentian.ca

Abstract

Modelling and investigating thermal transport in nanosystems in detail can be computationally intensive. In this work, the Lattice Boltzmann Method is used to simulate heat transport in kinked nanowires: nanowire systems with a bend. The Lattice Boltzmann Method serves as a computationally inexpensive method to examine kinked nanowires and compare them with the results of simulation methods used in a previous work. The method provides an effective way to study individual factors such as system geometry, surface parameters and transport lengths and may be used as a tool for understanding transport effects.

Keywords

NanowiresEngineered nanostructuresThermal modellingRadiative modellingLattice Boltzmann method

1 Introduction

Thermal transport in nanoscale systems is a critical problem with vast applications in the modern, semiconductor-device driven world. Recently, there has been an interest in the behaviour of heat transport when kinks or bends are introduced into nanowire systems [1, 6, 10]. Such geometric manipulations add another factor influencing transport to nanoscale systems. While the effects of system roughness, dimension, temperature, impurities and defects are oft-studied, bending or kinking produces its own effects on transport and, furthermore, may couple interestingly with the aforementioned ones.

Previous works [1, 8] have focused on the introduction of one or more 90∘ bends into nanowires and the effects on transport thereof. Recently, we have studied the effects of kink angles other than 90∘ using 3D Molecular Dynamics simulations and 2D phonon Monte Carlo simulations [9]. In this work, we build on our previous work [9] by using the Lattice Boltzmann Method (LBM) to study various kink angles. We will show that this method will allow the reproduction and investigation of the main effects of bulk Mean Free Path (MFP) and surface specularity with little computational effort.

2 Computational Methods

The LBM can be used to find an approximate solution to the Boltzmann Transport Equation within the relaxation time approximation. It simulates distribution functions of carriers flowing on a discretized lattice. For heat transport, the carriers are phonons. While the LBM is general and can be used for a wide variety of carriers, there have been more and more recent studies focusing on the behaviour of phonons [3–5].

This method only allows for a small set of discrete velocities, typically chosen so that during one time step, particles move from one lattice cite to the next. As suggested in [7], we use the D2Q7 scheme for the LBM: the lattice is built in 2 dimensions with 6 discretized velocities (see Fig. 1 for a simplified example: simulated systems are much more finely grained). The hexagonal structure of the lattice may result in imperfect reconstruction of the shape for different angles, though with a sufficiently fine simulation resolution the discrepancies are minimal. As in [7], we use the gray phonon approximation, hence only one group velocity is simulated. This results in a very easy to implement method that requires few details of the phononic properties of the system.

Fig. 1

Simplified example sketch of 2D lattice on 45∘ wire for D2Q7 LBM, indicating 6 velocities

Implementing the LBM involves iteration using the Bhatnagar-Groos-Krook method [2]:

fi(x+viδt,t+δt)-fi(x,t)=-α(fi(x,t)-feq(x,t))

(1)

where α is a scattering rate. fi is a distribution function of carriers travelling with a velocity vi at position x. The fi evolve as a function of neighbouring distribution functions and some equilibrium distribution, feq. In this case, feq has the carriers equally distributed among all 6 velocities. The scattering rate is related to the bulk MFP of carriers, lMFP∝α-1. Thus, changing α in simulations is equivalent to changing the MFP. A large scattering rate results in a small MFP, and vice versa. The ratio of the MFP to characteristic system lengths (often called the Knudsen number) is useful for understanding the critical behaviours of heat transport in a system. Note that α affects only the bulk MFP: a path length that carriers on average travel before scattering, but not including restrictions due to dimensions of the system. α does not influence how surface scattering changes the MFP. This is a simulation parameter that is related to the simulated mechanics of heat transport, not the overall MFP that is reduced due to the effects of surface scattering.

Our implementation allows control over the specularity of surface reflections. We use two specularities in this work: 0 (fully diffuse) or 1 (fully specular). A fully diffuse surface specularity redistributes incident carriers equally among all valid velocity directions near the surface, while a fully specular one reflects all incident carriers based on the direction of incidence in a mirror-like reflection.

Figure 2 shows a sketch of the general system design for the kinked nanowires, including terminology that simplifies discussion of the systems. The wires consist of straight segments of length l≈510.64 lattice constants joined to angled segments of length 2l. These are joined by intersecting rectangles, which results in wider cross-sections at the bends and knee depending on the kink angle. The kink angle, θ, denotes the angle of deflection from a straight wire. The wires have a width of 2r, with r=l/3. Distributions fi throughout the simulation are initialized to a normalized value of 1. From the distributions of carriers, the temperature of a given point is

T(x,t)=∑ifi(x,t)6

(2)

Note that the simulation is unitless, including distribution functions. Consequently temperatures are merely averages of the distributions of each velocity direction.

At each end of the wire are thermostatted regions Th=1.1 and Tc=0.9. Thermostats are implemented by resetting the distribution functions at each time step according to the temperature. Simulations are run until a steady-state is reached. System dimensions and number of lattice constants are built to reflect systems simulated in [9].

Fig. 2

Sketch of kinked wire system design with nomenclature

3 Results

The LBM yields insightful temperature and heat flux data at all points throughout the simulation. Heat flux is the more useful quantity for investigation of heat transport using colour maps. While temperature colour maps must span the entire range of system temperatures on their colour bar, the heat flux, being the gradient of temperature, can predominantly show areas of variation of the flux over their colour range.

3.1 Effects of Mean Free Path

Figure 3 shows heat fluxes for systems with a 45∘ kink angle, for different specularities (either fully specular or fully diffuse) and scattering rates. The scattering rates are chosen so that the MFPs are approximately 4r, 2r, and r, which correspond to twice the wire diameter, equal to the wire diameter and half the wire diameter, respectively. In general the heat flux is not uniformly distributed through the system, and is in fact along a path that may be narrower than the diameter of the wire itself. This narrow heat path is also seen in similar systems using phonon Monte Carlo simulations [9].

This effect is due to a number of phenomena, the first of which we will discuss is the “corner-cutting” effect. This effect has been seen in other works [1, 8] and is an expected result from classical heat transport, as can be seen in results from [9]. Essentially, a larger fraction of the heat flow takes a shorter path through the system, flowing tightly around the corners and bends in the system. This effect becomes more evident as the scattering rate becomes larger (MFP becomes smaller) in simulated systems: heat flows more diffusively and as such is able to react more quickly to changes in the geometry. The converse effect to this is the overshooting of heat around corners that occurs when the scattering rate, α, is small. In this case, heat travels a comparatively large distance with minimal scattering in the central volume of the system, and so reacts slowly to local variation of the temperature. This results in shadowed regions around the corners where the heat flux is low. In addition to the shadowing on the inside of the angle, overshooting results in corresponding areas of increased heat flux on the outside of the angle. Essentially, when transport is more ballistic (long MFP), overshooting results in shadowing around the corners and the corner-cutting effect is reduced. When transport is more diffusive (short MFP), corner-cutting becomes prevalent. These effects are also clearly seen in the previous work of [9], where simulations emphasizing ballistic behaviour show overshooting and shadowing while those that are more diffusive show stronger corner-cutting.

It is worth noting that due to the finite number of transport directions in the LBM (6 velocity vectors in our case), there are strong “beam-like” behaviours in the LBM results. Transfer of heat is reconstructed more effectively along these directions, which can exaggerate “optical” phenomena like long distance beams and reflections. This is somewhat reminiscent of a result from [9], where 3D molecular dynamics simulations found relationships between heat transport and the orientation of the lattice in the kinked portion of the wire. Furthermore, the shadowed regions and overshooting here conform much more strongly to angles than in the previous work. In methods that allow more direction for transport, the regions are much more radial.

Fig. 3

Heat Flux Colour Maps for 45∘ kinked systems. Top row: systems with fully diffuse surface scattering. Bottom row: systems with fully specular surface scattering. MFP for systems is decreasing to the right

Also seen in Fig. 3 is the effect of surface specularity. Diffuse reflections at the surface result in a layer of low flux along the surface of the system. Such behaviour is also seen in the previous results of [9]. Essentially, when the surfaces scatter diffusely, there is a more uniform distribution of heat flux along the allowable directions near the surfaces, which washes out any strong directionality in the heat flux incident on the surface and slows transport near the surface. Conversely, specular surfaces result in a more continuous heat flux along the surfaces. The scattering rate tends to somewhat change the behaviour of the low flux regions for diffuse surfaces. When the scattering rate is high, the low flux regions become prominent.

3.2 Effects of Kink Angle

Changing the kink angle in these systems can have a significant effect on the transport phenomena. Both the MFP and the specularity of the system can interact with the kink. Figures 4 and 5 show the flux colour maps for systems with 30∘ and 60∘ kink angles, respectively. Both sets of systems exhibit similar trends as the 45∘ systems when the scattering rate and specularity are changed. Corner-cutting is more prominent for higher scattering rates (shorter MFPs), while for lower scattering rates (longer MFPs) overshooting and shadowing are more evident. For systems with diffuse surface scattering, higher scattering rates result in more prominent flux reduction at the surface.

The size of the low flux region in the knee of any kinked system is tied to the kink angle: at the steady state, a significant portion of the heat can flow through a minimized path through the system. The low flux region also depends on particularities of the geometry. For the 30∘ systems when the surfaces are reflective and the scattering rate is low, heat scattering off of the system boundaries travels into the far corner of the knee. For the 60∘ system, due to the system geometry, very little heat flows through the corner of the knee, even with the same scattering rate. Similar changes of the low and high flux areas with the kink angle have been observed in phonon Monte Carlo simulations [9]. However, the corner-cutting effect remained more visible in the previous work than in our results here. The reduction of the corner-cutting here appears to be caused by the limited number of velocity directions visible as the beam-like streaks.

Once again, due to the 2D hexagonal lattice and its discretized velocities, there are symmetry effects for different kink angles. Certain angles (like 60∘) have their kink along one of the velocity directions which may strongly influence thermal transport. For instance, the 30∘ systems have overall larger fluxes throughout the system, while the 60∘ strongly emphasize corner-cutting effects. There is a broad trend of higher flux through systems with low kink angles also seen in [9].

Fig. 4

Heat Flux Colour Maps for 30∘ kinked systems. Top row: systems with fully diffuse surface scattering. Bottom row: systems with fully specular surface scattering. MFP for systems is decreasing to the right

Fig. 5

Heat Flux Colour Maps for 60∘ kinked systems. Top row: systems with fully diffuse surface scattering. Bottom row: systems with fully specular surface scattering. MFP for systems is decreasing to the right

4 Discussion

The LBM used provides insight into heat transport phenomena seen in a previous work [9]. Regions of high and low heat flux seen in phonon Monte Carlo simulations also occur in LBM simulations, which use a much less complete picture of phonon transport. Corner-cutting, low flux shadowed regions, ballistic overshooting and reduction of flux along the surface can be seen both in the figures shown here and in [9]. The trend of these effects with the length of the MFP shown here broadly reflects similar investigations done in that previous work. Some of these effects have been previously documented in other works [1, 6, 8]. The classical effect of corner-cutting for the flux dominates when transport is more diffusive in nature, while when ballistic effects are dominant corner-cutting becomes less visible and instead overshooting and shadowed regions appear.

Qualitatively, the trends of thermal conductance of kinked systems with angles ranging from 0 to 70∘ are similar for both the LBM used here and the results of the previous work. A more detailed comparison remains difficult as the limited number of velocity vectors in the LBM has strong interplay with geometric effects of the kink angle, which results in artifacts in the calculated thermal conductance.

For lower scattering rates, the limited set of velocities can exaggerate the appearance of beam-like behaviour in the LBM. As the methods used in this work use a singular phonon group velocity and only 6 discretized velocity directions, reconstructing a highly detailed picture of heat flow is not possible. Despite these limitations, the LBM has an inherent strength: it allows for very fast investigation of different combinations of MFP and specularity. Transport modes of interest can be examined based on the desired application or thermal phenomena of interest and which transport lengths may be most impactful. Furthermore, precise details of phonon properties are not needed for this method. Larger systems can be investigated relatively quickly, and when properly implemented, entire sets of parameter combinations can be simultaneously investigated over the course of hours.

5 Conclusion

Overall, the LBM implemented in this study was able to bring qualitative understanding to heat transport phenomena that have previously been examined through more computationally intensive techniques. The different regions and behaviours of the heat flux seen in phonon Monte Carlo simulations of kinked nanowires are visible even in simpler LBM simulations. As discussed, there are limitations with the simplicity of the LBM, but that it captures many of the effects well highlights the value this much less computationally intensive method.

Further study might investigate more sophisticated representations of phonon transport that maintain the low computational cost of the LBM. The effects shown here also raise interesting questions about deliberate inclusion of defects or other materials phenomena in specific locations in a nanosystem. Identifying regions of high flux can be useful for identifying where phenomena like inclusions might have the largest impact.

References

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Anufriev, R., Gluchko, S., Volz, S., Nomura, M.: Quasi-ballistic heat conduction due to Lévy phonon flights in silicon nanowires (2018). arXiv:​1809.​09808 [cond-mat, physics:physics]

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Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94(3), 511–525 (1954)

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Chattopadhyay, A., Pattamatta, A.: A comparative study of submicron phonon transport using the Boltzmann transport equation and the lattice Boltzmann method. Numer. Heat Transf., Part B: Fundam. 66(4), 360–379 (2014)

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Chattopadhyay, A., Pattamatta, A.: Energy transport across submicron porous structures: a lattice Boltzmann study. Int. J. Heat Mass Transf. 72, 479–488 (2014)

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Guo, Y., Wang, M.: Lattice Boltzmann modeling of phonon transport. J. Comput. Phys. 315, 1–15 (2016)MathSciNet

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Heron, J.S., Bera, C., Fournier, T., Mingo, N., Bourgeois, O.: Blocking phonons via nanoscale geometrical design. Phys. Rev. B 82(15), 155,458 (2010)

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Nabovati, A., Sellan, D.P., Amon, C.H.: On the lattice Boltzmann method for phonon transport. J. Comput. Phys. 230(15), 5864–5876 (2011)MathSciNet

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Park, W., Romano, G., Ahn, E.C., Kodama, T., Park, J., Barako, M.T., Sohn, J., Kim, S.J., Cho, J., Marconnet, A.M., Asheghi, M., Kolpak, A.M., Goodson, K.E.: Phonon conduction in silicon nanobeam labyrinths. Sci. Rep. 7(1), 6233 (2017)

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Robillard, A.N., Gibson, G.W., Meyer, R.: Thermal transport in kinked nanowires through simulation. Beilstein J. Nanotechnol. 14(1), 586–602 (2023)

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Zhang, Q., Cui, Z., Wei, Z., Chang, S.Y., Yang, L., Zhao, Y., Yang, Y., Guan, Z., Jiang, Y., Fowlkes, J., Yang, J., Xu, D., Chen, Y., Xu, T.T., Li, D.: Defect facilitated phonon transport through kinks in boron carbide nanowires. Nano Lett. 17(6), 3550–3555 (2017)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_18

Telling Oil Temperature for Frying from Audio and Video Signals Based on Multimodal Learning

Maksym Veremchuk1  , Zhao Pan1   and K. Andrea Scott1  

(1)

University of Waterloo, Waterloo, ON, Canada

Maksym Veremchuk (Corresponding author)

Email: mrveremc@uwaterloo.ca

Zhao Pan

Email: zhao.pan@uwaterloo.ca

K. Andrea Scott

Email: ka3scott@uwaterloo.ca

Abstract

The paper investigates the classification of oil temperature ranges during the frying process through the utilization of audio and video modalities, as well as their fusion. Data was collected in a kitchen, using smartphone audio and video sensors. Wet chopsticks were used as a controlled stimulus to cause a bubble production in the hot oil, with temperatures ranging from 140 to 240 ∘C in 10 ∘C increments. Corresponding data were recorded at each temperature interval. Pre-processing of the collected audio and video data was performed to eliminate irrelevant information, facilitating the training of a deep learning model. The findings highlight the significance of bubbles, including their number and size, as main features for classification, with a more significant presence of numerous and more giant bubbles observed at higher temperatures. Ultimately, the study demonstrates that the accuracy of audio and video-only modalities can be enhanced by applying multimodal learning techniques.

Keywords

Multimodal machine learningDeep fryingSequential data processingVideo and audio classificationTemperature measurement

1 Introduction

Frying is a common cooking technique shared by all cultures, and is considered to be one of the oldest methods of food preparation. Oil serves as the agent for heat transfer and flavour carrier. In addition, a well-controlled process of frying can give food aperitive textures. Besides cooking time, oil temperature is perhaps the most important parameter for frying. There are distinct recommended cooking temperatures for different cuisines. For example, for tempura in Japanese cuisine, the recommended oil temperature is around 170–180 ∘C; while in Chinese cuisine, for most stir-fry dishes, the oil temperature should be around 190–200 ∘C [1]. However, french fries are cooked with an oil temperature of 160–170 ∘C [2].

The culinary compendium often classifies the oil temperature into ten ranges from room temperature to smoking point. These temperature ranges were classified as appropriate for specific types of cuisine. For example, 140–160 ∘C temperature range is good for the initial stage of french fries cooking, 160–180 ∘C is good for falafel, 180–200 ∘C usually used for doughnuts, and frying at higher than 200 ∘C is considered unhealthy [3]. A resolution of 20 ∘C is precise enough for common cooking practices.

One may wonder how a chef measured the oil temperature before cooking thermometers were invented. In fact, a temperature sensor does not have to be complex. For example, a cooking trick has been widely applied in at least East Asia, which consists of dipping a chopstick into the hot oil and watching the bubbles around the chopstick and/or listening to the noise. With this technique, an experienced cook can tell the oil temperature with sufficient accuracy. The exact origin of this cooking tip is unknown, as it has been present for an indeterminate period of time. However, it has been passed down from generation to generation as ancestral knowledge and one can find different versions of the tip in many Chinese cookbooks. Despite the inconsistency in the knowledge, it is suggested that either auditory or visual perception of the bubbles can tell the temperature. The combination of these two sensing mechanisms may even provide more accurate measurements. In the context of human sensory perception, it has been observed that the auditory and visual modalities exhibit analogous characteristics. Specifically, individuals demonstrate enhanced estimation abilities when integrating auditory and visual cues compared to relying solely on either sensory modality in isolation. This ability is perhaps related to the McGurk effect, where people watching the same video but with different audio could hear different syllables [4].

The human brain in some sense is a neural network with biological sensors. The practice of multi-modal learning in classrooms suggests that using different senses when learning helps the students remember and understand information better. In the context of artificial intelligence, multi-modal deep learning (MMDL) includes co-learning when learning from two or more modalities. This contextually connected information from several sources occasionally gives one another the complementary details, showing aspects that are not visible when working with separate modalities. MMDL models incorporate heterogeneous data from several sources, enabling predictions that are more accurate. A driving force behind MMDL research continues to be the extraction and presentation of pertinent information from multimodal data. The challenge of combining modalities to increase effectiveness remains attractive. Furthermore, the accuracy and flexibility of multimodal systems are not optimum due to the insufficiency of the labeled data [5]. The present study aimed to investigate the use of multiple modalities, including audio, video, and their fusion, to classify oil temperature ranges during the frying process.

2 Methodology

2.1 Collection of Data

Experiments were conducted to collect data for hot oil temperature prediction using smartphone audio and video sensors (12 megapixel camera with f/1.8 aperture) in a kitchen environment. Moist chopsticks were dipped into the hot oil to generate the bubbles and the associated acoustic signal. The oil was heated in a pot, and the temperature was monitored using an instant cooking thermometer with a precision of 0.1 ∘C. The temperature of the oil was varied from 140 to 240 ∘C in increments of 10 ∘C. Both video and audio data were recorded at each temperature increment. This experimental setup allowed for the collection of a comprehensive dataset that captures changes in bubble production in different temperatures of oil. Examples of the data are shown in Fig. 1 for video and Fig. 2 for audio. The complete data set is available via the link: https://​drive.​google.​com/​drive/​folders/​1yiTCsNF1W-Ulf6Z1-U2o5rSayiGLWvfy?​usp=​sharing.

Fig. 1

Example frames from the video show the high- (top) and low-temperature oil (bottom). Bubble production becomes faster in the hotter oil, resulting in a bigger number of bubbles

Fig. 2

Example of Audio wave: high temperature (left) and low temperature (right). Different amplitudes and numbers of bubbles make distinctive features

2.2 Preprocessing and Cleaning of Collected Data

To ensure that significant information was extracted for training a deep learning model for hot oil temperature prediction, the acquired audio and video data were preprocessed and cleaned in the following steps:

1.

Frame Extraction: To extract the visual cues, the video data was divided into individual frames.

2.

Frame Reduction: Only the minimum number of frames was used to keep the necessary information. As the difference between the neighbouring frames was not significant, in order to reduce the amount of data while keeping the key visual elements, the frame rate was decreased to 20. We chose the frames using stratified sampling. Experiments showed that increasing the number of frames does not increase overall accuracy.

3.

Frame Cropping: To retain all significant visual elements, such as the bubbles, within the frame, the frames were cropped to a size of 224 by 224 pixels. This emulates the fact that when a chef practices the chopstick-based temperature estimation, his or her attention is focused on the small bubbling area on the free surface of the oil.

4.

Grayscale Transformation: To decrease the amount of data needed to train the deep learning model while keeping the crucial visual attributes, the frames were converted to grayscale. Color (associated with the visible light spectrum) does not contain significant information regarding the physics of the problem, and the number and shape of the bubbles and other visual cues do not change when the frames are converted to grayscale. This conversion also reduces the number of channels in the input data from three to one, which decreases the computational cost as well.

Through different time-frequency representations, including spectrograms, visual displays of an audio signal can provide a rich picture of the temporal and spectral characteristics of the original signal. To further pre-process the collected audio data for training the deep learning model, it was transformed into a Mel spectrogram as it shows better performance among other representations: Mel-frequency cepstral coefficients (MFCC), continuous Wavelet transform (CWT), constant-Q transform (CQT), and short-time Fourier transform (STFT) [6]. The Mel spectrogram is carried out by the Fourier transform [6].

2.3 Deep Learning Model for Audio, Video, and Data Fusion

We used a combination of audio and video data for our deep learning method to forecast the temperature of hot oil in a kitchen setting. To begin, we used a pre-trained ResNet18 model for video analysis, which showed the best performance among simple LSTM, convolutional LSTM [7], 3D ResNet [8] and pre-trained VGG [9]. This involved passing each video frame through the ResNet18 model and stacking together all output feature vectors. We then used three fully connected layers (2048×2048×4) to output the predicted class.

We used four convolution layers and three fully connected layers to extract features from the mel spectrogram for audio analysis. The main architecture and hyperparameters for the model were inherited from [10]. The final fully connected layers were taken to be the same as those for the video model.

We evaluated various data fusion methods to combine the audio and video data. For decision fusion, we simply added up the softmax layers produced by the audio and video models. For raw fusion, the resized audio mel spectrogram was added to a run of video frames before being processed by the video model. For feature fusion, the feature vectors from both models were stacked before being passed through the fully connected layers. In this case, the same number and size of fully connected layers as the single-modality models [11] was used. Models architecture shown in Fig. 3.

Fig. 3

Different types of modality fusions: raw fusion (top), feature fusion (center), late fusion (bottom)

2.4 Feature Extraction from the Audio and Video Data

We created feature maps (see Fig. 4) from the deep learning model during the feature extraction process from the audio and video data to visualize the features the model learns for high (200 ∘C) and low (140 ∘C) temperatures. This gave us insight into the features that the algorithm considers important when categorizing the oil temperature.

For the video data, the feature maps show that the presence of bubbles as well as the size and number of bubbles were significant features for classification. The model was able to accurately identify the presence of bubbles and their size in the images, and use this information to predict the hot oil temperature. The bubble distribution was also distinct between high and low temperatures, with more and larger bubbles present at higher temperatures, as demonstrated by the feature map in Fig. 4. This observation recovers the qualitative intuition that high-temperature frying generates more bubbles by more intense evaporation of the moisture in the porous media of the chopsticks. The relevant physics can be found in [12].

Fig. 4

High (left) and low (right) temperature video feature maps

Fig. 5

High (top four rows) and low (bottom four rows) temperature audio feature maps. Red boxes highlight the example of distinctive features—loud bursting bubbles

The feature map for the audio data revealed that the bursting of large bubbles is a crucial characteristic for classification (see the top image of Fig. 5). The left image of the wave in Fig. 2 illustrates the same audio data as the corresponding feature map in Fig. 5, where the network identified loud sounds (1.6 and 2.4 s in Fig. 2). In contrast, the right images in Fig. 2 show the absence of these loud bubbles, and the corresponding feature map in the bottom image of Fig. 2 does not display any distinctive features. Thus, the model can predict the oil temperature using the audio signal that corresponds to the sound of bursting bubbles.

Additionally, the feature map indicated a difference between high and low temperatures in the frequency content’s number of high peaks. Higher temperatures showed more and taller peaks, as illustrated in the feature map in Fig. 5, with an example of these peaks highlighted by the red boxes.

The actual physics is little known and preliminary investigation can be found in [12]; however, this does not prevent MMDL or humans from telling the temperature by listening to the noise from frying. Overall, the feature maps gave us insight into the key features the algorithm considers when classifying the hot oil temperature. By implementing these various data fusion techniques, we were able to improve the accuracy of our temperature prediction model as compared to using a single modality.

3 Experiments and Results

The models were trained and tested on a dataset of 137 video samples (34, 27, 28 48 samples of each class respectively), split into 60% for training, 30% for testing, and 10% for validation.

In order to select the optimal multimodal architecture for our specific problem, we propose to evaluate multiple approaches using two performance metrics. The first metric, commonly referred to as accuracy, measures the proportion of temperature measurements that are correctly classified by the model. This metric was calculated by dividing the number of correctly classified temperatures by the total number of temperature measurements. The second performance metric that we propose to use is a measure of the classifier’s ability to predict misclassified labels as close as possible to the true temperature class. Given that our temperature classes are ordered, it is more desirable for the classifier to mistake neighbour classes rather than those classes that are further apart. This metric will be calculated by measuring the distance between the predicted and true classes and ranking these distances based on the degree of similarity between the classes. For this, the mean squared error (MSE) could be a good choice.

Table 1

Accuracy and MSE metrics on different approaches

Metrics

Audio

Video

Feature fusion

Decision fusion

Raw fusion

Accuracy

83.3

61.9

85.7

74

83.3

MSE

0.2381

0.5238

0.1429

0.3333

0.2381

Feature fusion outperforms the other fusion models as well as the audio-only and video-only models (Table 1). The video-only model could not outperform the audio-only model.

This is perhaps due to the higher level of variability in the video data, and the strong sensitivity of the video data to the wetness of the chopsticks.

At the same time, audio data is not so dependent on the wetness as the values of the feature map would have the same relation but with different amplitudes. We compare the audio-only and feature fusion models, as our best single and multimodal models, using k-fold cross-validation. It can be seen the feature fusion model performs better than the audio-only model for four out of five folds (Table 2).

Table 2

K-fold for audio model and feature fusion model

Feature fusion

1 fold

2 fold

3 fold

4 fold

5 fold

Average

0.7857

0.7857

0.8889

0.9259

0.8889

0.8550

Audio

1 fold

2 fold

3 fold

4 fold

5 fold

Average

0.8214

0.7500

0.8148

0.7778

0.8889

0.8105

From Fig. 6 we can see that in the multimodal feature fusion confusion matrix for all available classes, the elements are closer to the main diagonal compared to the audio- and video-only learning. This means that more labels are correctly predicted by the multimodal feature fusion.

Fig. 6

Comparison of Audio (top left) and Video (top right) confusion matrices with multimodal confusion matrix (bottom)

As we can see, the prediction accuracy for “falafel” and “doughnut” is lower, and it is because they are more likely to be confused with each other. This confusion arises because they both have a moderate number of bubbles, compared to the high number in the “unhealthy” class and the low number in the “french fries” class. Furthermore, the lower prediction accuracy can also be attributed to the smaller number of data samples for “falafel“and “doughnut“compared to “french fries” and “unhealthy”, partially contributing to the confusion.

4 Conclusion

In conclusion, this study focused on the creation of a dataset of audio and video data for an experiment aimed to classify different ranges of frying oil temperatures in the kitchen. A comparison of different data fusion models for classifying oil temperature during the frying process were evaluated. Our findings revealed that by using multiple modalities, including audio and video, and their fusion, we can improve the accuracy of oil temperature classification during the frying process. Particularly, we discovered that the feature fusion model worked better than other data fusion models and could produce more accurate classification than single modalities. Overall, our findings demonstrate the potential of using multiple modalities and their fusion to improve the accuracy of models. This approach has potential applications in the food industry for ensuring consistent quality in fried food products. Further research could investigate the use of other modalities and their fusion for improving classification accuracy.

References

1.

McGee, H.: On Food and Cooking. Scribner (2004)

2.

Carrapiso, A.I., Otero-Bazago, E.M., Gil-Amado, J.A., Martín, L.: The effect of deep frying French fries and two types of Iberian pork on the characteristics of extra virgin olive oil and the fried products. J. Foods (2022). https://​doi.​org/​10.​3390/​foods11213394Crossref

3.

Here’s How Long to Fry Every Food, Available via DIALOG. https://​www.​tasteofhome.​com/​article/​deep-frying-temperature-chart/​

4.

Nath, A.R., Beauchamp, M.S.: A neural basis for interindividual differences in the McGurk effect, a multisensory speech illusion. J. NeuroImage (2011). https://​doi.​org/​10.​1016/​j.​neuroimage.​2011.​07.​024Crossref

5.

Summaira, J., Li, X., Shoib, A.M., Abdul, J.: A Review on Methods and Applications in Multimodal Deep Learning (2022). ArXiv:​2202.​09195

6.

Huzaifah, M.: Comparison of Time-Frequency Representations for Environmental Sound Classification using Convolutional Neural Networks (2017). ArXiv:​1706.​07156

7.

Shi, X., Chen, Z., Wang, H., Yeung, D., Wong, W., Woo, W.: Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting (2015). ArXiv:​1506.​04214

8.

Tran, D., Wang, H., Torresani, L., Ray, J., LeCun, Y., Paluri, M.: A Closer Look at Spatiotemporal Convolutions for Action Recognition (2017). ArXiv:​1711.​11248

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Simonyan, K., Zisserman, A.: Very Deep Convolutional Networks for Large-Scale Image Recognition (2014). ArXiv:​1409.​1556

10.

Inceoglu, A., Aksoy, E.E., Ak, A.C., Sariel, S.: FINO-Net: A Deep Multimodal Sensor Fusion Framework for Manipulation Failure Detection (2020). ArXiv:​2011.​05817

11.

Sun, J., Yin, H., Tian, Y., Wu, J., Shen. L., Chen, L.: Two-Level Multimodal Fusion for Sentiment Analysis in Public Security. J. Secur. Commun. Netw. (2021) https://​doi.​org/​10.​1155/​2021/​6662337

12.

Kiyama, A., Rabbi, R., Pan, Z., Dutta, S., Allen, J.S., Truscott, T.T.: Morphology of bubble dynamics and sound in heated oil. Phys. Fluids 34(6) (2022). AIP Publishing

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_19

Gravitational Models with State-Dependent Delay: Gravitating Binaries

Erik I. Verriest1  

(1)

Georgia Institute of Technology, Atlanta, GA, USA

Erik I. Verriest

Email: erik.verriest@ece.gatech.edu

Abstract

Einstein’s field equations are approximated, keeping a notion of simultaneity and of flat space, based on a finite propagation speed of the (Newtonian) gravitational force. This leads to the modification of Newton’s equation by an implicit state-dependent delay. We first present the gravitational potential due to a point mass in motion, prescribed by a trajectory, rp(t). The gravitational force at (r,t) is obtained as the (usual) gradient of the potential and a gravito-magnetic effect emerges. Next we predict the kinematics of gravitating binaries with unequal masses, thus generalizing our findings in [9, 12]. In particular, Newton’s results for the orbital speeds hold if the masses are replaced by the relativistic ones.

Keywords

State-dependent delayPost-Newtonian approximationGravitationRetarded fieldsBinary starsOrbital speedsRelativity

1 Introduction to Implicit State-Dependent Delay

Linear systems with fixed delay are well-studied, and transcend the theory of ordinary differential equations (ODE) to the realm of functional differential equations (FDE). The problem gets more involved progressively in the case of time-variant delays, and state-dependent delays where the delay is a functional of the unknown function itself. See [3]. Further complexity enters if such dependency is only implicitly given. An example is the soft landing problem described by Walther [13]. Consider a submersible descending to the bottom of the ocean. It sends out a sonar signal which bounces of the bottom and gets received by the vessel after the sonar transmission delay, at which time the vessel is at a new location specified by its dynamics r˙=f(r,u), where u is the control. With a time-stamped sonar signal, the observation at time t is the transmission delay τ(t) which satisfies

cτ(t)=y(t)-y(t-τ(t)),

where c is the speed of sound (see Fig. 1).

Fig. 1

Soft-Landing: The delay τ(t) is observed at time t, and one wants to update the state (vertical position and velocity) at t of the mobile unit from the known dynamics and τ(t)

Geometric and analytic approaches to such problems were given in [8, 10, 11]. A good approach to a problem starts by introducing a non-ambiguous and convenient notation. The dynamical equation presents the evolution of the system under its external influences as a controlled finite dimensional differential equation. Represent the states at time t as components of a vector r(t). An observation at time t, is usually denoted by y(t) and supplies information about the state r(t). Thus the classical systems model is of the form

r˙=f(r,u),y(t)=h(r(t)).

In the case of the implicit delay equation for the soft-landing problem, one can discern two formulations. The forward formulation is from the perspective of the transmission. If at time t′ the signal is transmitted from position r′, then it will be received at some future time t=tf at position r where

cτf(t′):=c(tf(t′)-t′)=|r-r′|,

is the forward delay. On the other hand, a signal that is received at time t at position r, was transmitted from r′ at time t′=tb(t) before t. The corresponding delay (with respect to the time of reception) is the backward delay satisfying

cτb(t)=c(t-tb(t))=|r-r′|.

Clearly, if r and r′ are fixed, τb(t)=τf(t′)=t-t′. Confusion is possible if one or both of r and r′ are function of time. To clarify, denote transmission events (emitted from r′ at time t′) by T(r′,t′), and reception events (at r and time t) by R(r,t). We denote the combined event of T(r′,t′)∩R(r,t) by [(r′,t′)→(r,t)]. The set of all forward (in time) events compatible with T(r′,t′) is the null cone with vertex (r′,t′)

FT(r′,t′)=[(r′,t′)→(r,t)]:|r-r′|=c(t-t′),t>t′.

Likewise, the set of backward compatible events from R(r,t) is the backward cone

BR(r,t)=[(r′,t′)→(r,t)]:|r-r′|=c(t-t′),t′<t.

In a space-time diagram with receiver fixed in time and space, the set B is a single cone with vertex at (r,t) and extending towards earlier times. This implies that for a moving transmitter rT(t) compatible transmission events are intersections of the backward cone BT(r,t) with the trajectory (world-line) (rT(s),s). Conversely, for a moving receiver rR(t), compatible transmitting events from a stationary transmitter are intersections of FT(r′,t′) with (rR(s),s). This implies:

Lemma 1

If |r˙T|<c, then BR(r′,t′) and FT(r,t) are singletons and for all t<t′ it holds that τ˙B(t′)<1 and τ˙F(t)<1.

We note that the conditions of the delays corresponds to the well-posedness condition for systems with time-variant delay [7]. Further applications of observability, observation, and Kalman-filtering occur in problems of sonar-based positioning and stream field estimation in the ocean, deep space network based spacecraft navigation, and mascon identification (Project GRAIL) on the moon. In what follows, we apply the approach to transmission and reception of "gravitational signals" in a post-Newtonian (pN) gravitation problem. See also [4, 5, 14] for other pN theories.

2 Physical Assumptions

Einstein’s tensorial field equations relating space-time geometry to matter and energy distribution are so unwieldy that solutions are only known in some very specific cases. A semi-relativistic approximation is desirable: One where space-time may still be considered as flat, but where Newton’s equations (with gravity acting instantaneously) are replaced by a post-Newtonian theory, involving the propagation of gravity at the speed of light. As this retardation depends on the geometry of the point masses, a dynamical system with state dependent delay results, where delay and state are implicitly related as in the signalling problem described in Sect. 1. Section 3 establishes the formulas for the retarded potentials.  From it, the retarded field, which determines the motion, is obtained as the gradient of the potential in Sect. 4. This results in a functional differential equation (FDE) with state dependent delay (See [3]). In particular, interesting new phenomena (entrainment, dynamic friction, orbital speeds) emerge from this pN theory.

Let at time t′ the mass distribution be represented by the scalar function ρ(r′,t′), the density field. This assumption requires already a notion of simultaneity and therefore cannot be consistent with the theory of relativity. We postulate that this density field generates a gravitational field,  which assumes at a fixed position, r, and time, t, the value due to the superposition of the fields of all infinitesimal contributions. However the contribution from r′ must have traveled a (straight line) distance |r-r′|, which takes a propagation time τ(r′)=|r-r′|/c. Hence only the past value, ρ(r′,t-τ(r′)), contributes to the field at r at time t. General relativity implies the warping of space and time, and hence geodesics (along which signals propagate) are not straight lines in general. Since our assumptions do not imply that c is a speed limit, we explicitly require v≪c. The following assumption will be approximately valid for weak fields and small speeds relative to c, the speed of light.

Assumption: Given the time varying mass density ρ(r′,t′), the gravitational potential at a fixed point r and time t is given by the linear superposition

ϕ(r,t)=G∫R3ρ(r′,t-τ(r′))|r-r′|dV(r′),

(1)

where G is the gravitational constant, and V(r′) denotes the volume element at position r′. This equation is coupled with an implicit position-dependent delay relation

∀r′:cτ(r′)=|r-r′|.

(2)

If ρ is continuous in its arguments, the field at a point r where ρ(r,·)≠0 can be obtained by taking the integral over an ϵ-small ball around r away, since that part will not contribute to the field by virtue of the asymptotic isotropy.

3 Retarded Gravitational Potential Due to a Point Mass in Motion

Let us now assume that the mass distribution is due to a unit point mass in motion, with prescribed trajectory, Xp:={(rp(s),s)}, not passing through r. This means that the mass density in (1) is given by a three-dimensional Dirac delta ρ(r′,t′)=δ(r′-rp(t′)). Under the assumption |r˙p|≪c, it follows from lemma 1 that there is only one event (r′,t′) on the trajectory Xp, so that at the fixed point and time (r,t):

ϕ(r,t)=G∫R3δ(r′-rp(tb))|r-r′|dV(r′),

(3)

where tb=t-τf(r′) is the retarded time satisfying (2). Since r is considered fixed, we omitted this argument for notational simplicity. We recall a theorem in [9] for which a simpler proof, analogous to the derivation of the Liénard-Wiechert potentials (see [6]) was later given in [8]. The main idea is the reduction of the three dimensional Dirac in (3) to a one-dimensional time Dirac.

Theorem 1

The gravitational potential of a unit point-mass in motion is given by

ϕ(r,t)=G|r-rp(tb)|1-1c1r-rp(tb)⊤r˙p(tb)

(4)

4 Retarded Gravitational (Force) Field

The gravitational (force) field of a mass M is the gradient of the gravitational potential, F(r,t)=∇rϕ(r,t), obtained in Sect. 2. First, consistent with the usual duality, gradients with respect to column vectors are represented as row vectors. Thus, the following three terms arrive from (4).

F(r,t)=-GM1-(r-rp(tb))⊤r˙p(tb)c|r-rp(tb)|1|r-rp(tb)|2∇r|r-rp(tb)|+-GM|r-rp(tb)|11-(r-rp(tb))⊤r˙p(tb)c|r-rp(tb)|2∇r1-(r-rp(tb))⊤r˙p(tb)c|r-rp(tb)|.

(5)

We use the well-known standard identity:

Lemma 2

∇r|r-r′|=-(r-r′)⊤|r-r′|.

(6)

We will also make use of the identities:

Lemma 3

For h(r): a differentiable scalar function of r, and sgn the sign function:

∇r|h(r)|=sgnh(r)∇rh(r).

(7)

Lemma 4

∇r1|r-r′|=-(r-r′)⊤|r-r′|3

(8)

It follows from these lemmas that for |v|<c,

∇r1-(r-r0)⊤v0c|r-r0|=∇r1-(r-r0)⊤v0c|r-r0|=-v0⊤c∇rr-r0|r-r0|.

Putting this all together, proves the following theorem [12].

Theorem 2

The gravitational field due to a moving mass M is given by

F(r,t)=-(r-rp(tb))⊤|r-rp(tb)|3GM1-(r-rp(tb))⊤r˙p(tb)c|r-rp(tb)|++GM|r-rp(tb)|211-(r-rp(tb))⊤r˙p(tb)c|r-rp(tb)|2r˙p(tb)⊤cI-(r-rp(tb))(r-rp(tb))⊤|r-rp(tb)|2.

(9)

The matrix between the square brackets is a projection operator (projection onto the plane perpendicular to the vector (r-r0)). If the gravitating mass moves in such a way that r˙p is parallel to (r-rp(tb)), then the second term in (9) is zero and the field is directed along r-rp(tb). Aligning the x-axis with this direction, force, position and velocities can be expressed in the x-coordinate. In particular, the x-component of the force acting on a unit mass particle at x is

Fx=-sgn(x-xp(tb))GM|x-xp(tb)|21-sgn(x-xp(tb))x˙p(tb)c,

(10)

where, as always, tb denotes the retarded time. Equation (10) displays two deviations from Newton’s law:  First the delayed position of the gravitating particle appears, and second, there is an aberration which depends on the (delayed) velocity.

As a caution, a naive generalization to Newton’s law in one dimension, of the form

Fx=-sgn(x-xp(tb))GM|x-xp(tb)|2,

instead of (10) cannot be correct. This is actually a good thing: For two equal mass binaries in a perfect circular orbit, the resultant of this naive gravitational force and the centrifugal force necessarily points outside the orbit. This invalidates the assumed existence of a stable circular orbit.

4.1 Field Expressed in Terms of the Predicted Position

Equation (9) lacks simplicity. A more appealing formula follows by introducing

r^p(t)=rp(tb)+τ(tb)r˙p(tb)

(11)

as a prediction (expected position) of the mass M in motion at time t, if for tb(t)=t-τ(tb)≤s≤t its velocity were held constant to vp(s)=r˙p(tb).

Theorem 3

The gravitational force exerted by a point mass M in motion is directed towards the predicted position based on a uniform motion given the delayed information (of position and velocity)

F(r,t)=∇rϕ(r,t)=-GM(r-r^p)⊤|r-r^p|3⏟=Newtonr-r^pr-rp11r-rp⊤1r-r^p2,

where 1r-rp⊤1r-rp^=cosϕ. with ϕ is the angle between the retarded and the predicted direction. See Fig. 2.

Fig. 2

Retarded and Predicted Position at time t

Proof

Reorganize (9) in the form

F(r,t)=GM|r-rp(tb)|211-(r-rp(tb))⊤|r-rp(tb)|r˙p(tb)c2rp˙(tb)⊤c+-GM(r-rp(tb))⊤|r-rp(tb)|31-(r-rp(tb))⊤|r-rp(tb)|rp˙(tb)c2×1-(r-rp(tb))⊤|r-rp(tb)|rp˙(tb)c+(r-rp(tb))⊤|r-rp(tb)|rp˙(tb)c.

(12)

With r^p from Eq. (11) it follows that

1-(r-rp(tb))⊤|r-rp(tb)|r˙p(tb)c=1r-rp(tb)|1r-rp⊤(r-r^p)|,

where 1r-rp=(r-rp(tb))⊤|r-rp(tb)| denotes the unit vector in the direction of r-rp. Relativity imposes |r˙p|<c, which with the Cauchy-Schwarz inequality reduces (12) to

∇rϕ(r,t)=-GM|r-rp(tb)|1r-rp⊤(r-r^p)2(r-r^p)⊤.

(13)

   □

Thus the gravitational force exerted by a point mass M in motion is directed towards the predicted position based on a uniform motion given the delayed information (i.e. delayed position and velocity). In other words, the post-Newtonian approximation implies the entrainment by a gravitational current (masses in motion).

4.2 Interpretation as a Gravito-Magnetic Effect

For a particle moving with uniform velocity, r^p(t) is the exact position of the particle at time t, r^p(t)=rp(t). By Theorem 3 a particle in motion will exert a force that is directed towards its future position, hence there is a force-component in the direction of the motion. It was shown in [9] that this implies that a gravitating rod of infinite extent generates a field due to the gravitational current it passes. As this is reminiscent of a magnetic field generated by an electric current (moving charges), we will call this a gravito-magnetic effect, although the analogy with EM theory is not exact. This gravito-magnetic effect gives a correction to the classical dynamic friction which slows a wandering star down when it enters a galaxy [2]. It also gives rise to a larger escape velocity for separating bodies [9]. In [12] it was shown that equal-mass gravitating binaries orbit with a larger orbital velocity than predicted by Newton’s theory. In the next section we derive the kinematics and dynamics for orbiting binaries of unequal mass, which recently have gained a lot of attention due to an observed anomaly [1].

5 Gravitating Binaries 

Consider two masses, M≥m, orbiting about their common center of mass, taken to be the origin. Let their orbits have radii R and r respectively, so that their separation is R+r. The equilibrium condition postulates RM=rm, hence R≤r. See Fig. 3.

Fig. 3

Geometry and force balance for orbiting binary with masses M and m

At time t, mass m ‘feels’ the force of the mass M when it is at its delayed position rp(tb). Equilibrium mandates that the centrifugal force, mvm2r, on m must be balanced by the (pN) gravitational force, which therefore is directed towards the origin since the predicted (or linearly extrapolated) position is diametrically opposite. This predicted position be r^p(t) is the intersection of the tangent at rp(tb) with the diameter Or. Let d denote the distance of this point from the origin O. With Theorem 3, this boils down to a simple problem in Euclidean geometry.

GMm(r+d)2r+dδ1cos2ϕ=mvm2r,

(14)

The geometric constraints in the upper right triangle of Fig. 3 are

dsinθ=vMτ,dcosθ=R.

Finite propagation speed of gravity yields the constraint cτ=δ, and the geometry in the lower triangle gives respectively by the cosine and the sine rule

δ2=R2+r2+2Rrcosθ,sinϕR=sinθδ.

Introduce the mass-ratio μ=mM=Rr≤1, and angular frequency parameters vMR=vmr=ω, where vM and vm are respectively the orbital velocities of M and m. Then if ϵ=defvmc≪1 by our assumption we find tanθ=ϵ1+μ2+2μcosθ, from which approximately

θ≈ϵ(1+μ),ϕ≈ϵμ,δr=1+μ2+2μcosθ,dr=μ1+ϵ2(1+μ2+2μcosθ).

(15)

Eliminating d,δ and θ from the above leads to the approximate relation between the orbital speed, vm, and r

vm2≈GMr(μ+1)21(1-ϵ2μ22)≈Gr(μ+1)2M1-ϵ2μ2.

(16)

Note that ϵμ=vMc so that if one defines the relativistic mass Mrel=M1-ϵ2μ2, then (16) coincides with Newton’s prediction for binaries. Likewise, one deduces the complementary (recall that ϵ=vmc)

vM2=Gμr(μ+1)2mrel,mrel=m1-ϵ2.

There are two special cases:

(i) If μ→0 ⇒ R→0, vM→0, and thus vm2=GMr, which is the Newtonian case. With a central force field M does not move and there is no entrainment.

(ii) The case μ=1 with circular orbit was reported in [12] and reduces to

vm2=Gm4r11-ϵ2=Gmrel4r,mrel=m1-vorb2c2,

(17)

It is interesting to see that the kinematics of a binary under the weak field and slow speed assumptions reduces to Newton’s if the (Newtonian) masses are substituted by the relativistic masses.

6 Conclusions and Beyond

We derived an approximation to general relativity under the assumption of weak fields and slow speeds which results in dynamical equations with implicit state-dependent delay. This post-Newtonian theory introduces a gravitational entrainment in the direction of the motion of the gravitating mass, which for gravitating binaries predicts an orbital speed exceeding the one predicted by Newton. Recent observation [1] has hinted at such an anomaly in gravitating binaries under low acceleration (i.e. large separation). As expected, stable orbits exist, which is not the case had one considered the naive generalization (delayed Newton’s law). The next step involves the generalized Keplerian laws of motion.

References

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Chandrasekhar, S.: Dynamical friction. I. General considerations: the coefficient of dynamical friction. Astrophys. J. 97, 255–262 (1943)

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Hartung,F., Krisztin, T., Walther, H.O., Wu, J.: Functional differential equations with state-dependent delays: theory and applications. In: Handbook of Differential Equations, vol. 3, pp. 435–545. Elsevier (2006)

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Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman (1973)

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Poisson, E., Will, C.M.: Gravity: Newtonian, Post-Newtonian. Relativistic. Cambridge University Press (2014)

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Smith, G.: Classical Electromagnetic Radiation. Cambridge University Press (1997)

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Verriest, E.I.: Inconsistencies in systems with time-varying delays and their resolution. IMA J. Math. Control. Inf. 28, 147–162 (2011)MathSciNet

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Verriest, E.I.: Inversion of state dependent delay. In: Karafyllis, I., Malisoff, M., Mazenc, F., Pepe, P. (eds.) Recent Results on Nonlinear Delay Control Systems, pp. 327–346. Springer (2016)

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Verriest, E.I.: Post-newtonian gravitation. In: Bélair, J., Frigaard, I.A., Kunze, H., Makarov, R., Melnik, R., Spiteri, R.J. (Eds.) Mathematical and Computational Approaches in Modern Science and Engineering, pp. 153–164. Springer (2016)

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Verriest, E.I., Lampe, B.P.: Range-only position estimation with delay correction via Lagrange-Bürman inversion, IFAC World Congress, Toulouse, France, July 9–14, 2017. IFAC-PapersOnLine 50(1), 2310–2316 (2017)

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Verriest, E.I.: Implicit state dependent delay in range-based position estimation and navigation. In: Marc Kilgour, D., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds.) Recent Advances in Mathematical and Statistical Methods, vol. 259, pp. 637–646. Springer (2018)

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_20

Self-similar Heat Transfer in a Turbulent Particle-Laden Free Flow

Hamid Reza Zandi Pour1   and Michele Iovieno1  

(1)

Politecnico di Torino, Dipartimento di Ingegneria Meccanica e Aerospaziale, Torino, Italy

Hamid Reza Zandi Pour (Corresponding author)

Email: hamid.zandipour@polito.it

Michele Iovieno

Email: michele.iovieno@polito.it

Abstract

The intermediate stage of the thermal interaction between two homothermal regions in a particle-laden turbulent flow is analyzed using data from direct numerical simulations (DNSs). The fluid and particle relevant thermal budget equations are provided and a self-similar analysis has been carried out. We show how particle thermal inertia introduces a lag in the dynamics which allows for self-similarity only in the asymptotic limit.

Keywords

Particle-laden flowsTurbulent mixingHeat transferFluid-particle thermal interactionSelf-similarity

1 Introduction

The thermal interaction between turbulent flows and suspended particles is a multidisciplinary topic with implications for a wide range of natural and industrial processes. Ongoing research continues to refine our understanding of the phenomenology of these interactions. The main contributions arrive from the analysis of fluid-particle thermal interaction in homogeneous and isotropic turbulence. Bèc et al. [1] studied the effects of particle clustering on heat transfer in a turbulent flow, how particles tend to concentrate in strong front-like structures of the fluid temperature field in which particles sample high levels of temperature fluctuations resulting in fluid-particle heat transfer enhancement at small scales. Carbone et al. [2] showed that non-local thermal dynamics of particles play a significant role and illustrated through direct numerical simulations how the formation of particle thermal caustics dominates small-scale thermal dynamics. Saito et al. [6] derived an analytical prediction for modulation of fluid temperature fluctuations by particles by using a Langevin model for fluid temperature fluctuations along each particle path.

Since the study of classical transport processes is crucial both from scientific and from engineering standpoints, in order to study the thermal interaction in an inhomogeneous flow, in our previous works [7–9] we have used direct numerical simulations to investigate the role of monodisperse spherical inertial particles in heat transfer in a simple archetypal situation, the developing mixing layer between two homothermal regions in a homogeneous and isotropic velocity field. A large set of direct numerical simulations has provided insights into the dynamics of the suspended inertial heavy particles on heat transfer in the time-developing thermal mixing layer which develops at the interface between the two homothermal regions, allowing to quantify the effect of inertia and thermal inertia and their interaction with background fluid turbulent field for a wide range of flow parameters: Stokes and thermal Stokes numbers up to 10 at a Taylor microscale Reynolds number from 56 to 124. Single point statistics of fluid and particle temperature fields and their correlations with velocity field have been provided in both the one- and two-way coupling regimes. Numerical results have shown, up to the numerical accuracy, it is possible to identify a self-similar stage of evolution of the thermal interaction zone, in which moments collapse when rescaled with a layer thickness, determined by the mean temperature, which grows as t1/2 for all performed simulations. Self-similar solutions play a crucial role in understanding physical problems, particularly in the field of fluid dynamics and other dynamical systems, because they allow not only for a simplifying framework for analyzing complex phenomena, but also to identify universal behaviors and study the scaling properties of a system, facilitating the development of general models and insights that can be applied across a wide range of scales.

Therefore, in this work, we show, starting from the transport equations for particles and fluid, that a quasi self-similar solution can be obtained in the intermediate stage of evolution of the thermal interaction layer, where the large-scale Stokes number of the particles determines the asymptotic convergence to a self-similar solution in this transport problem. The non self-similar terms decay in time. The theory is confirmed by comparison with the statistics obtained from direct numerical simulations.

2 Physical Model

We consider, as in [7], two unbounded regions at uniform temperatures T1 and T2 in a homogeneous and isotropic velocity field with zero mean velocity. Let us indicate the x<0 the region at temperature T1 and with x>0 the region at temperature T2. We assume that the flow is seeded by a set of initially uniformly distributed identical inertial particles. Direction x is the only inhomogeneity direction, so that when collisions are not considered, the probability density function f that a particle is, at time t, in the plane x with velocity v and temperature ϑ is given by the following equation

∂f(t,x,v,ϑ)∂t+∂∂x[f(t,x,v,ϑ)⟨Vp|x,v,ϑ⟩]+∂∂v[f(t,x,v,ϑ)⟨Ap|x,v,ϑ⟩]+∂∂ϑ[f(t,x,v,ϑ)⟨Θ˙p|x,v,ϑ⟩]=0

(1)

where the acceleration Ap and thermal acceleration Θ˙p are given, for particles much denser than the fluid, by Ap=(u(t,Xp)-Vp)/τv and Θ˙p=(T(t,Xp)-Θp)/τϑ, respectively [5]; u and T are the fluid velocity and temperature at the particle position Xp, while τv=(2/9)(ρp/ρ0)R2/ν and τϑ=(1/3)(ρpcpp)/(ρ0cp0)R2/κ are the relaxation and thermal relaxation times of the particles, which depend on particle radius R, fluid momentum and thermal diffusivities ν and κ as well as on the particle to fluid density ratio ρp/ρ0 and specific heat ratio cpp/cp0. By multiplying times ϑ and integrating over (v,ϑ), it is possible to obtain an equation for the particle mean temperature,

∂⟨ϑ⟩∂t+∂∂x⟨vϑ⟩=1τϑ⟨T⟩-⟨ϑ⟩

(2)

At the same time, the temperature of the carrier flow is given by an ensemble average of the passive scalar advection-diffusion equation

∂⟨T⟩∂t+∂∂x⟨uT⟩=κ∂2∂x2⟨T⟩

(3)

No particle thermal feedback is considered in this work.

3 Self-similarity

In this section we analyze the possibility that the system (3-2) has self-similar solutions. Let us introduce a length-scale δ(t), proportional to the thickness of the interaction zone, and we assume that the solution can be expressed in terms of η=x/δ(t), so that we define

⟨T⟩(t,x)=T1+T22-T2-T12φf(η)

(4)

⟨uT⟩(t,x)=T1-T22κδ-1ψf(η)

(5)

so that ψf vanishes for η→±∞ while φf(-∞)=1, φf(∞)=-1. By introducing these relations into (3) one obtains

-δ′(t)δ(t)ηφf′(η)+κδ2(t)ψf′(η)=κδ2(t)φf′′(η)

(6)

which allows for a self-similar solutions if all time dependent coefficients are proportional, i.e. if δ′/δ∝κ/δ2, i.e. when δ(t)=2κt by setting the proportionality constant equal to 2. With this assumption, equation (6) allows to determine the convective heat flux as a function of the mean fluid temperature,

ψf′(η)=2ηφf′(η)+φf′′(η)

(7)

which, by considering that ψf(±∞)=0, φf(-∞)=1, φf(∞)=-1, can be integrated to obtain ψf. Note that Eq. (7) is invariant under the transformation η→-η, φf→-φf, ψf→ψf, so that φf is an odd function of η and ψf is an even function. The unknown value of ψf(0) determines the convective heat flux between the two regions and, thus, the convective Nusselt number of the flow, which, by using the definition in [7], Eq. 7, can be expressed in self-similar variables as Nuc=ψf(0)/|φf′(0)|. Let us now consider the particle temperature equation (2). The right hand side of the equation prevents to have a solution which depends only on η, because, while all the other terms would scale as 1/t, it would scale as t0. i.e. since the convective heat flux reduces in time as 1/t, this implies that the difference between the fluid and particle temperature reduces as 1/t, and therefore the particle mean temperature cannot be function of η alone. Therefore, we relax the self-similarity hypothesis, and look for a solution in the form

⟨ϑ⟩(t,x)=T1+T22-T2-T12φp(t,η),⟨vϑ⟩(t,x)=T1-T22κδ-1ψp(t,η),

where δ(t) is the same length-scale already defined. The introduction of these expressions of the solution into (2) together with the definition of δ leads to

∂φp∂t(t,η)+14t-2ηφp′(t,η)+ψp′(t,η)=1τϑφf(η)-φp(t,η),

(8)

where the sign ′ indicates the derivative with respect to η. Equation (8) allows to determine the particle heat flux as a function of the particle mean temperature, which in turn depends on the carrier flow temperature. In the tracer limit, when particle inertia vanishes, τϑ→0+, the mean particle temperature becomes equal to the mean fluid temperature in the same position x and time t, so that φp(t,η)=φf(η) and the particle temperature-velocity correlation are given by (8) as ψp′=2ηφf′(η). In this limit the ratio between particle and fluid temperature-velocity correlation becomes

⟨vϑ⟩⟨uT⟩=ψpψf=1+12φ′(η)/∫-∞ηsφf′(s)ds-1.

When the Reynolds number increases, the difference between particle and fluid the velocity-temperature tends to vanish because the difference is only in the diffusive term. A finite inertia, i.e. finite τϑ, introduces a lag between the between the temperature of the carrier flow and the particle temperature, proportional to τϑ. Since the timescale of the process, δ/δ′∝t, increases with time, the ratio τϑ and the process timescale becomes smaller and smaller, so that the particle thermal inertia has a smaller and smaller effect on particle thermal dynamics. Therefore, let us expand φp and ψp in a power series of τϑ/t, i.e. let us write

φp(t,η)=∑k=0∞(t/τϑ)-kφk(η)=φ0(η)+(t/τϑ)-1φ1(η)+(t/τϑ)-2φ2(η)+⋯ψp(t,η)=∑k=0∞(t/τϑ)-kψk(η)=ψ0(η)+(t/τϑ)-1ψ1(η)+(t/τϑ)-2ψ2(η)+⋯

By introducing these expansions into (8) and by equating to zero the coefficients of all powers of t one obtains the following equations,

φ0-φf=0

(9)

12ψ0′-2ηφ0′=-φ1

(10)

-(k-1)φk+12ψk′-2ηφk′=-τϑk-1φk+1∀k≥1

(11)

This set of equations allows to determine the non self-similar part of the heat flux from the non self-similar part of the mean temperature distribution. Since the system is not closed, it is not possible to solve the problem unless a parametrization of velocity-temperature correlations is introduced. Anyway, any parametrization should be compatible with Eqs. (7) and (9–11). This is what was done, for example, in [3], where a gradient parametrization was introduced and thus a self-similar solution for particle concentration could be analytically obtained.

4 Numerical Simulations

The results of the previous section can be compared with information extracted by direct numerical simulations [7–9], carried out within the point-particle approach in a parallelepiped domain with periodic boundary conditions by decomposing temperature in a linear part and a residual part.

In those simulations, fluid temperature and velocity moments have been found to collapse when rescaled with the thickness of the thermal interaction layer δ∗, conventionally defined on the basis of the inverse temperature gradient as δ∗=(T1-T2)/max|∂T/∂x|. This thickness grows as δ∗/ℓ∼(t/τ)1/2, where τ=ℓ/u′ is the large-scale eddy turnover time, ℓ is the integral scale of the turbulence and u′ is the root mean square value of the velocity fluctuations. When the position x is rescaled with δ∗, i.e. in terms of the variable η∗=x/δ∗(t), the temperature profile is independent from the Reynolds number. This shows that the thermal mixing is driven mainly by the large scales of the flow [4, 7], which can correlate velocity and temperature fluctuations over longer distances. Since that δ∗/δ∼Pe1/2/2, where Pe=u′ℓ/κ is the Peclet number of the flow based on the integral scale. When data at the same Reynolds number are considered, this scaling factor is constant, so that we will continue to use the same variables introduced in Sect. 3. To use δ∗ as a scale for the self-similar equations is equivalent to the transformation η→ηPe1/2/2 and ψf→ψfPe1/2/2, ψp→ψpPe1/2/2. φf→φf, φp→φp. The main results of the simulations were the Nusselt number and the ratio between particle and fluid temperature-velocity correlation, which gives a measure of particle contribution to the overall heat flux across the thermal layer. With present variables, the convective Nusselt number is given by Nuc=ψf(0)/|φf′(0)|, so that it is contained in the values of the rescaled flux ψf. While the ratio between particle and fluid temperature-velocity correlation is ψp(0)/ψf(0)=F(Re,Pr,St), where Pr=ν/κ is the Prandtl number of the fluid.

Fig. 1

Mean temperature moment ratio Ip/If for a particles with St≤1; b particles with St≥1

A direct comparison of the temperature moments is made difficult by the intrinsic uncertainty in data coming from numerical simulation, where the noise due to numerical constraints and the limited sample size prevents the computation of derivatives. Therefore, a better comparison can be obtained by deriving integral quantities which describe the thermal mixing layer. We can observe that the following integrals, which represent the spatial moment of the mean temperature,

If=∫0+∞x(⟨T⟩-T2)dx=T1-T22δ2∫0+∞η(1+φf(η))dηIp=∫0+∞x(⟨ϑ⟩-T2)dx=T1-T22δ2∫0+∞η(1+φp(t,η))dη

scale as δ2, i.e. grow linearly in time when the solution behaves in a self-similar way. Therefore, by replacing the expansion of φp and using (9)-(11), we have

Ip=If+T1-T22δ2τϑt∫0+∞ηφ1(η)dη+oτϑt=If-T1-T22δ2τϑt12∫0+∞ψ0dη-∫0+∞η(1+φ0)dη+oτϑt,=If-T1-T22δ2τϑt12∫0+∞ψpdη-∫0+∞η(1+φ0)dη+oτϑt=If-τϑtδ22Jp-If+oτϑt

where Jf and Jp are the integral fluxes

Jf=1κ∫0+∞⟨uT⟩dx=T1-T22∫0+∞ψf(η)dηJp=1κ∫0+∞⟨vϑ⟩dx=T1-T22∫0+∞ψp(t,η)dη

We can therefore write

IpIf=1+τϑt1-δ22JpIf+oτϑt.

(12)

Fig. 2

Integral flux to mean temperature moment ratio, δ2Ip/If, for a particles with St≤1; b particles with St≥1

Equation (12) can be easily verified from the numerical simulations, because integral If, Ip, Jf, and Jp can be obtained directly by the flow statistics without the introduction of any hypothesis about the solution. Note that also the Ip, Jp admit an expansion in series of the inverse time when the expansion of φp and ψp are introduced in the integrals. The first terms of such series, corresponding to the integrals of φ0 and ψ0, can be obtained from the limits of Ip and Jp for large t.

Fig. 3

Integral flux ratio Ip/If for a particles with St≤1; b particles with St≥1

We use the results for direct numerical simulations of the thermal layer advected by a homogeneous and isotropic velocity field in one-way coupling and dilute collisionless flow regime with a Taylor microscale Reynolds number equal to 56, when the ratio between the thermal relaxation time τϑ to the Stokes relaxation time is fixed and equal to 4.43 and the Prandtl number is equal to 0.71 (see [7] for more details about the simulated flow). Figure 1 shows the ratio Ip/If. After an initial transient of about two eddy turnover times τ, the ratio evolves showing a gradual decay toward a constant value, according to Eq. (12). Since the fluid temperature evolution is driven by the large-scales of the motion, t scales over τ, so that t/τϑ scales as the inverse of the large-scale thermal Stokes number Stϑ,ℓ=τϑ/τ, and the corrections due to the particle thermal inertia are small, so that the non self-similar terms tend to be affected by the noise unavoidably present in any experimental statistics, and we cannot check higher order terms in (12). The ratio increases when the particle inertia increases, reaches a maximum when the thermal Stokes number of the particles, Stϑ=τϑ/τη, where τη is the Kolmogorov timescale, is equal to one, then reduces for larger inertia. This is because, notwithstanding the correction is proportional to τϑ, the coefficient of τϑ/t depends on the particle flux, which reduces when the Stokes number becomes larger than one [7–9]. We can check this behaviour by looking at the second term in expansion (12), the leading term of the non self-similar correction, which is shown in Fig. 2.

After the initial transient, the ratio δ2Jp/If slowly decays in time according to the inverse power series expansion in time, showing the same non monotonic dependence on particle inertia, with a maximum when the particle Stokes number is equal to one. However, the time variations appear to be smaller and after about five eddy turnover times the ratio is already constant. Indeed, if we look at the integral flux ratio Jp/Jf (see Fig. 3), we can observe that it remains almost constant in time. This suggests that the first term in the series expansion of ψf(t,η) is zero. This implies also that the higher terms in the expansion of φp(t,η) do not depend on the particle heat flux. Simulations at higher Stokes numbers present more marked fluctuations due to the reduced number of particles in the simulations, which were carried out at a constant volume fraction.

5 Conclusions

We have carried out an analysis of the heat transport at the interface between two homothermal regions advected by a homogeneous and isotropic velocity field in presence of suspended particles and fluid-particle thermal interaction. We have shown that particle inertia prevents a full self-similar evolution of the system, but that a quasi self-similar stage of evolution is achieved. Comparison with direct numerical simulations confirm the theoretical analysis.

Acknowledgements

The authors acknowledge the CINECA award IsB26_DroMiLa, HP10BU46TS, under the ISCRA initiative, for the availability of high performance computing resources and support. Additional computational resources provided by hpc@polito (http://​www.​hpc.​polito.​it) are also gratefully acknowledged.

References

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Béc, J., Homann, H., Krstulovic, G.: Clustering, fronts, and heat transfer in turbulent suspensions of heavy particles. Phys. Rev. Lett. 112, 234503/1–5 (2014). https://​doi.​org/​10.​1103/​PhysRevLett.​112.​234503

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Carbone, M., Bragg, A.D., Iovieno, M.: Multiscale fluid-particle thermal interaction in isotropic turbulence. J. Fluid Mech. 881, 679–721 (2019). https://​doi.​org/​10.​1017/​jfm.​2019.​773MathSciNetCrossref

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Chamecki, M., Meneveau, C.: Particle boundary layer above and downstream of an area source: scaling, simulations, and pollen transport. J. Fluid Mech. 683, 1–26 (2011). https://​doi.​org/​10.​1017/​jfm.​2011.​243MathSciNetCrossref

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Iovieno, M., Di Savino, S., Gallana, L., Tordella, D.: Mixing of a passive scalar across a thin shearless layer: concentration of intermittency on the sides of the turbulent interface. J. Turbulence 15(5), 311–334 (2014). https://​doi.​org/​10.​3390/​fluids4030144MathSciNetCrossref

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Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26(4), 883–889 (1983). https://​doi.​org/​10.​1063/​1.​864230Crossref

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Saito, I., Watanabe, T., Gotoh, T.: Modulation of fluid temperature fluctuations by particles in turbulence. J. Fluid Mech. 931, A6 (2022). https://​doi.​org/​10.​1017/​jfm.​2021.​939

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Zandi Pour, H.R., Iovieno, M.: Heat transfer in a non-isothermal collisionless turbulent particle-laden flow. Fluids 7(11), 345 (2022). https://​doi.​org/​10.​3390/​fluids7110345

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Zandi Pour, H.R., Iovieno, M.: The impact of collisions on heat transfer in a particle-laden shearless turbulent flow. J. Fluid Flow Heat Mass Transf. 10, 140–149 (2023). https://​doi.​org/​10.​11159/​jffhmt.​2023.​018

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Zandi Pour, H.R., Iovieno, M.: The role of particle inertia and thermal inertia in heat transfer in a non-isothermal particle-laden turbulent flow. Fluids 9(1), 29 (2024). https://​doi.​org/​10.​3390/​fluids9010029

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_21

A Tire Side Slip Model with Dynamic Friction Distribution over the Contact Patch

Shengxuan Zhao1  , Xianchao Zhang2   and Nan Xu¹

(1)

Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, Jilin, China

(2)

FAW Volkswagen Co., Ltd. Qingdao Branch, Qingdao, Shandong, China

Shengxuan Zhao (Corresponding author)

Email: zhaosx22@mails.jlu.edu.cn

Xianchao Zhang

Email: xianchao.zhang@faw-vw.com

Abstract

The tire forces and moments come mainly from the shear stress and sliding friction at the tread-road interface. The friction coefficient isn’t uniform over the contact patch because of the pressure and velocity distribution. However, most existing tire models don’t consider the influence of friction coefficient distributions. This paper aims to present a tire model considering friction distributions over the contact patch, which can be applied to steady-state and non-steady-state conditions. A discrete analytical tire model is proposed in this paper considering compliance of carcass, then the velocity distribution is discussed. Finally, a rubber friction model considering velocity and pressure distributions is introduced. The model adequately expresses the high-speed tire mechanical characteristics through simulation.

Keywords

Tire modelSide slipDynamic friction coefficient distributionLateral forceAligning torque

1 Introduction

With the increasing development of vehicle dynamics and control technology, the contact force between tires and road has gained increasingly attention. The tire forces and moments come from the rubber friction over the contact patch, which requires us to have a detailed understanding of the dynamic friction coefficient distribution characteristics. As observed from the friction tests of rubber materials, the dynamic friction coefficient is related with many factors, like sliding velocity, pressure, rubber materials and temperature. Due to the influence of tire structure, the tire pressure distribution is uneven and the sliding velocity in the contact patch at different position is also different. Although the pressure distribution characteristics of tire have been able to be accurately measured with the advanced measurement, the tread sliding velocity distribution and the dynamic friction coefficient distribution still cannot be measured. To solve these problems, many research work has proposed effective models. The influence of the dynamic friction coefficient is introduced in the commonly used tire models, such as UniTire and MF [1, 2]; articles [3–5] discussed the deformation of carcass and tread; article [6] explored the effect of the contact patch pressure distribution and sliding velocity on the side force distribution and tire mechanical characteristics; article [7] proposed an idea that tire tread can be discretized into several tread units to facilitate the analysis of tire deformation process and also applied to analyze the turn slip condition [8].

This paper develops a simulation model for cornering conditions considering the dynamic friction coefficient distribution characteristics. The discrete simulation model discretizes the tire rolling process and calculates the tire force by accumulating carcass and tread deformation. Firstly, the discrete model is deduced based on the carcass deformation and the pressure distribution in the contact patch; secondly, the distributions of velocity and friction coefficient in the contact patch are obtained from the analysis of kinematics between discrete steps. Finally, the analytical model is solved and validated.

2 Key Parts for Developing the Tire Model

2.1 Model Assumptions

Figure 1 shows the physical structure of tire model, there tire is simplified into the combination of rigid hub, flexible carcass and tread. The tread elements are simplified into separated massless elastic beams that adhere to the circumference of tire carcass.

Fig. 1

The physical structure of tire model

Fig. 2

Tire coordinate systems

2.2 Tire Coordinate System

Tire coordinate systems, tire forces and motions are depicted in Fig. 2. Various force of a moving car comes from the adhesion of tire and road, only the deformations of tread and carcass in the contact patch are considered. xoy is a dynamic coordinate system describing the deformations of carcass and tread with the origin ‘o’ at the center of the contact patch [10]. x axis is the wheel plane and y axis is the wheel spin plane.

Ω is the wheel speed of revolution, and Vr is the updating speed of the contact path, defined as the product of the wheel speed of revolution Ω and the effective rolling radius Re. V is the speed of the wheel center. The sliding speed under side slip condition of tire can be expressed as: Vsy=V·sinα [11].

Fig. 3

Deformation of carcass under cornering conditions

2.3 Deformation of Carcass

As depicted in Fig. 3, the deformation of carcass yc(x) is generalized, where a is the half length of the contact patch and θ is the carcass twisting angle. The deformation of carcass is divided into three parts: lateral translating part yc0, lateral bending part ycb and twisting part yθ, which can be expressed as:

yc=yc0+ycb+yθ,yc0=Fy/Kc0.ycb=FyKcbξxa,yθ=θ·x=Mz/Nθ·x,

(1)

where Fy is the lateral force, and Mz is the aligning torque. Kc0, Kcb and Nθ are carcass lateral translating stiffness, lateral bending stiffness and twisting stiffness respectively, ξ(x/a) is a symmetrical bending deformation function of carcass. In the paper, ξ(x/a) is expressed as:

ξxa=321-xa2,-a≤x≤a

(2)

2.4 Pressure Distribution

The deformations of tread are restricted by the tire vertical load Fz, friction coefficient μ and pressure distribution function η(x/a). The pressure distribution function η(x/a) in this paper is expressed as [1]:

ηxa=541-xa4,-a≤x≤a.

(3)

qz(x) is the vertical force per unit area and could be expressed as [10]:

qz(x)=Fz2a·ηxa,-a≤x≤a.

(4)

2.5 Kinematic Analysis of Tread Particle

The tread deformation of a rolling tire without considering the flexibility of carcass is shown in Fig. 4. xoooyo is a fixed coordinate system on the ground, xoy is a dynamic coordinate system with the origin ‘o’ at contact center C. Vc denotes the speed of point C; (u, v) is the displacement of a contact point; φ is the yaw angle; ex→ and ey→ are the unit direction vector of coordinate xoy. Then the velocity of the tread particle Vg can be expressed as:

Vg=Vc+(xo˙+u˙)ex→+(yo˙+v˙)ey→+φ˙(xo+u)ey→-(yo+v)ex→.

(5)

Fig. 4

Analysis of tread particle velocity

In the cornering condition, the rotating angular velocity φ˙=0, the rolling velocity component at the contact center C is xo˙=-ro·Ω, yo˙=-tanα·ro·Ω=-Vcy. Then Eq. (5) can be simplified to:

Vg=v˙=dvdt=∂v∂xo·dxodt+∂v∂yo·dyodt+∂v∂t.

(6)

Considering ∂v/∂xo=-tanα shows the orientation of the tread deformation, dxo/dt=vc·cosα and dyo/dt=vc·sinα are the longitudinal and lateral speed of the tread particle. The tread velocity under side slip condition used in this paper is expressed as Eq. (7), and vsy of this equation is the lateral sliding speed of the whole tire.

Vg=vsy+∂v∂t.

(7)

3 Discrete Analytical Tire Model

In the discrete analytical tire model, the tire’s contact patch is divided into several infinite elements. The discrete step of the model is Δe=2a/n, as the contact patch is equally divided into n parts. After the tire rolling a certain step ΔX (usually ΔX=Δe), the tire force can be calculated through superposition of the deformation of each infinite element.

3.1 Deformation of Tread

The deformation of tread bristles in the contact patch without considering sliding and carcass flexibility is shown in Fig. 5. The initial tire rolling position is X. The abscissa and ordinate of one point P(xt,yt) of the contact patch increased Δx and Δy respectively at ΔX from the tire rolling, which can be expressed as:

Δx=-ΔX,Δy=-ΔX·tgα.

(8)

Fig. 5

The deformation of tread bristles without considering sliding

If the point is still in the contact, and its coordinate are:

xt(X+ΔX)=x(X)-ΔX,yt(X+ΔX)=y(X)-ΔX·tgα.

(9)

Then the deformation of this discrete point in the contact patch is expressed as:

yt(X+ΔX)=y(X)-ΔX·tgα,-a≤x≤a-ΔX;-(a-x)·tgα,a-ΔX≤x≤a.

(10)

3.2 Distribution of Dynamic Friction Coefficient

The adhesion or sliding state of each point is determined by the pressure distribution over the contact patch, and the maximum lateral deformation of tread is restricted by the maximum friction force, which can be expressed as:

yt(X+ΔX)=sgn(y(X+ΔX))·min(|y(X+ΔX)|,|μ·qz(x)/kty),

(11)

where kty denotes the lateral stiffness of tread per unit length and μ is the friction coefficient. In the previously proposed tire models, the friction coefficient is a determined constant or even in the contact patch. However, the friction coefficient is considered to be dynamic and uneven over the contact patch in this paper. The friction coefficient distribution is obtained in this paper by velocity analysis in the contact patch, which agrees with the actual situation.

Fig. 6

The deformation of tread considering sliding in the discrete model

The deformation of tread in the contact patch considering sliding is shown in Fig. 6. In this figure, Δyc is the lateral displacement of the tire, whose derivative on time is the lateral sliding velocity vsy; the deformation of tread is limited by the pressure distribution in the slip region, then a slip displacement Δypressure arises due to the update of the pressure distribution. The lateral deformation of trend is based on the lateral deformation of carcass considering carcass flexibility. Similar to the pressure distribution, the lateral deformation of trend will update during tire rolling, then a displacement of carcass Δycarcass arises. Above all, the displacements of tread grids can be expressed as Eq. (12), which can be used to deduce tread slip velocity by Eq. (7).

Δy=Δyc+Δypressure-Δycarcass.

(12)

To solved the velocity distribution in the contact patch, the friction model used in this paper can be expressed as:

μd=μs+(μm+μs)·exp-μh2·log2(VsVsm+N·exp(-VsVsm)),

(13)

where μm, μs, μh and Vsm are friction characteristic parameters influenced by the contact pressure qz; N(usually N=0.8) is a factor to make the friction coefficient increase slightly around the origin; Vs is the slip speed between the road and tire.

3.3 Analytical Model Considering Carcass Flexibility

As descripted in the induction and Fig. 7, the deformation of carcass can be divided into translation, bend and twist. The curve AD depicts the deformation of carcass, which is the upper side of tread bristles, and AB is the maximum deformation of tread without considering sliding. Hence the deformation of tire is expressed as:

Δy=yt(x)-yc(x).

(14)

Fig. 7

The deformation of tire considering complex deformation of carcass

Then, the lateral force Fy and the aligning torque Mz of tire are integrated from the deformation between tread and carcass:

Fy=∫-aakty·(yt+yc)dx,Mz=∫-aakty·(yt+yc)xdx.

(15)

With the contact patch discretized and considering Eq. (1), the equation of the lateral force Fy and the aligning torque Mz can be expressed as:

Fy(X)=aktyn·∑i=1n(yti+yti+1)/(1+ε0+εb),Mz(X)=aktyn·∑i=1nxi+12Δe(yti+yti+1)/(1+εθ),

(16)

where ε0 is the translating characteristic ratio; εb is the bending characteristic ratio; εθ is the twisting characteristic ratio. They are expressed in Eq. (17),

ε0=2akty/Kc0,εb=2akty/Kcb,εθ=2a3/3Nθ.

(17)

4 Solution of the Model

The tire lateral force and aligning torque are gradually formed during the rolling process, the flexibility of carcass makes the formation of tire force have a relaxation process. In any rolling position of the tire, the deformation of carcass results from the balance between tread force and carcass force. Therefore, a loop iteration process can be used to calculate the deformation of tire and tire force. In the computation procedure the model only needs to confirm the slip angle α, wheel speed of revolution Ω and effective rolling radius Re.

Fig. 8

The calculating frame chart of discrete model

The calculating process of the model considering the general deformation of the tire is shown in Fig. 8. First, tire forces and aligning torques without sliding are calculated directly based on the coordinates of tread using Eq. (10). Second, the corresponding coordinates of carcass and the deformations of tread with sliding are determined by using Eq. (16). Meanwhile through the analysis of velocity, the friction coefficient is updated through Eq. (13). Finally, the changes of tire forces and aligning torques before and after considering sliding are compared. If these changes are large, the coordinates of tread considering sliding and the updated friction coefficient should be returned to the model, and tire forces and aligning torques are recalculated. When these changes are slight and satisfy the convergence criterion of the model, the calculation is stopped and the tire forces are outputted.

5 Simulations and Analysis

Table 1

Parameter values of the model used in the simulation

Parameter

Values

Parameter

Values

Parameter

Values (m)

Fz

4000 N

Kcy

1.25×105 N/m

Δe

0.001

a

0.0665 m

Kcb

8.0×106 N/m

Re

0.3

b

0.05 m

Nθ

4000 N m/rad

Kty

2.0×108N/m3

μy

1.0

The values of parameters used for the simulation in this paper are obtained from Ref. [12], which are listed in Table 1.

Figures 9 and 10 show the lateral force and aligning torque on the steady state separately with different wheel speeds of rotation Ω. As α increases, the lateral force Fy increased first to its peak and then decreased, and the downward trend of the lateral force became more evident as the velocity increases. The tire aligning torque Mz became negative at large slip angles, and the negative trend of the aligning torque became more evident as the velocity increases.

Fig. 9

Lateral force simulation results at different wheel rolling speeds

Fig. 10

Aligning torque simulation results at different wheel rolling speeds

Fig. 11

Tread velocity distribution (Vc=10 m/s, Fz=4000 N)

Fig. 12

Friction coefficient distribution (Vc=10 m/s, Fz=4000 N)

Figures 11 and 12 show the tread velocity distribution and the friction coefficient distribution calculated by this model. It is evident from these two figures that the sliding velocity in the contact patch is gradually increased from front to back, when the tire is in a full sliding condition. The velocity change is faster in the front and back of the contact patch, while relatively gentle in the middle. As the slip angle increases, the sliding velocity in the contact patch decreases and the sliding region also becomes narrower. Because the dynamic friction coefficient of rubber within certain velocity is less than the maximum static friction coefficient, the friction coefficient will decrease with the increasing of the slip angle, and the point of the resultant force of the friction is gradually moving to the back-end of the contact patch. Hence, the peak value of the side force and the back of the aligning torque appear around α=-6∘ in the macro curve of the side force and aligning torque. The result of the simulation is consistent with the former test data, which suggests that this model correctly reflects the mechanical characteristics of the tire.

6 Summary

A tire model considering sideslip was established, which effectively considered the flexibility of the tread and carcass as well as the dynamic friction, and the model was used to simulate and analyze the lateral force and aligning torque under different sideslip angles and wheel speeds. The results show that the model can not only calculate the lateral force and self-aligning torque under different sideslip angles and wheel speeds, but also analyze the influence of the dynamic friction coefficient of the tire contact patch on the tire force and aligning torque.

Acknowledgements

The authors would like to thank the support of National Nature Science Foundation of China (Grants No. 51405185). The authors also thank the staffs in State Key Laboratory of Automotive Simulation and Control for their contribution to this study.

References

1.

Guo, K.H.: Handling Dynamics of Automobiles. Jilin Press of Science and Technology (1991) (in Chinese)

2.

Pacejka, H.B.: Tyre and Vehicle Dynamics. Butterworth Heinemann (2002)

3.

Guo, K.H., Sui, J.: A tire side slip model with flexible tire carcass and the identification of tire structural parameters. In: Proceedings of 13th IMAC (1995)

4.

Guo, K.H., Liu, Q.: A generalized theoretical model of tire cornering properties in steady state condition. SAE Paper 973191

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Guo, K.H., Sui, J.: A theoretical observation on empirical expression of tire forces. In: Proceedings of the 9th IAVSD Symposium (1995)

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Guo, K.H., Zhuang, Y., Lu, D., Chen, S.K., Lin, W.: A study on speed-dependent tire-road friction and its effect on the force and the moment. Veh. Syst. Dyn. 43(Supplement), 329–340 (2005)

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Zegelaar, P.W.A.: The Dynamic Response of Tyres to Brake Torque Variations and Road Unevennesses. Delft University of Technology (1998)

8.

Bai, F., Guo, K.H., Lu, D.: Tire model for turn slip properties. SAE Int. J. Commer. Veh. 6(2) (2013). https://​doi.​org/​10.​4271/​2013-01-2371

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Guo, K.H., Sui, J.: The effects of longitudinal force and vertical load distribution on tire slip properties. SAE Technical Paper 945088

10.

Guo, K.H., Lu, D.: UniTire: unified tire model for vehicle dynamic simulation. Veh. Syst. Dyn. 45(Supplement), 79–99 (2007). https://​doi.​org/​10.​1080/​0042311070181674​2

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Ning, L.: The Modification of Tire Simulation Models for Side Slip Property Under Complex Condition. Master Thesis, Jilin University, Changchun (2007) (in Chinese)

12.

Lu, D., Guo, K.H.: Effect of tire rolling speed on cornering characteristics of vehicle. Automob. Technol. (9), 10–13 (2009) (in Chinese)

Part IVMathematical and Statistical Modelling in Biological, Medical and Health Sciences

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_22

Stability and Qualitative Analysis of a Switched SQEIAR Epidemic Model

Zohreh Abbasi¹ and Xinzhi Liu1  

(1)

University of Waterloo, Waterloo, ON, Canada

Xinzhi Liu

Email: xinzhi.liu@uwaterloo.ca

Abstract

This study introduces a switched SQEIAR epidemic model that accounts for variability in the contact rate. The basic reproduction number (R0) of the proposed model is calculated, and its equilibria are determined, followed by a stability analysis. A qualitative analysis is presented, including a theorem that examines the positive invariance property of the model’s solutions. To ensure globally uniformly exponential stability (GUES), a contact-rate-dependent average dwell time (CRDADT) strategy is proposed. This strategy is supported by a Lyapunov function, and sufficient conditions, expressed as linear matrix inequalities (LMIs), are derived. The strategy’s effectiveness is demonstrated through a numerical example.

Keywords

Switched SQEIAR epidemic modelContact rate variabilityEquilibrium analysisStability analysisContact-rate-dependent average dwell time (CRDADT)

1 Introduction

The importance of understanding disease spread is underscored by outbreaks of infectious diseases, which pose significant challenges to global health. Mathematical models have been employed to provide insights into this spread. Notably, basic models like SIS [1] and SEIR [2] offer perspectives on disease progression based on existing data, while more intricate models, such as SQEIAR [3], consider factors including vaccination and the quarantine of susceptible individuals. Such analytical instruments are considered critical for well-informed decision-making.

This study emphasizes the importance of fluctuating contact rates in the spread of diseases [4–7], employing the SQEIAR model to account for these variations. This includes considering the contact rate as switching between different phases of the disease’s spread. Furthermore, a positive invariance set for the SQEIAR model is established. Both the disease-free and endemic equilibrium points have been calculated, and the local stability of the disease-free equilibrium point has been investigated, offering valuable information on the conditions needed to maintain a disease-free state. Additionally, the basic reproduction number, R0, a critical measure of the potential for an epidemic’s growth [8, 9], has been calculated for the model.

In this study, the contact rate is modeled to vary based on the phase of the outbreak, recognizing that different stages require different levels of intervention. Initially, strict measures are implemented to reduce interactions and contain the spread of the disease. As the situation evolves, these measures are often relaxed, leading to a gradual rise in the contact rate, particularly evident when institutions like schools reopen. Although analyzing these variable rates presents challenges, it is vital. Moreover, as slight alterations in contact rates can significantly impact disease spread, understanding them becomes even more crucial [1, 10].

Previous studies, for instance, have associated contact rates with elements like seasonal changes and weather conditions [5, 10]. In contrast, this research employs a switched-systems methodology, where each subsystem corresponds to a distinct phase of the outbreak, characterized by varying contact rates. Societies under an epidemic are viewed as dynamic systems that switch between stable and unstable subsystems, influenced by government actions. In stable subsystems, government measures effectively curb disease spread, while easing these measures leads to unstable subsystems, increasing the risk of transmission. The strategic switching between these subsystems, involving timely interventions, hygiene promotion, and potential vaccination, is crucial for effective disease control and maintaining system stability.

Additionally, analyzing the stability of switched systems is a crucial area of study, focusing on how systems behave when transitioning between various operational modes and assessing whether the system can maintain stability during these transitions. In this study, the goal is to refine the mode-dependent average dwell time (MDADT). This approach helps achieve system-wide stability and addresses challenges posed by traditional ADT methods [11, 12]. In this study, the mode is interpreted as the contact rate, introducing the concept of contact-rate-dependent average dwell time (CRDADT).

This approach allows for switching (based on CRDADT) to a situation where the disease-free solution is stable by implementing strict restrictions. The system remains in stable modes for a specific amount of time, usually long enough to dissipate transient effects. Then, it switches to a subsystem by relaxing restrictions in which the endemic solution is stable, staying for a shorter, specified period to let society engage in normal activities without stringent measures, promoting economic growth and social well-being. This process aims to ultimately eradicate the disease and achieve stability for the entire system.

2 Model Description and Qualitative Analysis

In this section, the switched SQEIAR epidemic model is introduced, along with a detailed description of its parameters and variables. The mathematical properties of the model, including positive invariance, are then presented. Finally, the model’s equilibria are identified, and their stability is assessed using a threshold criterion.

2.1 Switched SQEIAR Epidemic Model

The switched SQEIAR epidemic model is governed by the following dynamics:

S˙t=μ-βiΛt+μ+ξSt,

(1-a)

Q˙t=ξSt-μQt,

(1-b)

E˙t=βiStΛt-μ+kEt,

(1-c)

A˙t=1-pkEt-μ+ηAt,

(1-d)

I˙t=pkEt+1-zηAt-μ+αIt,

(1-e)

R˙t=zηAt+fαIt-μRt.

(1-f)

The model categorizes individuals into several groups. Initially, susceptible individuals are denoted by St. Some get quarantined at the rate of ξ, moving into the quarantined group, Qt. Upon exposure to the virus, susceptible individuals move to the exposed group, Et, at a rate βi, which varies for each phase. As the disease progresses, exposed individuals become contagious, with a fraction p moving into the symptomatic group It, and a fraction (1-p) becoming asymptomatic At at a rate k. Symptomatic and asymptomatic individuals eventually recover at rates η and α, respectively. Additionally, Λt=εEt+δAt+1-qIt is utilized to monitor the potential spread of the disease within a population over time. As shown in Fig. 1, the disease spreads through contact between susceptible individuals and infectious groups E(t), A(t), and I(t).

Fig. 1

Flow diagram of a switched SQEIAR epidemic model

2.2 Positive Invariance Set

In this subsection, it is demonstrated that the set Ω is a positively invariant set for system (1).

Theorem 1

The set Ω=S,Q,E,I,A,R∈R+6;0≤S+Q+E+I+A+R≤1 is a positively invariant set for system (1).

Proof

Consider the following equations: S˙tS=0=μ>0,Q˙tQ=0=ξSt≥0,E˙tE=0=βiStδAt+1-qIt≥0,A˙tA=0=1-pkEt≥0, I˙tI=0=pkEt+1-zηAt≥0,R˙tR=0=zηAt+fαIt≥0, and S˙t+Q˙t+E˙t+A˙t+I˙t+R˙tS+Q+E+A+I+R=1=-1-fαIt≤0.

By examining them, it can be observed that all the derivatives for St,Qt,Et,It,At, and Rt are non-negative when their respective variables are equal to zero. This implies that, starting from any point within the set Ω, the trajectories will either remain within Ω or enter it. They will not leave the set, thereby confirming the positive invariance of Ω for system (1).

2.3 Equilibrium Points Analysis

In this subsection, the equilibria of the model are evaluated under two scenarios: one where no infections are present and another where infections persist. The stability of these equilibria is then assessed based on a threshold criterion.

The equilibrium in the absence of disease, referred to as the Disease-Free Equilibrium (DFE), is derived from the conditions E(t)=A(t)=I(t)=0 on ∂Ω. This leads to:

QDFEi=μξ+μ,ξξ+μ,0,0,0,0.

(2)

Subsequently, for the Endemic Equilibrium (EE) of system (1), we set the right side of system (1) equal to zero, yielding:

St=μβiΛt+μ+ξ,

(3)

Qt=ξμSt,

(4)

Et=βiμ+kStΛt,

(5)

At=1-pkη+μEt,

(6)

It=pkμ+αEt+1-zημ+αAt,

(7)

Rt=zημAt+fαμIt.

(8)

Furthermore, using (6) in (7) yields:

It=pkμ+α+1-pk1-zηη+μμ+αEt,

(9)

substituting (6) and (9) into (3) results in:

St=μβiΨEt+μ+ξ,

(10)

with Ψ being ε+1-qkμ+αp+η1-z1-pη+μ+δk1-pη+μ. Inserting (6), (9), and (10) into (5), we obtain:

βiμΨ=μ+kβiΨEt+μ+ξ,

(11)

which can be reformulated as: E∗=μμ+k-μ+ξβiΨ,A∗=1-pkη+μE∗,I∗=pkμ+α,E∗+1-zημ+αA∗,S∗=μβiΨE∗+μ+ξ,Q∗=ξμS∗, and R∗=zημA∗+fαμI∗. Concluding, the Endemic Equilibrium is: QEEi=S∗,Q∗,E∗,I∗,A∗,R∗.

Moreover, a key metric called R0 is used to evaluate a new pathogen’s potential to infiltrate a population. It denotes the average number of infections one infected person might cause in an entirely susceptible population during their contagious period [13]. If R0 is below 1, the disease is unlikely to spread widely. Conversely, if R0 is greater than 1, it indicates a potential outbreak risk [13]. The basic reproduction number for the model discussed is derived from [9] as follows:

R0i=βiμΨk+μμ+ξ.

(12)

Theorem 2

The equilibrium QDFEi of system (1) is confirmed to be locally asymptotically stable when the condition.

R0i<1,i=1,2,⋯,ι,

(13)

is met. conversely, it is unstable if R0i>1.

Proof

Considering the QDFEi defined by (2) and aligning it with the system (1) meets criteria (A1) through (A5) as highlighted in Theorem 2 from [9]. Thus, the proof stands validated. This implies that when R0i exceeds 1, the disease remains and becomes prevalent in the community. On the other hand, when R0i is less than 1, the disease dies out, provided the initial count of each group in the model falls within the attraction domain of the QDFEi. □

3 Managing Disease Spread by Adjusting the Contact Rate

The goal is to help the government manage the spread of diseases by implementing and relaxing restrictions. This involves determining specific times to impose or lift restrictions, depending on the situation. The restrictions can be applied for a certain period to reduce contact between individuals and control the disease’s spread, or they can be relaxed, allowing more contact for a limited time. This study uses the concept of “contact-rate-dependent average dwell time (CRDADT)” to determine how long these restrictions should last based on the current contact rate. The term “dwell time” refers to the duration of time spent in a particular state (either restricted or relaxed). This method enables the government to manage the disease by switching between strict restrictions (stable mode with less disease spread) and more relaxed measures (unstable mode with disease spread). The overall aim is to maintain control over the disease’s spread while balancing the need for normalcy.

Remark 1

If R0 is greater than one for a subsystem, the subsystem is unstable at its disease-free equilibrium point. Conversely, if R0 is less than one, the subsystem is considered stable at its disease-free equilibrium point. A stable subsystem (s) is achieved when interventions effectively lower the contact rate, steering the system toward disease eradication. In contrast, an unstable subsystem (u) arises when restrictions are relaxed, leading to a higher contact rate and the continued presence of the disease.

Below, some notations related to the switched system are presented. The switching signal σts is a piecewise constant function that operates over an infinite sequence of time intervals ts,ts+1 where s∈N, starting at t1=0. Each ts is a switching instant, indicating the time at which the system transitions to a different mode. During the interval ts,ts+1, the subsystem σtsth is active.

Definition 1

[14] For the switched system (1), the DFE is considered globally uniformly exponentially stable (GUES) if, given a particular switching signal σ(t), there exist positive constants κ>0 and ε>0 such that the inequality ‖X(t)‖≤κe-ε(t-t0)‖X(t0)‖ holds for all t≥t0 with any initial conditions.

Theorem 3

Consider the switched nonlinear system denoted as (1). Suppose φi>0, 0<ϕi<1 for i∈u and φi<0, ϕi>1 for i∈s. If there exists a set of C1, non-negative functions ViXt:Rn→R+ where Xt=[S(t)Q(t)E(t)A(t)I(t)Rt]T, and functions α and α′ from class K∞, such that:

α‖Xt‖2≤ViXt≤α′‖Xt‖2,i∈ι,

(14)

V˙iXt≤φiViXt,i∈ι,

(15)

ViXts≤ϕiVqXts-,i,q∈s×ι,i≠q,

(16)

ViXts≤ϕiVqXts-,i,q∈u×s,i≠q.

(17)

Then, the DFE of system (1) achieves GUES under any CRDADT switching signals that satisfy the following conditions:

τai≤-lnϕiφi,i∈u,

(18)

τai≥-lnϕiφi,i∈s.

(19)

Proof

Assume that within the time interval [0,T], the number of switches that occur is denoted by NσT,0, where ∑i∈uNσiT,0=NσT,0. By integrating inequality (15), it can be deduced that:

VσtsXT≤eφσtsT-tsVσtsXts

≤eφσtsT-tsϕσtsVσts-1Xts-

≤eφσtsT-tsϕσtseφσts-1ts-ts-1Vσts-1Xts-1

≤eφσtsT-tsϕσtseφσts-1ts-ts-1ϕσts-1Vσts-2Xts-1-

⋮

≤∏i=1NσϕσtieφσtsT-ts×⋯×eφσt0t1-t0Vσt0Xt0

=∏i∈uϕiNσi(T,0)∏i∈sϕiNσi(T,0)e∑i∈uφiTi+∑i∈sφiTiVσt0Xt0.

(20)

Based on the definitions of slow and fast MDADT switching mentioned in [11], we can express (20) as follows:

VσtsXT≤e∑i∈uN0i+Tiτailnϕi+φiTi+∑i∈sN0i+Tiτailnϕi+φiTiVσt0Xt0

=e∑i∈uN0ilnϕi+∑i∈sN0ilnϕie∑i∈uTiφi+lnϕiτai+∑i∈sTiφi+lnϕiτaiVσt0Xt0.

(21)

If conditions (18) and (19) are satisfied, then φi+lnϕiτai≤0,∀i. Therefore:

VσtsXT≤e∑i∈uN0ilnϕi+∑i∈sN0ilnϕiemaxi∈ιTiφi+lnϕiτaiVσt0Xt0.

(22)

Thus, VσtsXT converges to zero as T→∞, provided that the CRDADT satisfies conditions (18) and (19). Finally, from (14) and Definition 1, the DFE of system (1) is proven to be GUES. □

Subsequently, Theorem 4 is introduced to explore the stability criteria of the system (1) using the Lyapunov function proposed in Theorem 3. Initially, the switched model (1) is restructured as follows:

fiX=AXt+χit,i∈ι,

(23)

where A=-(μ+ξ)00000ξ-μ000000-μ+k00000(1-p)k-(μ+η)0000pk(1-z)η-(μ+α)0000zηfα-μ,

and χit=μ-βiΛtSt0βiStΛt000T.

Theorem 4

Under the scalars given in Theorem 3, if there exist positive definite matrices Pi,Pq,Ti∈S++6 satisfy the following conditions:

PiA+ATPi-φiPi+Ti≤0,i∈ι,

(24)

0Pi∗-Ti≤0,i∈ι,

(25)

Pi-ϕiPq≤0,i,q∈ι,i≠q,

(26)

then the DFE of system (1) is GUES.

Proof

Consider the Lyapunov function:

ViXt=XTtPiXt.

(27)

The derivative of ViXt along the trajectories of (23) is:

V˙iXt=2XTtPiAXt+χit.

(28)

Substituting (27) and (28) into (15) gives:

V˙iXt-φiViXt

=2XTtPiAXt+χit-φiXTtPiXt

=XTtPiA+ATPi-φiPiXt+2XTtPiχit≤0.

(29)

Applying Young’s inequality, we obtain:

V˙iXt-φiViXt≤ZTWiZ≤0,

(30)

where Z=XTtχiTtT, and Wi=PiA+ATPi-φiPi+Ti00PiTi-1Pi.

To ensure (30) holds, the matrix Wi must be negative semidefinite. Applying the Schur complement, the condition PiTi-1Pi≤0 transforms into: 0Pi∗-Ti≤0, validating conditions (24) and (25). From inequalities (16) and (17), we also have:

ViXts-ϕiVqXts-≤0.

(31)

Substituting (27) into (31) yields:

XTtsPiXts-ϕiXTts-PqXts-≤0.

(32)

Thus, since Pi-ϕiPq≤0 by condition (26), and with conditions (24) to (26) satisfied, the DFE of system (1) is confirmed to be GUES.

4 Numerical Analysis

The purpose of this section is to validate the preceding theoretical findings through numerical analysis. The model parameters are adopted from [3, 15], with the initial values set as follows: S0=0.85, Q0=R0=0,E0=A0=0.025,I0=0.1. The switching between subsystems is examined under two scenarios. In the first scenario, the switching signal, derived based on the CRDADT criterion, determines the timing of the switching between stable and unstable systems, as illustrated in Fig. 2a. The second scenario considers cases where the CRDADT condition is not fulfilled, as shown in Fig. 2b. Figure 3 illustrates that when the CRDADT condition is satisfied, the disease can be eliminated from the community. However, if this condition is violated, the disease persists and spreads within the population.

Fig. 2

The switching signal derived in accordance with/contrary to the proposed CRDADT

Fig. 3

The impact of adhering to or violating the CRDADT condition on disease spread or eradication

5 Conclusion

This study introduced a switched SQEIAR epidemic model that accounts for varying contact rates during different stages of infection onset. The duration of the disease was segmented into phases based on these contact rate variations, resulting in a switched system. A strategy was developed to calculate the average dwell time for each phase, ensuring system stability and minimizing the need for continuous quarantine measures. This model provides a framework that can assist policymakers in optimizing resource allocation and designing targeted responses to the dynamics of the infection.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_23

The Influence of Refuges, Fear, and Velocities on Predator-Prey Dynamics

Kawkab Abdullah Nabhan Al Amri1   and Qamar J. A. Khan1  

(1)

Sultan Qaboos University, Muscat, Oman

Kawkab Abdullah Nabhan Al Amri (Corresponding author)

Email: kawkab.alamri222@gmail.com

Qamar J. A. Khan

Email: qjalil@squ.edu.om

Abstract

In this article, we investigate the dynamics of interactions between predators and prey in two distinct habitats: the refuge and areas outside the refuge. As the predation pressure decreases, the prey emerges from its hiding place. Predators engage with the prey in the unprotected areas and can even kill them. The behavior of the prey is influenced by the number of predators present, causing them to shift to safer regions. This suggests that when the predator population increases, more prey will seek protection within the refuge, while they will choose to remain outside the refuge when there are fewer predators. By considering the impact of predator density on prey velocity, we incorporated both predator and prey speeds, as well as their anti-predator behavior. The prey has the ability to reduce its velocity to evade attacks. We examined the relationship between anti-predator behavior and prey density and found that prey density increases with a higher level of anti-predator behavior, whether it is due to an increase in predator or prey velocities. Furthermore, we presented evidence that, under specific conditions, the coexisting equilibrium will be globally stable. Additionally, we identified the occurrence of a saddle-node bifurcation at the coexisting equilibrium point.

1 Introduction

The study of predator-prey interactions within an ecological community subsystem is a popular field of research for mathematicians, ecologists, and biologists. Predator-prey modeling and analysis have gained considerable attention in recent years. Predation influences the population of prey both directly through slaughter, and indirectly through modifications of behavior on fear [1, 2]. Prey species modify their behavior and physiology in response to predator-induced fear. They often relocate from their preferred habitats to safer environments, driven by the fear of predators. While this may initially increase their short-term survival, it ultimately reduces their long-term viability. Predators exert pressure on prey in all types of habitats, leading to the adoption of various anti-predator behaviors. These behaviors include changes in foraging patterns, switching habitats, physiological adaptations, and more. When prey species are fearful of predators, they tend to consume less food. Fear-induced responses in prey can manifest in different defense mechanisms, such as reducing mating activities, experiencing hunger, or migrating from vulnerable to safer locations. Interestingly, these defense strategies can actually slow down the growth rate of prey populations. Recent research studies have highlighted the significant impact of indirect predator effects on population dynamics [3–7]. These studies provide insights into how predators influence prey populations through non-direct means, shedding light on the complex dynamics of predator-prey interactions.

Another intriguing and significant aspect of ecosystems is the fear-induced refuge adopted by prey. This refuge behavior has a profound impact on the dynamics of food webs and plays a crucial role in maintaining a balance in predator-prey interactions. Prey individuals have the ability to seek sanctuary when they perceive a threat from a predator. The refuge serves as an anti-predator behavior that can prolong the duration of predator-prey encounters, reducing the risk of extinction caused by excessive predation pressure.The concept of refuge behavior was initially demonstrated by Gause in his experiments using the protozoan species Didinium (predator) and Paramecium (prey) [8]. Subsequent studies, such as the research conducted by Wang et al. [9], have further highlighted the advantages of prey refuge in promoting the coexistence of predator and prey populations. However, there is a threshold of fear beyond which the effectiveness of refuge behavior may diminish. Elmojtaba et al. [10] and Kar [11] conducted studies on prey-predator models where the proportion of prey refuge remained constant and depended solely on the prey population, without considering the predator population.

The change in consumer behavior in response to the presence of predators is known as antipredator sensitivity. Prey face a trade-off between hiding and foraging, as they can reduce their velocity to evade predators but this also limits their ability to gather resources. The theoretical basis that acceleration is influenced by a stimulus gradient is supported by field experiments that analyze individual movement characteristics. For example, studies on fish schools [12] and flying insect swarms [13, 14] have shown that individuals’ acceleration vectors align towards the centroid of dynamically stable formations. Arditi et al. [15] investigated the directed movement of predators and the emergence of density-dependent patterns in predator-prey models. Their model assumed that the spatial and temporal variations in predator velocity are determined by the prey gradient. Sadowski and Grosholz [16] developed a fundamental tritrophic food chain model that incorporated predator and prey velocities. According to their theory, prey can adapt to predators by exhibiting antipredator behavior, which involves slowing down in response to predator density. They examined how prey density and the amplitude of predator-prey cycles in equilibrium were influenced by both fast-moving (mobile) predators and slow-moving (sit-and-wait) predators. The advantage of active search behavior diminishes when predators and prey exhibit more directional movement, as shown by Scharf [17]. Gerritsen [18] found that in a search mode model based on movement in a three-dimensional space, active-seeking predators are more likely to target slow-moving species, while ambush predators are predicted to target fast-moving animals. Ross and Winterhalder [19] recently developed a two-dimensional predator-prey encounter model with random prey and predator movement.

2 Model Formulation

In recent studies, the importance of fear, refuge, and velocity in predator-prey dynamics has been highlighted. However, there is a need to investigate the combined effects of these factors. This study aims to examine the interactions among fear, refuge, and velocity in a prey-predator system. The research focuses on a three-species model consisting of a predator species (y) and two types of prey species occupying different habitats. The first habitat is the refuge habitat, denoted as prey species (x1), while the other habitat represents prey species outside the refuge, denoted as (x2). In this model, we consider the growth rate of the prey species inside the refuge habitat as parameter a, and the carrying capacity of the prey in the refuge habitat as parameter k. It is assumed that the prey species are initially present within the refuge habitat, and their population follows a logistic growth pattern. Furthermore, the refuge habitat is believed to provide protection for the prey species against predators. When the fear of predation decreases, the prey species relocate to a second habitat outside the refuge, where they are susceptible to predation according to the law of mass action. The fear parameter is denoted as e, and the rate at which prey leave the refuge habitat is represented by α. When predation fear increases due to the presence of predators, the prey species migrate to the refuge habitat, with the immigration rate into the refuge habitat denoted as p. The decision of prey to migrate to the refuge habitat depends on the predator population, meaning that more prey will seek refuge when the predator population is high, while they will prefer to stay outside the refuge when the predator population is low. The predators feeding rate on prey outside the refuge habitat is represented by b, and the conversion rate of prey to predator is denoted as c. In the absence of prey, predators experience exponential mortality, with the predators mortality rate indicated by μ. In this model, the velocities of both predators and prey are taken into account. The predator velocity is denoted as w, while the prey velocity is given by γe-θy, where γ represents the maximum prey movement rate in the absence of a predator. The prey velocity decreases as the predator population grows, indicating that a higher velocity increases the likelihood of encountering a predator. Therefore, by incorporating the influence of predator density on prey velocity, we include the antipredator behavior parameter θ in our model. Antipredator sensitivity refers to the variation in prey behavior as the predator population changes. The predator-prey relationship is described by the following system of differential equations:

dx1dt=ax1(1-x1k)-αx11+ey+px2ydx2dt=αx11+ey-px2y-bx2yw2+γ2e-2θydydt=cx2yw2+γ2e-2θy-μy

(1)

3 Positivity of Solutions

Our goal is to demonstrate that all solutions to model (1), given positive initial data, will remain positive for all time t>0. This can be easily verified by analyzing the model equations.

dx1dt|x1=0=px2y≥0dx2dt|x2=0=αx11+ey≥0dydt|y=0=0≥0

Hence, all solutions will remain nonnegative for all time with positive initial values that is ( x1(0),x2(0),y(0)) ∈R+3. [20]

4 Boundedness

Theorem 1

All the solutions of model (1) with non-negative initial conditions, with initiate in R+3 are uniformly bounded.

Proof

Let w=cx1+cx2+by. Take the time derivative along the solution of model (1), dwdt=c(dx1dt+dx2dt)+bdydt≤cax1-bμy.

For any positive q, we have dwdt+qw≤cax1-bμy+qcx1+qcx2+qby. Since the maximum population of prey in the refuge is determined by the carrying capacity, k, then x1<k and x2<k. We are assuming that the prey species in the refuge grow logistically but never grow outside the refuge because most resources are stored inside the refuge. Therefore, dwdt+qw≤(a+2q)ck-(μ-q)by, for q<μ, dwdt+qw≤(a+2q)ck. Let (a+2q)ck=L. Using theory of differential inequality for w(t), we get 0<w(x1,x2,y)≤Lq(1-e-qt)+w(0)e-qt

where w(0)=w(x1(0),x2(0),y(0)). Moreover, for t⟶∞, 0<w≤Lq. Hence all the solutions for the system (1), (initiating in R+3) are confined in the region Δ=(x1,x2,y)∈R+3:w(t)≤Lq+ϵ,foranyϵ>0. This shows that the solution of the system represented by (1) is bounded.

5 Equilibrium Points of the System

We find three biologically meaningful equilibrium points:

1.

The trivial equilibrium point E0=(0,0,0).

2.

The equilibrium point Ex2=(0,μcw2+γ2,0).

3.

The coexistence equilibrium point E∗=(x1∗,x2∗,y∗) where x1∗=(1+ey∗α)x2∗y∗(p+bw2+γ2e-2θy∗), x2∗=μcw2+γ2e-2θy∗ and y∗ is the positive root of the following equation

ay∗μk1+ey∗αp+bw2+γ2e-2θy∗kcw2+γ2e-2θy∗-1+ey∗αp+bw2+γ2e-2θy∗y∗μ-by∗μw2+γ2e-2θy∗=0

6 Stability Analysis

6.1 Local Stability Analysis

The stability matrix for the system (1) is

J=a-2ax1k-α1+eypyαx1e(1+ey)2+px2α1+ey-py-byw2+γ2e-2θy-αx1e(1+ey)2-px2-bx2w2+γ2e-2θy[w2+γ2e-2θy(1-θy)]0cyw2+γ2e-2θy-μ+cx2w2+γ2e-2θy[w2+γ2e-2θy(1-θy)],

The stability matrix for E0 is J0=a-α00α0000-μ, The eigenvalues of J0 are 0,a-α, and -μ. E0 is neutrally stable if α>a i.e if the migration rate from the refuge habitat is more than the growth rate of prey inside refuge habitat. The stability matrix for Ex2 is Jx2=a-α0pμcw2+γ2α0-bμc000, The eigenvalues of Jx2 are a-α,0, and 0. Ex2 is neutrally stable if α>a i.e if the migration rate from the refuge habitat is more than the growth rate of prey inside refuge habitat. The stability matrix for the equilibrium point E∗ is J=ABCDEF0GH, where A=-ax1∗k-px2∗y∗x1∗<0,

B=py∗>0, C=αex1∗(1+ey∗)2+px2∗>0, D=α(1+ey∗)>0, E=-(p+bw2+γ2e-2θy∗)y∗<0,

F=-αx1∗e(1+ey∗)2-px2∗-bx2∗w2+γ2e-2θy∗[w2+γ2e-2θy∗(1-θy∗)]<0 if y∗<1θ, G=cy∗w2+γ2e-2θy∗>0 and H=-μγ2θe-2θy∗<0.

The characteristic equation is

λ3+a1λ2+a2λ+a3=0

(2)

where a1=-(A+E+H), a2=AE+EH+AH-BD-FG and a3=-GDC-HAE+GAF+HBD.

From Routh-Hurwitz criteria, the equilibrium E∗ is locally asymptotically stable if a1>0, a3>0 and a1a2>a3.

Here a1=-(A+E+H)>0,

a3=GAF-GDC-HAE+HBD=a(2x1∗k-1)[cy∗αx1∗ew2+γ2e-2θy∗(1+ey∗)2+cy∗px2∗w2+γ2e-2θy∗-bx2∗cy∗(w2+γ2e-2θy∗(1-θy∗))-μγ2θe-2θy∗py∗-μγ2θe-2θy∗by∗w2+γ2e-2θy∗]+α1+ey∗[bx2∗cy∗(w2+γ2e-2θy∗(1-θy∗))+μγ2θe-2θy∗by∗w2+γ2e-2θy∗].So,a3>0ifx1∗>k2.

a1a2-a3=-(A+E+H)(AE+EH+AH-BD-FG)-GAF+GDC+HAE-HBD=-A2E-AEH-A2H+ABD+AFG-AE2-HE2-AEH+BDE+EFG-AEH-E2H-AH2+BDH+FGH-AFG+CDG+AEH-BDH=-2AEH-A2E-A2H+ABD+AFG-A2E-HE2+BDE-EH2-AH2+FGH+CDG

Here -2AEH-A2E-A2H+AFG-A2E-HE2-EH2-AH2+FGH+CDG>0 but ABD+BDE<0.

Take ABD+BDE-A2E-AE2=-A(AE-BD)-E(AE-BD)=(-A-E)(AE-BD). Here -A-E>0. we want to show that AE-BD>0.

AE-BD=-px2∗y∗x1∗-ax1∗k-(p+bw2+γ2e-2θy∗)y∗-αpy∗1+ey∗=px2∗y∗x1∗+ax1∗k(p+bw2+γ2e-2θy∗)y∗-αpy∗1+ey∗=px2∗y∗α(1+ey∗)(p+bw2+γ2e-2θy∗)x2∗y∗+ax1∗k(p+bw2+γ2e-2θy∗)y∗-αpy∗1+ey∗=py∗α(1+ey∗)+ax1∗k(p+bw2+γ2e-2θy∗)y∗-αpy∗1+ey∗=ax1∗k(p+bw2+γ2e-2θy∗)y∗>0.Therefore,a1a2>a3.

6.2 Global Stability Analysis

Theorem 2

The coexistence equilibrium E∗=(x1∗,x2∗,y∗) is globally stable if

-ax1∗+a4k(k+x1∗)2+αx1∗1+ei3-px1∗i2i3s1-αi1x2∗s2(1+es3)+px2∗s3+bx2∗s3w2+γ2-ci2y∗w2+γ2-μi3y∗≤0

Proof

From (1)1,dx1dt≤ax1(1-x1k)+px2y. At supremum of x1, dx1dt=0, then ax1(1-x1k)+px2y≥0. Using (1)2, dx2dt≤αx1-px2y. At infimum of x2, dx2dt=0, then αx1-px2y≥0.

Therefore, ax1(1-x1k)+αx1≥0 or x1≤ka(α+a). So, supremum of x1=ka(α+a)=s1. px2y≤αx1 or px2y≤αka(α+a), then x2≤αkapy(α+a). If i3 is the infimum of y, then, x2≤αkapi3(α+a). Therefore, supremum of x2=αkapi3(α+a)=s2. Using (1)3, at supremum of y, dydt=0 and so cx2w2+γ2e-2θy-μ=0. Then, cαkapi3(α+a)w2+γ2e-2θy≥μ or w2+γ2e-2θy≥μapi3cαk(α+a).

Let L=μapi3cαk(α+a). So,w2+γ2e-2θy≥L⟹w2+γ2e-2θy≥L2, -2θy≥ln(L2-w2γ2).

y≤-12θln(L2-w2γ2)⇒y≤12θln(γ2L2-w2). Therefore, supremum of y is 12θln(γ2L2-w2)=s3. From (1)1, dx1dt≥ax1(1-x1k)-αx11+ey≥ax1(1-x1k)-αx1. At infimum dx1dt=0, then ax1(1-x1k)-αx1≤0. Therefore, x1≥ka(a-α) and the infimum of x1=ka(a-α)=i1. Now using (1)3, dydt≤cx2yw2+γ2-μy. At infimum of y, dydt=0, then cx2w2+γ2-μ≥0 or x2≥μcw2+γ2. Therefore, infimum of x2=μcw2+γ2=i2. Using (1)2, dx2dt≥αx11+ey-px2y-bx2yw2+γ2 and then dx2dt≥αx11+ey-αk(α+a)ap(p+bw2+γ2).

Let αk(α+a)ap(p+bw2+γ2)=M, then dx2dt≥αx11+ey-M. At infimum of x2, dx2dt=0, then αx11+ey≤M. So, αi11+ey≤M or y≥αi1-MeM. Therefore, infimum of y=αi1-MeM=i3.

Now, we construct a Lyapunov function as follows:

V=x1-x1∗-x1∗ln(x1x1∗)+x2-x2∗-x2∗ln(x2x2∗)+y-y∗-y∗ln(yy∗)dVdt=(1-x1∗x1)dx1dt+(1-x2∗x2)dx2dt+(1-y∗y)dydt=(x1-x1∗x1)[ax1(1-x1k)-αx11+ey+px2y]+(x2-x2∗x2)[αx11+ey-px2y-bx2yw2+γ2e-2θy]+(y-y∗y)[cx2yw2+γ2e-2θy-μy]dVdt=(x1-x1∗)[a(1-x1k)-α1+ey+px2yx1]+(x2-x2∗)[αx1x2(1+ey)-py-byw2+γ2e-2θy]+(y-y∗)[cx2w2+γ2e-2θy-μ]dVdt=-ak(x1-k+x1∗2)2-ax1∗+a4k(k+x1∗)2+αx1∗1+ey-px2x1∗yx1-αx1x2∗x2(1+ey)+px2∗y-(b-c)x2yw2+γ2e-2θy+bx2∗yw2+γ2e-2θy-cx2y∗w2+γ2e-2θy-μy+μy∗dVdt≤-ax1∗+a4k(k+x1∗)2+αx1∗1+ey-px2x1∗yx1-αx1x2∗x2(1+ey)+px2∗y+bx2∗y-cx2y∗-μy+μy∗-gy2+gyy∗

where -(b-c)x2yw2+γ2e-2θy<0 from model (1).

dVdt≤-ax1∗+a4k(k+x1∗)2+αx1∗1+ei3-px1∗i2i3s1-αi1x2∗s2(1+es3)+px2∗s3+bx2∗s3w2+γ2-ci2y∗w2+γ2-μi3y∗.Therefore,dVdt≤0if-ax1∗+a4k(k+x1∗)2+αx1∗1+ei3-px1∗i2i3s1-αi1x2∗s2(1+es3)+px2∗s3+bx2∗s3w2+γ2-ci2y∗w2+γ2-μi3y∗≤0

7 Saddle-Node Bifurcation (LP)

The coexistence equilibrium point E∗ of the system (1) encounters saddle-node bifurcation when parameter α passes through α(LP). The Jacobian of system (1) around the coexistence point E∗ has an eigenvalue equal to zero (i.e. det(JE∗)=0) at the critical parameter α(LP). From equation (2), if det(JE∗)=0 then

a2x1∗k-1(cy∗αLPx1∗ew2+γ2e-2θy∗(1+ey∗)2+cy∗px2∗w2+γ2e-2θy∗-bx2∗cy∗(w2+γ2e-2θy∗(1-θy∗))-μγ2θe-2θy∗py∗-μγ2θe-2θy∗by∗w2+γ2e-2θy∗)-αLP1+ey∗bx2∗cy∗(w2+γ2e-2θy∗(1-θy∗))+μγ2θe-2θy∗by∗w2+γ2e-2θy∗=0

and so, α(LP)=-ξ1ξ2ξ1(cy∗x1∗ew2+γ2e-2θy∗(1+ey∗)2)-ξ3(11+ey∗), where ξ1=a2x1∗k-1,

ξ2=cy∗px2∗w2+γ2e-2θy∗-bx2∗cy∗(w2+γ2e-2θy∗(1-θy∗))-μγ2θe-2θy∗py∗-μγ2θe-2θy∗by∗w2+γ2e-2θy∗ and ξ3=bx2∗cy∗(w2+γ2e-2θy∗(1-θy∗))+μγ2θe-2θy∗by∗w2+γ2e-2θy∗.

J(E∗,α(LP))=-ax¯1k-px¯2y¯x¯1py∗α(LP)ex1∗(1+ey∗)2+px2∗α(LP)(1+ey∗)-(p+bw2+γ2e-2θy∗)y∗-αLPx1∗e(1+ey∗)2-px2∗-bx2∗w2+γ2e-2θy∗[w2+γ2e-2θy∗(1-θy∗)]0cy∗w2+γ2e-2θy∗-μγ2θe-2θy∗,=π1π2π3π4π5π60π7π8,(say).

Let v=[v1,v2,v3]T be the eigenvector of J(E∗,α(LP)) and w=[w1,w2,w3]T be the eigenvector of JT(E∗,α(LP)) corresponding to the zero eigenvalue at α=α(LP). Now by solving J(E∗,α(LP))v=0 and JT(E∗,α(LP))w=0 implies that v=[v3π1(π2π8π7-π3),-π8v3π7,v3] and w=[-π4π1w2,w2,-w2π7(π2π4π1+π5)] where v3, w2 are any nonzero real number. Now rewrite system (1) as follow: dXdt=h(X,α(LP))=[h1(X,α(LP)),h2(X,α(LP)),h3(X,α(LP))]T where X=x1x2y,h1=dx1dt,h2=dx2dtandh3=dydt,hα=-x11+eyx11+ey0⟹hα(X,α(LP))=-x11+eyx11+ey0,

wThα(X,α(LP))=(π4x1π1(1+ey)+x11+ey)w2≠0,Dhα(X,α(LP))=-11+ey0ex1(1+ey)211+ey0-ex1(1+ey)2000,

wTDhα(X,α(LP))v=(π4w2π1(1+ey)+w21+ey)v3π1(π2π8π7-π3)-w2ex1(1+ey)2(π4π1+1)v3 ≠0. Now

D2hα(X,α(LP))=M1M2M3M4M5M6M7M8M9M10M11M12M13M14M15M16M17M18M19M20M21M22M23M24M25M26M27, where M1=-2ak, M2=0, M3=αe(1+ey)2, M4=M5=0, M6=p, M7=αe(1+ey)2, M8=p, M9=-2x1αe2(1+ey)3, M10=M11=0, M12=-αe(1+ey)2,M13=M14=0, M15=-(p+bw2+γ2e-2θy)+byθγ2e-2θyw2+γ2e-2θy, M16=-αe(1+ey)2,M17=-p-b(w2+γ2e-2θy(1-θy))w2+γ2e-2θy,

M18=2αx1e2(1+ey)3-bx2γ2[-2θγ2e-2θy(1-θy)-γ2θe-2θy]w2+γ2e-2θy-bx2γ2θ2(e-2θy)2(w2+γ2e-2θy)32, M19=M20=M21=M22=M23=0,

M24=cw2+γ2e-2θy-cyγ2θe-2θyw2+γ2e-2θy, M25=0, M26=c[w2+γ2e-2θy(1-θy)]w2+γ2e-2θy,

M27=-cx2[w2+γ2e-2θy]2(w2+γ2e-2θy)32+cx2[-2θγ2e-2θy(1-θy)-θγ2e-2θy]w2+γ2e-2θy.

wTD2hα(X,α(LP))(v.v)=M1w1v12+[(M3+M7)w1+(M12+M16)w2]v1v3

+[(M6+M8)w1+(M15+M17)w2+(M24+M27)w3]v2v3+[M9w1+M18w2+M27w3]v32≠0.

Thus the system satisfies all Sotomayor’s theorems saddle-node bifurcation [21] near the interior equilibrium E∗ at the parameter α=α(LP).

8 Numerical Analysis

To support the previous theoretical findings, numerical simulation data are provided in this section. The parameter values are listed in table 1, some of them may be altered to examine their effects.

Table 1

Parameter values used for simulations

Parameter

a

k

α

e

p

b

w

γ

θ

c

μ

value

0.7

0.8

0.035

5

0.0119

0.05

2

2

0.69

0.02

0.07

Figure 1 shows the convergence to the co-existence equilibrium point (x1∗,x2∗,y∗).

Fig. 1

The convergence to the co-existence equilibrium point, the parameters are a=0.7, k=0.8, α=0.035, e=5, b=0.05, p=0.0119, c=0.02, μ=0.07, θ=0.69, w=2, γ=2

Based on Fig. 2, it can be observed that increasing predator velocity does not have an impact on prey populations inside refuges, but it does lead to a decrease in prey populations outside of refuge habitats. Conversely, the population of predators increases with higher predator velocity. In Fig. 3, it is evident that as prey velocity increases, the predator population grows while the prey population outside the refuge declines. However, the increase in prey velocity does not affect the prey population within the refuge habitat. Consequently, prey that move quickly are more exposed to predators. Therefore, any increase in predator or prey velocity results in higher prey mortality, leading to a decrease in the prey population outside the refuge. The refuge habitat provides a safe haven for the prey population, protecting them from predators.

Fig. 2

The effect of predator velocity on the population densities, the parameters are a=0.7, k=0.8, α=0.035, e=5, b=0.05, p=0.0119, c=0.02, μ=0.07, θ=0.69, γ=2

Fig. 3

The effect of prey velocity on the population densities, the parameters are a=0.7, k=0.8, α=0.035, e=5, b=0.05, p=0.0119, c=0.02, μ=0.07, θ=0.69, w=2

Antipredator sensitivity describes the changes in prey behavior in response to increasing predator levels. When antipredator sensitivity is elevated, it leads to higher prey density outside of the refuge area, resulting in a decrease in predator density, as illustrated in Fig. 4. However, antipredator sensitivity does not have an impact on prey density inside the refuge habitat.

Fig. 4

The effect of antipredator sensitivity on the population densities, the parameters are a=0.7, k=0.8, α=0.035, e=5, b=0.05, p=0.0119, c=0.02, μ=0.07, w=2, γ=2

Figure 5 illustrates that as the refuge parameter increases, a greater number of prey individuals are able to find shelter within the refuge habitat. Consequently, the prey population inside the refuge gradually increases until it reaches its carrying capacity. As a result of this process, the prey population outside the refuge decreases. Since the predator population relies on the prey population, it also decreases and eventually becomes extinct.

Fig. 5

The effect of refuge on the population densities, the parameters are a=0.7, k=0.8, α=0.035, e=5, b=0.05, c=0.02, μ=0.07, θ=0.69, w=2, γ=2

Figure 6 demonstrates that as predator fear increases, leading to higher prey density inside the refuge habitat, this is due to the prey choosing to seek safety within the refuge. Outside of the refuge, both the predator population and the prey population experience a decline. With fewer prey available outside the refuge, predators have less food to consume. Consequently, due to intensified predation on a reduced prey population, both the prey population and the predator population eventually decline.

Fig. 6

The effect of fear on the population densities, the parameters are a=0.7, k=0.8, α=0.035, b=0.05, p=0.0119, c=0.02, μ=0.07, θ=0.69, w=2, γ=2

The coexistence equilibrium point E∗ of the system (1) undergoes a saddle-node bifurcation when the parameter α crosses the threshold α(LP), as depicted in Fig. 7.

Fig. 7

The saddle-node bifurcation (limit point (LP)) of the system at the coexistence equilibrium E∗ at α=2.351996, the parameters are a=0.7, k=0.8, α=0.035, e=5, b=0.05, p=0.0119, c=0.02, μ=0.07, θ=0.69, w=2, γ=2

9 Conclusion

This study focuses on investigating the dynamics of predator-prey interactions in two different environments: the refuge and the area outside the refuge. Inside the refuge, prey are protected from predators and provided with sufficient resources, leading to a logistic growth of the refuge population. On the other hand, outside the refuge, predators and prey interact, and predation occurs. Prey are constantly aware of the presence of predators, and as their fear decreases, they venture out of hiding. However, as the fear of being attacked intensifies, prey seek refuge in the protected area. The movement of prey to the refuge depends on the predator population. When the predator population is high, more prey seek shelter in the refuge regions, while they prefer to stay outside the refuge when the predator population decreases. To capture the complexity of the system, we incorporated predator and prey velocities, as well as antipredator behavior, into our model by allowing predator density to influence prey velocity. The suggested system’s equilibrium states are identified as being biologically feasible. The nonzero equilibrium will be asymptotically stable if x1∗>k2 and it is globally stable if -ax1∗+a4k(k+x1∗)2+αx1∗1+ei3-px1∗i2i3s1-αi1x2∗s2(1+es3)+px2∗s3+bx2∗s3w2+γ2-ci2y∗w2+γ2-μi3y∗≤0. When parameter α crosses through α(LP), the coexisting equilibrium point E∗ of the system (1) experiences saddle-node bifurcation. In our numerical analysis aimed at understanding the impact of various factors on our system, we made several key observations. Firstly, when the antipredator sensitivity is low, both predator and prey velocities have an equal influence on the equilibrium prey density. Additionally, any increase in predator or prey velocity leads to higher prey mortality. However, when the antipredator sensitivity is increased, it results in a decrease in predator population density and an increase in prey density outside the refuge area. Furthermore, we found that as the refuge parameter increases, the predator population and prey population outside the refuge habitat decrease and eventually go extinct. In contrast, the prey population inside the refuge continues to grow until it reaches the carrying capacity. Moreover, as the fear parameter increases, the density of prey inside the refuge habitat increases, indicating that more prey seek shelter. Simultaneously, the density of prey outside the refuge and the predator density both decline. Overall, our numerical analysis revealed the intricate dynamics of the system, highlighting the effects of predator and prey velocities, antipredator sensitivity, refuge parameter, and fear parameter on the populations of predators and prey in both refuge and non-refuge environments.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_24

A Multiscale Model for Protein Allostery: Side Chain Concerted Motions Initiated by Brownian Kicks

Forbes Burkowski1  

(1)

School of Computer Science, University of Waterloo, Waterloo, ON, Canada

Forbes Burkowski

Email: fjburkowski@uwaterloo.ca

Abstract

Allosteric regulation in proteins has been studied for more than 25 years but the fundamental mechanisms of allosteric signalling and the time scales of their propagations remain elusive. Here I describe allostery in the PDZ3 domain stressing the role of side chain atoms with lower degrees of freedom. Interactions are due to concerted (cooperative) motions involving changes in dihedral angles of the side chains. While Brownian motion studies usually emphasize the path taken by the large particle within a solvent bath. I focus instead on the collision event itself: Particle motion is impelled by surrounding molecules moving in a chaotic milieu (high dimensional specification of this motion). When the rare but certain collision event occurs, the particle moves in a linear jump (low dimensional specification of that motion). The multiple time scales are evident because the particle motion is a very rare event compared to the extremely fast motions of the surrounding smaller molecules. I argue that protein side chain interactions involve a similar generation of motion in atoms with lower degrees of freedom due to the action of atoms with higher degrees of freedom starting with the solvent itself.

Keywords

Multiscale modellingProtein allosteryBrownian kicksConcerted motionsSide chain fluctuations

1 Introduction

Many papers dealing with allostery have focused on dynamic allostery and in particular there have been many investigations of allosteric regulation in the PDZ3 domain because it should be the “easy” case study. This domain is small, only 119 residues in length, and the almost perfect structural alignment when overlapping the backbones of the protein in its apo and holo forms indicates that dynamic allostery [1] must be involved.

Reviews of allostery, especially in the context of conformational ensembles [2] along with discussions about entropy issues [3, 4] and the importance of considering side-chain motions in the investigation of residual entropy [5] are recommended. Of particular importance here, is the study of electrostatic redistribution [6] of charged residues in the dynamic allostery of the PDZ domain. Using a molecular dynamics study, they observed that PDZ3 allostery does involve various minor changes in side chain conformations but more significantly there are some very significant changes in the contact frequencies for neighboring charged residues. The observations described in their paper will dovetail with results reported here and in Sect. 4, I speculate on the importance of charged residues in the larger context of the synaptic protein PSD-95. Another important issue, often neglected in papers dealing with allostery, is the recognition that water should play a more prominent role in protein functionality, noting that there are often networks of conserved residues that extend from the surface of an enzyme to the active site [7] and that coupled water relaxations mediate catalytic reactions at actives sites [8].

2 Methods

The following points describe the steps done in computational experiments that use a network representation to describe fluctuation pathways that start at surface residues and subsequently spread into the protein core. The goal is to assess whether ligand binding can act as a manager of water initiated fluctuations in contrast to the role of being the originator of such an allosteric signal. Knowing that ligand binding will affect the hydrogen bonding patterns of charged residues [6], the analysis will focus on using these charged residues as the starting points for these fluctuations. The experiments implemented the following steps for each of the protein PDB files that were studied (1BFE and 1BE9):

1.

Do the initial set-up.

2.

Generate perturbation data.

3.

Calculate correlations of residue conformations.

4.

Initiate fluctuations at charged residues and evaluate fluctuation intensities for residues in the binding site (done by probabilistic percolation).

The next sections describe the steps in more detail. All the processing steps (implemented as Python scripts) were done using UCSF Chimera [9]. Displays of the residue network used graphics facilities provided by the StructBio packages [10].

2.1 Initial Set-Up

Protein files were fetched and hydrogen atoms were added to the structure followed by an initial energy minimization to get the “reference” structure. A list of neighbors was derived for each residue and saved for future steps. Residue rj was considered to be a neighbor of a given residue ri when the shortest distance between the residues fell below 5 Angstroms. This distance threshold was only applied to the “moving” atoms, i.e. backbone atoms and beta carbon atoms were excluded. Knowing the neighborhood structure, it was possible to display the residue network (Fig. 1). Graph nodes were placed at the centroids of the moving atoms and the edges define the neighborhood structure. In general, the graph is less dense than residue networks that work with alpha carbon atoms as nodes while using a larger threshold. It is easy to see that the graph has a small world network type of structure [11] with highly connected regions just below the binding site, illustrated in Fig. 1 as a cluster of hydrophobic residues represented by yellow nodes. Meanwhile, lattice-like structures float over beta sheets and serve to connect the binding site nodes with other regions of the protein such as the α3 helix positioned near the right side of the display.

Fig. 1

The residue network for PDZ3 (1BFE). Color legend for residue nodes: hydrophobic (yellow), polar uncharged (gray), polar charged negative (red), polar charged positive (blue) [colour figure online]

2.2 Generating Perturbation Data

In generating the data, every residue in the protein acted as a perturbed residue. Each perturbed residue was put through a series of conformational changes by modifying its side-chain chi angles. Each chi angle assumed n different initial values:

χ∈χr+d|d∈-12sn-1,-12sn-3,...,0,...,12sn-3,12sn-1.

(1)

In this definition, χr represents the initial χ angle in the reference structure and d is the increment applied to χr, while s represents a step size. In the computational study, n was set to 9 and s was set to 10. The chi angle modifications were applied in a nested fashion: two chi angles ranged over 81 initial settings, three chi angles produced 729 different settings, etc.

After each setting of the chi angles, I used the Minimize Structure command provided by Chimera to move the structure towards a conformation that corresponded to a local energy minimum. Amber parameters [12] were used in the energy evaluations. Default parameters were used in the Chimera minimization: Steepest descent (100 steps) removed highly unfavorable steric clashes and then conjugate gradient computations (10 steps) guided the system to a local energy minimum.

Naively, one might assume that the minimization procedure would simply bring the perturbed residue back to the conformation that it had in the reference structure. In almost all cases, this did not happen. Instead, the energy minimization algorithm, continually adjusted all residue conformations as it strived to get a low energy value. Since the perturbations were not so drastic as to introduce a complete rotameric change, the system went into an energy well presumably near the global minimum. This minimization behavior is consistent with the microstate model described by [13]: the many small energy wells found by the repeated energy minimizations were contained within a larger and wider energy well that also included the energy well corresponding to the reference structure.

After the completion of all the minimization calculations one is left with a set of final chi angles for the perturbed residue and a set of corresponding final chi angles for each neighboring residue. One can then combine the list of chi angles from the perturbed residue with the list of chi angles from a particular neighboring residue and use a correlation evaluation algorithm (see next step) to get an estimate of the concerted motions that would take place in a more extended simulation.

The minimization/correlation strategy has at least two advantages when compared with an approach that merely evaluates residue-residue contact: (1) The evaluation is not based on simple residue pairing since interaction evaluations incorporate the influence of other neighboring residues, and (2) the technique accommodates the asymmetry of residue-residue interactions. When the perturbed residue and responding neighbor residue reverse their roles, we typically find that the correlations are different. This generates a residue network involving directed edges and is more biophysically realistic [4].

2.3 Evaluating the Correlation Between a Perturbed Side-Chain and Its Responding Neighbor

The input for each correlation analysis is a matrix P of chi angle settings for the perturbed residue and a matrix R of chi angle settings for the neighboring residue that is responding to these perturbations. As expected in a correlation analysis, both matrices must be centered. Referring to the generation of these matrices as described in Step 2, we have P∈Rn×mp and R∈Rn×mr where n is the number of samples (a power of 9 as discussed earlier), mp is the number of chi angles in the perturbed residue and mr is the number of chi angles in the responding neighbor residue. In an earlier study [14], a kernel based partial canonical correlation analysis [15] was used to evaluate the correlation between P and R. This is a powerful technique, but because of the nonlinearities imposed by the kernel method, it is difficult to assess the importance of the various chi angles when inspecting a particular correlation. Here, I opt for a more straightforward Canonical Correlation Analysis (CCA) to measure the linear relationship between P and R. The evaluation is a modification of the technique described in [16]. The analysis reduces to the problem of deriving two canonical vectors: hp of size mp × 1 and hr of size mr × 1, such that the canonical variates zp = Php and zr = Phr are maximally correlated, that is:

argmaxhp,hrρ=zpTzrzpzr=hpTPTRhrhpTPTPhphrTRTRhr

(2)

This problem is equivalent to the following constrained optimization problem:

argmaxhp,hrρ=hpTPTRhrwithconstraintshpTPTPhp=hrTRTRhr=1.

(3)

The values of r, hp and hr can be derived by solving the following generalized eigenvalue problem [17]:

0PTRRTP0hphr=ρPTP00RTRhphr.

(4)

After some manipulation we can derive the following equations for ρ hp and ρ hr:

ρhp=PTP-1PTRhr,ρhr=RTR-1RTPhp.

(5)

Recognizing that (PTP)⁻¹PT is the pseudoinverse of P and (RTR)⁻¹RT is the pseudoinverse of R, the previous equations can be written as:

ρhp=P+Rhr,ρhr=R+Php.

(6)

It can be shown [16] that the two equations for the constraint are equivalent to the single constraint equation:

hpTPTPhp+hrTRTRhr=2.

(7)

Equations (6) and (7) can be solved using an iterative scheme. The matrix products P⁺R, R⁺P, PTP and RTR along with initial values for hp and hr can be evaluated prior to the iterative loop which cycles through the following updates:

P+RhrR+Phpt-1→ρhpρhrtρhpTtPTPhpρt+ρhrTtRTRhrρtρhpTtPTPhpρt+ρhrTtRTRhrρt2.2.→ρt+1ρhptρhptρt+1ρt+1ρhrtρhrtρt+1ρt+1→hphrt+1.

(8)

The update loop continues until the difference between two consecutive values of ρ falls below some small value, such as 0.000001.

After the iterations are complete, each interacting pair (perturbed residue and responding neighbor) will have their derived canonical vectors: hp and hr. Inspection of these vectors provides information about the importance of each chi angle in the calculation of the canonical correlation. In most cases, the vector entry with the largest absolute value will correspond to chi 1. This is especially true for residues on a beta sheet since the “DoF1” atoms bonded to the beta carbons are at roughly the same distance from the sheet and consequently they interact in a highly concerted fashion. More significantly, for a particular perturbed residue and its responding neighboring residue, the square of the correlation ρ² will represent the proportion of the variation in the chi values of the neighbor that is predictable from the chi values of the perturbed residue. These squared correlations are used as weights on the edges of the residue network described earlier.

Since 0≤ρ2≤1 we can use ρ2 as if it is a probability. It would represent the likelihood that the perturbed residue and its neighboring residue can jointly assume conformations that are consistent with concerted motions. Here are some examples of the correlations seen in PDZ3 (apo form 1BFE). The notation “ >> ” is used to represent the directionality associated with the correlation: perturbed residue >> responding residue.

A_408_SER >> A_409_SER: 0.94096 0.88540

A_409_SER >> A_408_SER: 0.93724 0.87842

A_386_VAL >> A_323_LEU: 0.98037 0.96113

A_323_LEU >> A_386_VAL: 0.61235 0.37497

In each line, the first number is the correlation and the second number is the square of that correlation. The first two lines are typical of two interacting residues each with a single chi angle. The probability of reaching a compatible conformation after a residue is perturbed is quite high and independent of the residue chosen from the pair to be the perturbing residue. The last two lines show a different interaction. While valine has a single chi angle, leucine has two. As expected, a perturbation of valine will induce a linear response in leucine with a reasonably high probability. However, going in the other direction, the correlation indicates that a perturbation of leucine has a rather low probability of inducing a correlated fluctuation in the valine neighbor, hence the directionality of the interaction. Note that this example, is consistent with the initial claim made in the abstract: The atoms with a higher degree of freedom induce motions in the atoms with a lower degree of freedom but less frequently. However, this less frequent interaction is ultimately the more important phenomenon because it is characteristic of the pattern of fluctuations that start at the protein surface and eventually induce changes in the binding site near the hydrophobic core.

The next section will discuss strategies to reveal network pathways made up of weighted edges designating the probability of concerted motion for residues reacting to an initial perturbation at the protein surface.

2.4 Evaluating Probabilistic Percolation

Earlier papers have presented several strategies for discovering interaction pathways, while often neglecting the influence of water [18]. Here, a percolation propagation algorithm was used to support a model that incorporates the characteristics that I considered to be biophysically valid while capitalizing on the communication capabilities of a small world network [11]:

Pathways are allowed to begin at residues with solvent accessibility, not just the binding site residues. In this fashion the model acknowledges the influence of water, and in particular, the role played by charged residues [6].

There are multiple pathways [19].

Correlations are subject to spreading from one or more surface residues into the protein core along multiple pathways in a simultaneous fashion as opposed to a path determined by a single random walker.

After initial propagation, the continued progression of resulting fluctuations should be subject to some type of attenuation [20].

Fluctuation paths should have a sense of direction [4].

In the past, several researcher have adopted the notion of percolation to describe activity in communication networks within a protein [21, 22].

A perturbation begins at a charged residue and induces fluctuations in a probabilistic fashion on its neighboring residues which in turn induce fluctuations on their neighbors, (again in a probabilistic fashion) in the next step of the spreading process. The fluctuations spread into the residue network in a breadth first fashion, but the fluctuation intensity (FI) steadily decreases in proportion to the product of probabilities encountered on the paths. Expressed mathematically, the FI of residue j (designated as fj) will depend on the fluctuation intensity of the all residues that claim residue j as a neighbor and the values of weights on the edges that join residue j to these perturbing “parents”:

fj=1-∏i∈Pj1-cij2fi.

(9)

cij2 In this equation, based on the OR-rule for mutually non-exclusive events, Pj is the index set specifying all residues that claim residue j as a neighbor and have just received fluctuations in the previous step of the percolation. The term is the square of the correlation calculated when residue i acted as a perturbing residue and residue j was the responding residue. Recall that the square of the correlation was considered to be a probability value.

3 Results

The computational experiments just described were done on the unbound (1BFE) and bound (1BE9) versions of the PDZ3 domain. Perturbations were initiated at charged residues with position numbers: 309, 310, 312, 313, 317, 318, 331, 332, 334, 348, 352, 354, 355, 357, 366, 368, 372, 373, 380, 393, 395, 396, 399, 401, 405, and 411. Final fluctuation intensities were recorded for each residue in the binding site: 323, 325, 326, 327, 328, 339, 372, 379, and 380. To avoid the inclusion of low intensity fluctuations, I deleted any line in Table 1 that had an FI value less than 0.25 for both unbound and bound states. This resulted in the elimination of 8 sub-tables, and hence Table 1 only presents data for 18 of the charged residues. Significantly, several charged residues are on the α3 helix or they are immediate neighbors of α3. This is consistent with the reported importance of α3 [23].

Table 1

The top line in each sub-table specifies the perturbing residue and a notation to represent secondary structure: strand =  =  =  = , coil ––––, helix ~  ~  ~  ~  ~ . For each residue in the binding site, the first column specifies the fluctuation intensity of the unbound state, and the next column specifies the fluctuation intensity of the bound state. The final column holds the ratio of these values.

4 Discussion

The percolation algorithm endeavors to find a sequence of consecutive residue interactions that maintain levels of contact (induced by the low 5 Angstrom neighbor constraint) while allowing some measure of motion (trying to preserve residual entropy) that I describe as fluctuations. The percolation of these interactions within the protein should not be viewed as some continuous process, but more realistically, as minor side-chain alterations that are essentially discretized as the system hops from one energy well to another. In particular, the conformational state of the binding site would be subject to such alterations and the site would assume different conformational arrangements, each with a momentary existence the duration of which depends on the depth of the energy well.

This would invite the idea that there is a type of feedback loop: In the apo state, the binding site is receiving various fluctuations that allow it to present a sample of conformations that are not entirely random (a type of controlled conformational selection process). Upon ligand binding the interactions of the charged residues change [6]. If this hypothetical control by charged residues is true, then we would have three mechanisms at work in the formation of the binding site conformations:

1.

The architecture of the network configuration establishes how the charged residues are connected to the binding site and this would define a base level role in the functionality of this scaffold domain.

2.

The interaction of the charged residues among themselves would alter the frequency of induced fluctuations determined by the rate of persistent making and breaking of hydrogen bonds [6].

3.

The interaction of the charged residues with the charged residues of neighboring proteins would also serve to modify the initiation of these perturbations.

The last point is especially important because PDZ3 is in the highly complex milieu of the PSD-95 protein [24] and the “sensing” of nearby proteins would influence its behavior. While this suggestion is different from the concept of a barcode system [25] it serves a similar purpose.

Returning to the issue of fluctuation propagation, Table 1 gives us a clue about the changes that may occur at the binding site when one charged residue is activated. But this is very limited information: It is similar to tweaking one input of a neural net in an attempt to reveal overall functionality. Nonetheless, the model gives us an indication about the propensity that two neighboring residues have in the establishment of cooperative conformations that define one link in a lengthier path to the binding site. This is consistent with the conformational selection hypothesis.

Finally, it has been observed that truncation of α3 decreases binding affinity by 21-fold [23]. Truncation of α3 is equivalent to removal of a substantial portion of the residue network. In this scenario, the important charged residues (393, 395, 396, 399 and 401) listed in the third column of Table 1 can no longer offer the necessary fluctuations going into the binding site. The protein becomes limited in the offering of a full repertoire of configurations for conformational selection and the binding of the ligand is remarkably decreased. If the system worked with induced fit, the removal of α3 would not be so significant. In a similar fashion, phosphorylation of TYR 397 leading to the unpacking of α3 from the rest of PDZ3 [26] would have a similar effect because of the disruption of links in the residue network.

Do we really need the complexity of a full residue network activated by a set of distributed inputs? I argue that the complexity of such an evolutionarily derived residue network is an essential aspect of its functionality. A recent paper on induced fit and conformational selection [27] stresses the necessity of studying a protein within its milieu while recognizing that proteins may have binding sites that are promiscuous in one setting and more selective in a different setting. The simple notion that allostery involves the transmission of a “signal” from a binding site to a more remote site may not match the extreme complexity of protein functionality.

Funding

This work has been funded by the Discovery Grants Program of the Natural Sciences and Engineering Research Council of Canada (Application Number: RGPIN-2016-04,660).

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Long, D., Brüschweiler, R.: Atomistic kinetic model for population shift and allostery in biomolecules. J. Am. Chem. Soc. 133, 18999–19005 (2011)

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_25

Near-Soma Activated Action Potential in Gonadotropin-Releasing Hormone Neurons

Marina A. Chugunova1  , Sue Ann Campbell1   and Matthew W. Harris1  

(1)

University of Waterloo, Waterloo, Canada

Marina A. Chugunova (Corresponding author)

Email: mchuguno@uwaterloo.ca

Sue Ann Campbell

Email: sacampbell@uwaterloo.ca

Matthew W. Harris

Email: matthew.harris@uwaterloo.ca

Abstract

Gonadotropin-releasing hormone neurons trigger the reproductive axis in mammals by exocytosis of the corresponding hormone. The release of the hormone has two modes: pulsatile and surge-like. Until recently, both modes were modelled as a result of the action potential that is initiated in the proximity of the soma and propagates to the distal part of a neuron where it facilitates one or another mode of a hormone release. New experimental data shows that the pulsatile mode persists even when the electrical activity in proximity of the soma is blocked. We model the propagation of the action potential initiated near the soma on the iSite to show that the significant loss of energy during propagation prevents the sufficient excitation of the membrane on the distal segment of GnRH dendrite to trigger the pulsatile exocytosis.

Keywords

Gonadotropin-releasing hormone neuronsAction potentialSoma-proximal initiation sitePropagation energy lossKisspeptin neurons effect

1 Function and Morphology of the Gonadotropin-Releasing Hormone Neurons

Gonadotropin-releasing hormone neurons (GnRH neurons) are the part of the endocrine hypothalamus structure. They are unevenly distributed over several sections of the hypothalamus, with the majority of the population being located in the medial septal, the rostral preoptic area, and the anterior hypothalamic area [6].

GnRH neurons produce and secrete a specific neuropeptide, called gonadotropin-releasing hormone (GnRH) with a similar decapeptide structure for all mammals. The secretion of GnRH by GnRH neurons is known to happen in more than one area. In this article, we consider the population of the GnRH neurons that releases a hormone into the pituitary portal circulation system via the median eminence.

Delivered into the pituitary, GnRH regulates the release of the luteinizing hormone (LH) and the follicle-stimulating hormone (FSH). When FSH and LH are produced in sufficient amounts and with the required pattern of release, they stimulate the reproduction function of mammals: FSH increases the estradiol secretion and the formation of ovarian follicles whereas LH induces ovulation [11]. Therefore, GnRH neurons in hypothalamus are considered to be a trigger of the reproductive axis.

The majority of GnRH neurons have a bipolar structure [14]. Two dendrites from the opposite sites of the soma project to a length up to 5000 microns. At least one of the dendrites, and in 60% of cases, both dendrites, project to the median eminence. A distal part of the dendrite morphs into the axon-like processes in the area of median eminence.

2 Two Modes of GnRH Exocytosis

Due to the difficulties of measuring the hormone concentration in the area of the median eminence, the patterns of the GnRH release into a portal circulation are determined by the observed corresponding patterns of LH release. By using this approach, during the rodent estrous cycle, two modes of GnRH exocytosis were discovered: pulsatile and a surge-like. The pulsatile mode with a period under 1 h is required for the basal LH and FSH secretion. The surge is described as a rapid 2–4 h increase in GnRH release which happens once every 5 days. A GnRH surge initiates a surge-like release of LH that triggers ovulation [11].

It is well-established that the patterns of GnRH exocytosis are closely connected to the electrophysiological activities in a GnRH neuron, namely, the voltage and calcium dynamics [4]. The classical approach to modelling these processes is to start with an action potential initiated on the soma of a neuron. Propagation of the action potential along the processes of a neuron is expected to generate certain oscillation patterns of the intracellular calcium ion concentration which leads to exocytosis [2, 3, 9, 10, 12]. Following the logic of classical modelling, to change the pattern of the exocytosis, it suffices to change characteristics of the action potential initiated on the soma. This approach to modelling in neuroscience has been developed for decades and provides reliable computational results. In most cases, modeling is simplified to a space-independent model represented by a system of ordinary differential equations.

In recent years, however, a series of experiments conducted independently by several research labs discovered that the pulse and surge modes of the GnRH exocytosis are regulated by two different dendritic domains of the GnRH neuron [5, 13]. These two domains are spatially well-separated: one is a soma-proximal zone, and another is a distal zone, the part of a dendrite located near the median eminence. Hence, these autonomous domains are separated by 1000–5000 microns along a dendrite of a GnRH neuron.

Moreover, experiments show that disabling the electrical activity on the soma-proximal zone does not influence the pulsatile mode of a GnRH release. Therefore, a distal zone electrophysiological activities underlie the pulsatile exocytosis during the estrous cycle. On the other hand, to initiate a surge-like GnRH release, the electrophysiological activity is required on both proximal and distal segments of a GnRH neuron.

Based on this experimental data, the distal zone of a GnRH neuron is, in fact, a non-stop GnRH pulse generator during the estrous cycle whose dynamics is determined by some other factors than the electrical activity on the soma-proximal zone. At the same time, the electrical activity on the soma-proximal zone, spontaneous or induced by synaptic transmission, initiates action potentials on a regular basis. The question arises: why is a near-soma initiated action potential not a trigger of a pulsatile exocytosis in this case? Why is the classical modelling approach described above is not applicable in this case? The question is difficult to address experimentally due to the inaccessibility of the distal zone of the GnRH neuron. Therefore, to answer this question we build a spatiotemporal model of a GnRH neuron.

3 Model Description

The main goal for development of our model is to analyze the amplitude (energy) changes in the action potential while it propagates towards the median eminence, from the soma-proximal zone to the distal zone of a GnRH dendrite. Our computational experiments explore the cases of spontaneous and induced action potentials.

We model the voltage dynamics by the partial differential equation

Cm∂Vm(x,t)∂t=DV∂2Vm∂2x-Iionic(V,x,t)+Iappl,

(1)

where

Iionic(V,x,t)=INa+ICaL+ICaT+IKDR+IM.

(2)

In Eqs. (1) and (2), Cm denotes the membrane capacitance, Vm denotes the membrane potential, Iappl denotes the current applied at the location of the iSite, DV denotes the diffusion coefficient, INa denotes a TTX-sensitive sodium current, ICaL and ICaT denote the L-type and T-type calcium currents, IKDR denotes a delayed rectifier potassium current, and IM denotes the M-type potassium current.

Our model is constructed for a bipolar GnRH neuron with the length of each dendrite being 1000 microns and both dendrites projected to the median eminence. The variable x denotes the distance from the distal part of one of the dendrites, namely, the location in the proximity of the median eminence, where a dendrite morphs into a short axon-like segment. Hence, soma is located at the position x=1000μm.

The majority of action potentials in GnRH neurons are initiated on the iSite, the segment of a dendrite located in a soma-proximal zone at a distance of 50–80 microns from the soma. In this segment, the density of sodium channels is significantly increased. This property of the iSite was modelled by the three-fold increase of the sodium channel conductance. All other parameters of the model are constant throughout the neuron. See the Appendix for the details.

The choice of the one-dimensional model for the action potential propagation along the dendrite was determined by the main goal of our research. That is, to demonstrate that in the case of GnRH neurons, the inevitable loss of the energy prevents the majority of the electrical activity initiated on soma and near-soma region from reaching and affecting the dynamics of the distal part of the neuron. The inclusion of more complex geometry, such as dendritic spines, or intracellular dynamics into the model increases the loss of the energy as the action potential propagates. At the same time, all the ion channels involved into the propagation of the action potential are included into the model [7].

4 Results

In this article, we discuss two particular computational experiments:

1.

Spontaneous, that is, with a zero applied current, action potential is initiated on the iSite. The results are presented in Figs. 1 and 2.

2.

Action potential is initiated on the iSite after an applied current of 100 pA, which represents the external synaptic input. The results are presented in Figs. 3 and 4.

Fig. 1

The membrane potential dynamics in three locations of the GnRH neuron: the iSite, the soma, and a distal portion of a dendrite close to the median eminence. The spontaneous action potential is initiated on the iSite

In Fig. 1, we observe that the spontaneous action potential decreases rapidly while propagating along the dendrite towards median eminence. From the initial maximum of approximately 38 mV on the iSite, the membrane potential drops to merely 11 mV on the soma and propagates further with the same level of maximum values to the distal zone of the GnRH neuron. The consequent oscillations in the membrane potential on the soma and near the median eminence represent potential transients rather than an action potentials.

Fig. 2

Left panel: a spatiotemporal representation of the action potential propagation along the dendrite of a GnRH neuron. The spontaneous action potential is initiated on the iSite. Right panel: relative loss of the energy of the action potential during propagation in comparison to a maximal energy at the point of its origin

In the left panel of Fig. 2, we present a spatiotemporal plot of the same action potential. It shows the loss of the energy in the voltage dynamics in both dendrites. This loss increases towards the median eminence. In the right panel of Fig. 2, the loss of energy is presented in terms of the relative decrease in comparison to the maximum energy of the membrane potential on the iSite. The calculations are conducted based on the energy of the oscillation being directly proportional to the square of the oscillation amplitude. The asymmetry of the plot is caused by the asymmetrical location of the iSite with respect to the dendrites. This plot demonstrates nearly five-fold loss of the oscillation energy during the propagation of the action potential along the dendrite of a GnRH neuron.

In the case where the action potential on the iSite is initiated by the synaptic transmission (Figs. 3 and 4), the maximum of the initial action potential is higher than in the first case due to the additional external energy provided. On the iSite, the membrane potential reaches 44 mV, and those values on soma and distal zone reach 18 mV. Nevertheless, we observe the similar rapid decrease of energy in consequent voltage dynamics.

Fig. 3

The membrane potential dynamics in three locations of the GnRH neuron: the iSite, soma, and a distal portion of a dendrite close to median eminence. The action potential is initiated on the iSite after applied current of 100 pA, which represents the external synaptic input. The action potential is enhanced by the positive estradiol feedback through the population of kisspeptin neurons

Fig. 4

On the left panel: spatiotemporal representation of the action potential propagation along the dendrite of a GnRH neuron. The action potential is initiated on the iSite after applied current of 100 pA, which represents the external synaptic input. The action potential is inhanced by the positive estradiol feedback through the population of kisspeptin neurons. On the right panel: relative loss of the energy of the action potential during propagation in comparison to a maximal energy at the point of its origin

5 Discussion

The drastic change of the amplitude of the membrane potential observed in our computational experiments is caused mainly by two factors: at least a three-fold decrease of the sodium channels distribution density and the energy dissipation due to diffusive nature of action potential propagation. This is likely because the diffusion coefficient in Eq. (1) includes the electrical resistance of the intracellular medium.

We now briefly discuss the possibility of multiple synaptic inputs along the dendrites of the GnRH neuron. Such inputs of energy might help sustain the action potential propagation. The possibility of synaptic inputs of this kind was suggested in some models as a cause for the action potential arriving to the distal zone from the iSite and initiating the pulsatile release of a hormone [3]. However, experimental data shows that the pulsatile pattern of the exocytosis is organized on the distal zone of a dendrite independently of the electrophysiological activity in the soma-proximal zone.

Moreover, the most plausible section of GnRH dendrites that would lead to synchronizing synaptic activity is the area of the bundling between dendrites of distinct GnRH neurons. That area is located closer to the median eminence. The bundled dendrites might share synaptic inputs. However, there is no experimental evidence so far for a direct communication between dendrites [1].

It is important to recall that our computational experiments were conducted under the condition that there is no additional electrophysiological activity in a distal zone of the dendrite. Thus, the results of computational modelling allow us to conjecture that the action potential initiated on the iSite is unable to initiate the corresponding action potential on the non-active distal zone due to a significant loss of energy during propagation. That is, an iSite initiated action potential is not a cause of the pulsatile exocytosis mode of the GnRH neuron. This agrees with the experimental data that blocking activity in the proximal zone does not affect pulsatile exocytosis [5, 13].

Conclusion. We conjecture that the distal part of a GnRH neuron has electrophysiological dynamics which are independent of the soma and iSite. The action potential initiated on the iSite changes this existing dynamics and switches the mode of exocytosis from pulsatile to a surge.

Our current model does not include calcium-activated channels. From the perspective of the energy change, the activation of the intracellular calcium stores such as endoplasmic reticulum and mitochondria would lead in total to further energy loss of the action potential. The addition of such channels is planned for future extensions of our model.

The next step in developing our model is to further research the distal zone dynamics and to combine it with the activities on the soma-proximal zone. This is a rare case in neuronal modeling where the activity in the soma area is secondary to the activity on the distal part of a dendrite.

Appendix

In our model, we use the following parameter values: the membrane capacitance, Cm=350 pF, and the diffusion coefficient, DV=30,000μm2/ms (as determined in [2]) for all segments of the GnRH neuron. All the rest of the parameter values and definitions are taken from [8].

In Eq. (2) for the ionic currents, we define:

INa is a TTX-sensitive sodium current.

This current is modelled by the reaction scheme with four possible states: activated (A), deactivated (D), deactivated-inhibited (D∗), and activated-inhibited A∗. Two of the reaction rates are voltage-dependent as follows:

α(Vm)=101+exp(-(Vm+6)/10),

(3)

β(Vm)=101+exp((Vm+54.4)/4.6).

(4)

The reaction rates α and β are also corrected for the microscopic reversibility by the coefficient

a=k1k2-k1-k20.5,

(5)

where k1=0.3, k1-=0.03, k2=0.001, and k2-=0.01 are the remaining reaction coefficients.

Therefore, the dynamics of the sodium channel is defined by the following three equations:

dDdt=β(Vm)A+k2-(1-D-A-A∗)-(α+k2)D,

(6)

dAdt=α(Vm)D+k1-A∗-(β(Vm)+k1)A,

(7)

dA∗dt=k1A+α(Vm)a(1-D-A-A∗)-k1-+β(Vm)aA∗.

(8)

The formula for the sodium current is

INa=gNaONa(Vm-ENa),

(9)

where the fraction of the conducting channels ONa=ONa3.

Ion channels are distributed along the full length of a neuron with the threefold increase of conductance gNa on the iSite.

ICaL and ICaT are the L-type and T-type calcium currents.

Ion channels are evenly distributed along the length of a neuron and defined as follows:

ICaL=gCaLmCaL2(Vm-ECa),

(10)

and

ICaT=gCaTmCaT2hCaT(Vm-ECa).

(11)

Two potassium currents are included into the model: IKDR, delayed rectifier, and IM, M-type potassium current.

The corresponding channels are evenly distributed along the length of a neuron. They are modelled as follows:

IKDR=gKDRnKDR4(Vm-EK),

(12)

IM=gMnM(Vm-EK).

(13)

The gating variables dynamics is represented by the generalized equation

dxdt=x∞-xτx,

(14)

with the steady-state functions x∞ expressed as follows:

mCaL∞=1+exp(-(Vm+40)/12)-1,

(15)

mCaT∞=1+exp(-(Vm+56.1)/10)-1,

(16)

hCaT∞=1+exp((Vm+86.4)/4.7)-1,

(17)

nKDR∞=1+exp(-(Vm+25)/15)-1,

(18)

nM∞=1+exp(-(Vm+37)/4)-1,

(19)

The corresponding time constants are:

τmCaL=5exp((Vm+15)/25)+exp(-(Vm+15)/25),

(20)

τmCaT=7exp((Vm+50)/9)+exp(-(Vm+50)/9)+0.8,

(21)

τhCaT=22,

(22)

τnKDR=15exp((Vm+30)/15)+exp(-(Vm+30)/15)+1,

(23)

τnM=80exp((Vm+30)/15)+exp(-(Vm+30)/15)+0.8.

(24)

References

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_26

Modeling Patterns of Sex-Dependent Neuroprotection Loss in Alzheimer’s Disease

Corina S. Drapaca1  

(1)

Pennsylvania State University, University Park, PA, USA

Corina S. Drapaca

Email: csd12@psu.edu

Abstract

Alzheimer’s disease (AD) is a degenerative disorder of the cerebral neuro-glial-vascular units that affects older adults. Clinical observations suggest that postmenopausal females are at higher risk of getting AD than males and females with AD experience a faster progression of the disease than aged-matched male patients. Presently, AD is incurable. Mathematical models of AD can highlight mechanisms of AD onset and progression, and, thus, inspire improved therapies. In a recent paper (Drapaca in Fractal Fract 6:457, 2022) a population model of AD that accounts for sex differences was proposed. Variable fractional order temporal derivatives are used where the fractional order models the variable fading memory due to neuroprotection loss caused by AD progression and has different expressions for the two sexes. Since the above-mentioned paper is a purely theoretical proof-of-concept study, in this paper the sex-dependent neuroprotection losses are assumed to be proportional to the age-related declines in female estrogen and male testosterone estimated from clinical data. Numerical and semi-analytical solutions of the population model are given. Computer simulations show that the results agree with clinical observations.

Keywords

Alzheimer’s diseaseFractional calculusPopulation modelSex hormonesIntegro-differential equations

1 Introduction

Alzheimer’s disease (AD), a progressive degenerative disorder of the cerebral neuro-glial-vascular units, is characterized by pathological accumulations of amyloid-β (Aβ) plaques and tau-containing neurofibrillary tangles. It leads to memory loss, cognitive decline, and behavioral changes [15, 18]. According to the World Health Organization, AD is the most prevalent kind of dementia that affects tens of millions of older adults worldwide [9]. Although AD has been known for more than a century [2], its treatment has been elusive and, presently, there is no cure for this disease.

Besides age and the presence of the APOE4 gene, sex is also a major risk factor for AD [12, 13, 15, 25, 28]. Clinical observations suggest that more postmenopausal females get AD than males [3] and females with AD experience a faster progression of the disease than aged-matched male patients [7, 8]. The interlinked factors making AD a sex-biased disease are hormonal state, activation of glial cells and neuroinflammation [14]. Estrogen, the main group of female sex hormones, is involved in numerous neuroprotection processes that require the participation of neurons, astrocytes, microglia, and endothelial cells. Specifically, estrogen protects neurons against Aβ toxicity and the formation of neurofibrillary tangles [5, 28]. During the transition to postmenopause, the amount of estrogen decreases drastically [16], which, together with other aging processes, leads to neuroprotection loss and increased neuroinflammation [26]. The neuroinflammation promotes AD progression and is also enhanced by the presence of the disease leading to the activation of proliferative reactive astrocytes, and the pro-inflammatory M1 and anti-inflammatory M2 activated microglia [6]. Animal models of AD showed that the estrogen-triggered M2 phenotype in adult females is more pro-inflammatory than in males and may be dysfunctional [8].

Mathematical models of AD can highlight mechanisms of AD onset and progression, and, thus, inspire improved, potentially sex-specific, therapies. The majority of published models of AD do not account for sex differences. However, in a recent publication [10], a population model of AD was proposed that incorporates the sex-biased feature of the disease. The model uses the signaling pathways proposed in [19] that happen among the populations of surviving neurons, Ns, dead neurons, Nd, activated microglia in pro-inflammatory state, M1, activated microglia in anti-inflammatory state, M2, quiescent (resting) astrocytes, Aq, proliferative reactive astrocytes, Ap, and aggregation-prone amyloid-β fibrils, Aβ. The pathways providing AD inhibition are Aq→M2,Aq⊣M1,M2⊣M1, while the pathways involved in the AD progression are M1→Nd,Aβ⊣M2,Aβ→M1 (→ denotes activation and ⊣ represents suppression). Variable-order fractional temporal derivatives are also used to model population dynamics. The fractional order models the variable fading memory due to neuroprotection loss caused by AD progression with age and has different expressions for the two sexes. Although the model is a mere proof-of-concept, the results confirm published clinical observations [20–22, 24]. This justifies the further mathematical explorations and improvements of the model in [10] that are presented in this paper.

The aim of this paper is twofold: to use clinical data on sex hormones levels for the expressions of the sex-dependent variable fractional orders, and to find semi-analytical solutions of the proposed linear system of variable fractional order differential equations. It is assumed that the sex-dependent neuroprotection losses are proportional to the percentages of the maximum mean (total) testosterone in males and the maximum mean estradiol (dominant type of estrogen) in females across the lifespan that are calculated from measurements of approximate mean sex-steroid levels in plasma in males and females [16]. The expressions for the sex-dependent variable fractional orders, which will be further called memory functions (express a mathematical property not related to the brain’s functions of memory) are then provided. The range of the memory functions is [0, 1], where values closer to 0 correspond to longer memory (greater neuroprotection) while values closer to 1 represent shorter memory (less neuroprotection). Next, semi-analytical solutions of the population model are found. Existence and uniqueness results for linear systems of fractional order differential equations are proven in [23] for variable fractional orders and in [4, 17] for constant fractional orders. Following [17], semi-analytical solutions using Mittag-Leffler functions are obtained for constant fractional orders. For the variable fractional order case, the memory functions are approximated by piecewise constant functions and the corresponding semi-analytical solutions are obtained from semi-analytical solutions for constant fractional orders. The piecewise constant approximations of the memory functions are obtained using the direct and inverse Haar one-dimensional wavelet transforms, a common approach used for signal/image processing [1]. Numerical solutions of the population model are also found using the numerical scheme in [10]. The results reinforce the findings in [10] namely that (1) postmenopausal females with AD experience a larger decrease in the population of surviving neurons than aged-matched male patients, and (2) the neuronal loss in female patients slows down at later times if a treatment preserving the estrogen-controlled neuroprotection levels is applied. Also, the results show that the semi-analytical solution agrees well with the numerical one for the variable fractional order case. For the constant fractional order case, perfect agreement between the numerical and semi-analytical solutions is observed, except for fractional orders close to 0 when the semi-analytical solution for the Aβ population shows better behavior at very short times than its corresponding numerical solution which suffers from spurious oscillations.

The paper is structured as follows. The next section reviews briefly the mathematical model and its numerical discretization, and presents the semi-analytical solutions. Computer simulations are shown in the Results section, and the paper ends with a Conclusion section.

2 Mathematical Model

The population model is [10]:

0Dtq(t)y=My,

(1)

where y=Ap,Aq,Aβ,M1,M2,Ns,NdT, and:

M=000α5-α400000-α5α40000-αr0-α16α1500-(α7+α12)(α8+α13)α9-α14-(α6+α11)α100(α7+α12)-(α8+α13)-α9α14(α6+α11)-α10-α2α10-α3000α2-α10α3000.

Numerical values for the elements of matrix M are taken from [19] and given in Table 1.

Table 1

Signalling pathways and corresponding rates [10, 19]

Rate (pathway)

Value [year-q(t)]

Rate (pathway)

Value [year-q(t)]

α1 (Aq→Ns)

10-5

α10 (Nd→M1)

10-2

α2 (Ap→Nd)

10-3

α11 (Ns⊣M1)

10-2

α3 (M1→Nd)

10-2

α12 (Aq⊣M1)

10-4

α4 (M2→Aq)

10-4

α13 (Aβ→M1)

10-2

α5 (M1→Ap)

10-2

α14 (M2⊣M1)

10-4

α6 (Ns→M2)

10-2

α15 (Ns→Aβ)

1

α7 (Aq→M2)

10-4

α16 (M2⊣Aβ)

10-2

α8 (Aβ⊣M2)

10-2

αr (M2⊣Aβ)

1

α9 (M1⊣M2)

10-2

The variable-order fractional Caputo derivative of order 0<q(t)≤1 is, by definition:

0Dtq(t)f(t)=1Γ(1-q(t))∫0tdf(τ)dτ(t-τ)q(t)dτ.

(2)

where Γ(z)=∫0∞τz-1exp(-τ)dτ is the gamma function.

The initial condition at t=0 is given at the time of AD onset which is close to the time when postmenopause starts in females (around age 55  [27]) and is taken from [19]:

y(0)=103,105,103,103,105,104,102T.

(3)

The study is run for a period of T=20 years.

Fig. 1

Normalized levels of female estradiol and male testosterone [16] and their best-fitted curves (a, b). Memory functions (4) and (5) associated to the best-fitted curves (c). Therapies (6) for females starting 2, 4, and 10 years after the AD onset are also shown (c)

Percentages of maximum levels of female estradiol and male total testosterone present in plasma of adults over 48 years of age are taken from [16] and normalized, and their corresponding best-fitted curves are found using Matlab’s built-in function lsqcurvefit (Fig. 1a, b). Assuming that the neuroprotection losses due to AD progression with age are proportional to the declines in sex hormones in older females and males, the sex-dependent memory functions defined on the time domain [0, T] are then obtained from these best-fitted curves as follows (Fig. 1c):

qf(t)=0.8082-0.7346×exp(-0.2048t)

(4)

for postmenopausal females, and:

qm(t)=0.0819+0.0065t

(5)

for age-matched males. As the sex-dependent neuronal protections decrease with the AD progression in manners similar to the declines in the levels of female estradiol and male testosterone with age, the memory of the dynamics among considered populations fades (shorter memory is represented by values of the memory functions closer to 1). Figure 1c shows that, over the time period T, the population model (1) experiences a larger memory reduction with memory function qf (maxqf(t)≈0.8,t∈[0,T]) than with memory function qm(t) (maxqm(t)≈0.2,t∈[0,T]).

A treatment for postmenopausal females with AD that starts at time t=ttreat and keeps the neuroprotection level constant is modeled as:

qtreat(t)=qf(t),t≤ttreat,qf(ttreat),t∈(ttreat,T],

(6)

Numerical Solutions

Let 0=t0<t1<…<tN-1<tN=T be an equally-spaced discretization of the interval [0, T] of constant step size h. Denote by yn=y(tn),qn=q(tn), for tn=nh and n=0,1,2,…,N. The discrete form of system (1) obtained using a forward difference approximation scheme of order O(h) for formula (2) is [10]:

I-Γ(2-qn+1)hqn+1Myn+1=yn-∑j=1n(j+1)1-qn+1-j1-qn+1yn+1-j-yn-j,

(7)

where I is the identity matrix, and n=0,1,…,N-1.

As in [10], the numerical solution of system (1) with initial condition (3) for a constant fractional order q is found using Matlab’s function fde_pi1_ex [11]. For a variable fractional order q(t), system (7) with initial condition (3) is solved using Matlab’s built-in function lsqnonneg. Computer simulations shown in the next section are obtained for a step size h=0.001.

Semi-Analytical Solutions

The direct and inverse one-dimensional Haar wavelet transforms are applied to memory functions (4) and (5) to obtain their associated piecewise constant approximants on the discretization of [0, T] introduced earlier. Matlab’s built-in functions haart and ihaart are used for this task. On each subinterval [tn,tn+1],n=0,1,…,N-1, the approximants are constants in (0,1] and thus a semi-analytical solution can be found on each subinterval as in [17]. If 0<q~n≤1 is the constant Haar approximant of an arbitrary variable order q(t) on the subinterval [tn,tn+1], then the solution of system (1) for q~n is a linear combination of the eigenvectors of matrix M multiplied by the Mittag-Leffler functions involving the eigenvalues of M. Using Matlab’s built-in function eig, the eigenvalues λi,i=1,…7 and eigenvectors vi,i=1,…7 of M are found. Since two eigenvalues (λ5 and λ6) are a complex conjugate pair, the semi-analytical solution for q~n and t∈[tn,tn+1] is:

y(t)=∑j=14cjvjEq~nλjtq~n+c5realv52Eq~nλ5tq~n+Eq~nλ6tq~n-c5imagv52iEq~nλ5tq~n-Eq~nλ6tq~n+c6imagv52Eq~nλ5tq~n+Eq~nλ6tq~n+c6realv52iEq~nλ5tq~n-Eq~nλ6tq~n+c7v7Eq~nλ7tq~n

(8)

where the Mittag-Leffler function is Eq(z)=∑j=0∞zjΓ(qj+1) (note E1(z)=expz). Thus, semi-analytical solutions are obtained by using expression (8) for the constant Haar approximants corresponding to the memory functions (4) and (5). Constants ci,i=1,…7 are found from the initial condition (3).

3 Results

Figures 2, 3 and 4 show the numerical solutions of system (1), and Fig. 5 compares numerical and semi-analytical solutions of the population model. The values of the constant fractional orders q∈{0.1,0.3,0.8} are chosen to be close to the minimum and maximum values of the memory functions qf(t) and qm(t) seen in Fig. 1. The solutions of system (1) for variable fractional orders show, as expected, intermediate behavior between the solutions corresponding to the constant fractional orders bounding the variable fractional ones (Figs. 2 and 5a). The case q=1 is shown for reference.

Postmenopausal females with AD have a larger amount of Aβ fibrils (Fig. 2) than age-matched male patients and thus they also see larger decreases (increases) in the populations of Ns (Nd) (Fig. 3a, b), M1 (M2) (Fig. 4a, b), and Ap (Aq) (Fig. 4c, d). For female patients, the results show that among the three starting dates (2, 4 and 10 years after the AD onset) of a treatment preserving the estrogen-controlled neuroprotection levels, the lowest amount of Aβ fibrils and, consequently, the largest population of Ns, are predicted for the therapy starting 2 years after the AD onset (Figs. 2 and 3c, d). Thus, the sooner a treatment starts and the longer it is applied the better the outcome is. Figures 3c, d also show that neuronal loss slows down at later times regardless of the starting time of the treatment. Due to space limitation, results for the other populations are not shown.

Lastly, Fig. 5 shows that the semi-analytical solutions agree well with the numerical ones for variable fractional orders. Perfect agreements between the numerical and semi-analytical solutions are observed for constant fractional orders (Fig. 5a), except for the case q=0.1 for which the semi-analytical solution for the Aβ population shows better (expected) behavior at very short times than its corresponding numerical solution which suffers from spurious oscillations which appear not to disappear by decreasing the step size h (Fig. 5b). The higher accuracy, along with the much shorter computational times, highlights the superiority of the semi-analytical solutions over the numerical solutions of system (1). Due to space limitation, results for the other populations are not shown.

Fig. 2

Numerical solutions of population Aβ for various constant fractional orders, and for qf(t), qm(t), and qtreat(t)

Fig. 3

Numerical solutions of populations Ns (a, c) and Nd (b, d) for qf(t), qm(t) and qtreat(t). Figures a, b share the legend inside Fig. b, and Figs. c, d share the legend located on the top of Fig. d

Fig. 4

Numerical solutions of populations M1 (a), M2 (b), Ap (c), and Aq (d) for qf(t) and qm(t). The figures share the legend inside Fig. d

Fig. 5

Numerical and semi-analytical solutions of populations Aβ (a, b), Ns (c) and Nd (d) for qf(t) and qm(t). Figure a also shows results for constant fractional orders. Figure b is a zoom-in of Fig. a for q=0.1. Figures c, d share the legend located on the top of Fig. d. In the legend maps, (n) and (a) stand for numerical and respectively semi-analytical solutions

4 Conclusions

This paper expands the work in [10] in the following two ways. First, measurements of approximate mean sex-steroid levels in plasma in males and females from [16] are used to define the sex-dependent memory functions of population model (1). Computer simulations, like the results in [10], show a larger decrease in the population of Ns in postmenopausal females with AD than in age-matched male patients and a treatment preserving the estrogen-controlled neuroprotection levels slows down the neuronal loss in female patients at later times. The sooner the therapy starts and the longer it is applied the better the outcome is which agrees with clinical evidence [21, 24]. Second, semi-analytical solutions of system (1) are obtained using piecewise constant Haar approximants for the memory functions and the semi-analytical solutions of (1) for constant fractional orders that involve the Mittag-Leffler functions. The semi-analytical and numerical solutions for variable fractional orders agree well. Also, the semi-analytical and numerical solutions for constant fractional orders agree very well except for the population of Aβ fibrils for which the numerical solution suffers from spurious oscillations for fractional orders close to 0 and at very short times, while the semi-analytical solution has the expected non-oscillatory behavior. This is an important advantage of semi-analytical solutions over the numerical ones.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_27

Deciphering Heterogeneity of ATP-Induced Ca2+ Responses in Osteoblasts Using the Flux-Balance Model

Anthony Quint1  , Nicholas Mikolajewicz1  , Svetlana V. Komarova1   and Anmar Khadra1  

(1)

McGill University, Montreal, QC, Canada

Anthony Quint

Email: anthony.quint@mail.mcgill.ca

Nicholas Mikolajewicz

Email: nicholas.mikolajewicz@mail.mcgill.ca

Svetlana V. Komarova

Email: svetlana.komarova@mcgill.ca

Anmar Khadra (Corresponding author)

Email: anmar.khadra@mcgill.ca

Abstract

When P2 receptors (P2Rs) are activated by ATP, they trigger an array of heterogeneous Ca2+ responses in osteoblasts. Previous mathematical modeling of this system has identified P2Y2R and P2X7R as the key receptors involved in generating Ca2+ responses that transition between transient and oscillatory when ATP concentration ([ATP]) is changed. Experimental recordings of these Ca2+ responses have also occasionally displayed low amplitude, high frequency Ca2+ oscillations (flickers). To further examine this phenomenon and heterogeneity, we expanded in this study the Ca2+ flux-balance model of this system by dividing the cell into three compartments: cytosolic shell (within 80 nm of cellular membrane) and bulk (rest of the cell), as well as the endoplasmic reticulum (ER). We further used an R script to cluster Ca2+ responses. Our results revealed that the model can maintain the [ATP]-induced transitions, and that the shell compartment is likely responsible for producing Ca2+ flickers, while the bulk compartment is responsible for the low frequency, high amplitude Ca2+ oscillations. The clustering analysis yielded 19 distinct clusters, with the mathematical model identifying the contributing fluxes for each.

Keywords

Osteoblastic calcium signalingHeterogeneity and clusteringATP stimulationFlux-balance modelBifurcation analysis

1 Introduction

The human skeleton can alter its morphology and metabolism to adapt to varying external mechanical forces exerted by its environment [1]. Extracellular ATP is released from bone forming osteoblasts into the extracellular milieu due to such mechanical forces, including stress and hydrostatic pressure [2], where the ATP then initiates intracellular signaling through P2 receptors (P2Rs) [3–5]. The presence of P2Rs on these cells suggests that purinergic signaling plays an important role in skeletal mechano-transduction [6, 7].

For osteoblastic cells of various origins, transcripts for P2Y2R and P2X7R are amongst the most highly expressed of the P2 family, an aspect that was further verified functionally [8]. Together, their combined response to [ATP] spans six orders of magnitude (10-8 to 10-2 M) [7, 8].

Fluorescent Ca2+ imaging techniques on osteoblasts demonstrated one key ATP-mediated dose-dependent feature manifested as a shift between three modes of activity [8]. These modes include a transient response with a semi-pronounced narrow peak at low [ATP], a slow oscillatory response superimposed on decaying Ca2+ profile at intermediate [ATP], and a transient response with a pronounced wide peak at high [ATP] [8]. Additionally, the data showed that at the plateau component of the dose response curve, the peak amplitude of Ca2+ response exhibits two local dips at a moderate (10-4 M) and elevated (10-2 M) [ATP] [8]. In [8], a Ca2+ flux-balance model was developed to establish the sufficiency of the P2Y2Rs and P2X7Rs in recreating the ATP-mediated shifts between the three modes of activity and the profile of the dose-response curve.

Fluorescent Ca2+ imaging also revealed another feature not previously accounted for by the flux-balance model or data analysis, namely, the high frequency, low amplitude oscillations (Ca2+ flickers) superimposed on the low frequency, high amplitude oscillations (Ca2+ spikes) [9]. Such flickers are likely deterministic and due to Ca2+ microdomain activity near the plasma membrane (PM) of the cell [9], i.e., not due to noise. These microdomains could represent the small volume pockets of high Ca2+ concentration that diffuse from the PM of the cell (shell compartment), towards the centre of the cell (bulk compartment) and contribute to the spatially averaged Ca2+ signal that is experimentally detected, i.e. the Ca2+ spikes. On a different note, analysis of Ca2+ signals suggests that the previously discussed [ATP]-induced transitions between transient and oscillatory regimes are not strict, in that Ca2+ signals produced at a given [ATP] can exhibit transient or oscillatory characteristics, or even complex combinations of both. This phenomenon is hypothesized to be due to variability in the kinetics of cell or ER membrane receptors.

In this study, we aim to: (1) explore which physiological mechanisms are at play in producing the Ca2+ flickers, and (2) classify all the Ca2+ responses detected and characterize the role of fluxes in defining each class. To accomplish this, we will implement a mathematical modeling approach, focused on expanding a previously developed Ca2+ flux-balance model [8], that accounts for new fluxes and the shell and bulk compartments, as well as using a machine classifier to explain the sources of heterogeneity in Ca2+ responses detected.

2 Mathematical Model

Ca2+ signalling in osteoblasts was previously described using a flux-balance model [8]. Here, we expand this model by dividing the cell into three compartments (Fig. 1): ER, cytosolic shell in the region 80 nm deep into the cell from the PM (denoted by S), and cytosolic bulk representing the remaining inner region of the cell (denoted by B). The respective Ca2+ concentration in each compartment is denoted by [Ca2+]ER, [Ca2+]S, [Ca2+]B, respectively. Based on this, the cytosolic Ca2+ concentration ([Ca2+]Cyt), is given by [Ca2+]Cyt=V¯s[Ca2+]S+V¯B[Ca2+]B/V¯Cyt, where V¯X represent shell (X=S), bulk (X=B) and cytosolic (X=Cyt) volumes.

Cytosolic and ER Ca2+ are assumed to be governed by eight fluxes across the PM and ER membrane as well as Ca2+ exchange between the shell and bulk compartments (Jca-ex=vCa([Ca2+]S-[Ca2+]B), where vCa=7 s-1 is the maximal exchange rate). Fluxes at the PM are either inward through L-type Ca2+ channels (JIL), P2X7Rs (JP2X7R-Ca), store-operated Ca2+ channels (JSOCC) and leak (JInLeak), or outward through the PMCA pumps (JPMCA), whereas those at the ER membrane are either outward through leak (JERLeak) and IP3Rs (JIP3R), or inward through the SERCA pumps (JSERCA).

The P2Y2-dependent JIP3R used in this model is taken from the Li-Rinzel formalism for IP3Rs [10]. In this study, the inactivation variable is split into a bulk (hB) and shell (hS) components, based on where the ER is located to account for their respective dependencies on [Ca2+]B and [Ca2+]S. As in [8], the Li-Rinzel model of IP3Rs is modified to include the dynamics of [ATP]-dependent intracellular IP3 concentration ([IP3]Cyt) governed by P2Y2R activity. Because of the bulk-shell formalism, [IP3]Cyt is split into bulk ([IP3]B) and shell ([IP3]S) concentrations, coupled together through the exchange between the two compartments JIP3-ex=vIP3([IP3]S-[IP3]B), where vIP3=7 s-1 is the maximal exchange rate.

The flux JP2X7R-Ca through P2X7Rs is identical to that in [8], where a 12-state Markov model of P2X7R kinetics is used [11]. A flux through L-type Ca2+ channels is also included due to their expression in osteoblasts [12]. This flux depends on the kinetics of these channels with one activation variable mL as described in [13]. Given that osteoblasts are passive cells, the voltage is assumed to be constant. This assumption is used to compute the currents produced by P2X7R and L-type Ca2+ channels and than converted to flux using the expression Jη=-αIη, where η=P2X7R,L and α=1/zV¯CytF (z=2 is Ca2+ valence and F is Faraday’s constant).

The ER Ca2+ leak flux JERLeak, as well as the Ca2+ and IP3 bulk/shell exchange fluxes Jca-ex and JIP3-ex, are assumed to be proportional to the concentration differences between their compartments [10]. The SERCA and the PMCA pumps are modelled using Hill functions [14]. Meanwhile, SOCCs, which are Ca2+ channels on the PM that are activated when ER Ca2+ stores are depleted [15], are modelled using a fourth order reverse Hill function [16]. Lastly, the inward Ca2+ leak JInLeak is assumed to be a small constant value.

The model describing Ca2+ handling in osteoblasts is thus given by

d[Ca2+]Sdt=fivs(JInLeak+JIP3R-S+JERLeak-S-JSERCA-S

(1)

-JCa-ex-JPMCA-S+JP2x7R-CA+JSOCC+JIL)

(2)

d[Ca2+]Bdt=fivB(JIP3R-B+JERLeak-B-JSERCA-B+JCa-ex)

(3)

d[Ca2+]ERdt=fERvER(JSERCA-S+JSERCA-B-JIP3R-S-JIP3R-B

(4)

-JERLeak-S-JERLeak-B)

(5)

d[IP3]Sdt=vS(αATP[ATP][ATP]+kATP-δATP[IP3]S-JIP3-ex)

(6)

d[IP3]Bdt=vB(JIP3-ex-δATP[IP3]B)

(7)

dhXdt=h∞X-hXTh-X(X=S,B)

(8)

dmLdt=m∞L-mLTL,

(9)

where fi (fER) is the fraction of free cytosolic (ER) Ca2+, vS=48, vB=1 and vER=9.4 are cytosol/shell, cytosol/bulk and cytosol/ER volume ratios, respectively, αATP is the maximum rate of IP3 production by P2Y2Rs, kATP is the half-maximal production of IP3 by P2Y2Rs, δATP is degradation rate for IP3, h∞ (m∞L) is the steady state inactivation (activation) of IP3Rs (L-type Ca2+ channels), and τh (τL) is the time constant of IP3R inactivation (L-type Ca2+ activation). The model was simulated using the ode15s solver in MATLAB (see link). ATP stimulation was performed starting from the “resting state” obtained by simulating the model for 2000 s prior to ATP application. With no ATP present, [IP3]cyt=0.

Fig. 1

A diagram summarizing the main fluxes and the three compartments included in the mathematical model describing Ca2+ handling in osteoblasts

3 Slow-Fast Analysis to Define the Oscillatory Regime

With the new modifications made to the model, it is imperative to verify whether [Ca2+]B is still able to exhibit the transitions from transient to oscillatory and back to transient responses due to the P2Y2R-dependent Ca2+-induced Ca2+-release through IP3Rs upon stimulation with increasing doses of [ATP]. Since [IP3]B and [Ca2+]ER are both slow variables in our system, we apply a slow-fast analysis to investigate if the new model is capable of producing an oscillatory regime that can induce such transitions. This is done by assuming that there is no Ca2+ flux through P2X7R and by setting the two variables [IP3]B and [Ca2+]ER to be parameters, allowing us to plot the two-parameter bifurcation of [Ca2+]B with respect to [Ca2+]ER and [IP3]B (Fig. 2). As expected, the P2X7R-knockout system (i.e., when JP2X7R=0) produced an oscillatory regime (gray-shaded area) bounded by two supercritical Hopf bifurcations (Fig. 2). This indicates that trajectories may or may not cross this regime as the [IP3]B and [Ca2+]ER change (see arrows in Fig. 2). It follows that Ca2+ responses can transition from regimes that are transient, to oscillatory and back to transient while [Ca2+]ER decreases.

Fig. 2

Two parameter bifurcation diagram of [Ca2+]B with respect to [Ca2+]ER and [IP3]B. The grey region represents the oscillatory regime with a boundary defined by two supercritical Hopf bifurcations. Trajectories that start at the red circle could either miss the oscillatory regime (red line) and generate a small transient response, go through the oscillatory regime (green line) and generate slow oscillations, or go across the oscillatory regime very briefly (blue line) and generate a pronounced transient response

4 Fast Versus Slow Oscillations

One of the most important features detected in the heterogeneous osteoblastic Ca2+ responses is the presence of Ca2+ flickers superimposed on transient Ca2+ responses when stimulated with 1 μM ATP (Fig. 3a). Upon stimulating the cell model with a step-pulse of 1 μM ATP, Ca2+ flickers are observed (Fig. 3b). These fast oscillations are mainly due to the high Ca2+ activity in the shell compartment induced by Ca2+ mobilization across the membrane (Fig. 3c); this translated into small amplitude, high frequency oscillations in the bulk compartment (Fig. 3d) that remained visible when combining both compartments to produce the [Ca2+]Cyt response (Fig. 3b). These simulated cytosolic Ca2+ responses resemble those obtained experimentally (Fig. 3a), suggesting that Ca2+ flickers seen in Ca2+ recordings may be due to PM channel activities near the cell membrane and not just stochastic fluctuations.

Fig. 3

Recorded and simulated cytosolic osteoblastic Ca2+ responses showing high-frequency small amplitude oscillations. a Time course of a recorded fluorescence Ca2+ signal in an osteoblastic cell stimulated with [ATP]=1μM. Time series simulations of b [Ca2+]Cyt, c [Ca2+]S, and (D) [Ca2+]B produced by the model when 1μM ATP pulse is applied

5 Clustering Analysis

For each given [ATP], a hierarchical clustering algorithm from the WGCNA R library is used to divide Ca2+ response signals into clusters based on similar features. The level of smoothing is chosen to eliminate white noise from the Ca2+ response signals, while dampening the presence of the Ca2+ flickers. This induced enough homogeneity in the signals to produce 19 distinct clusters.

These clusters are then subsequently analyzed manually to detect potential “outliers” whose features are more consistent with other clusters; the outliers are then assigned to these appropriate cluster. Afterwards, the minimum value of each signal is subtracted to standardize the signal’s baseline to zero. The same standardization is applied to the simulated signals to allow for better comparisons.

In Table 1, the clusters for each [ATP] are summarized. The parameter changes that are required for the mathematical model to produce a typical profile from each cluster, as well as their qualitative traits, are stated in the table.

Table 1

The osteoblastic Ca2+ signal clusters, along with model parameters specific to each cluster (i.e., the parameters capable of generating the signaling characteristics of their respective cluster)

[ATP]

Cluster #

Parameter changesa

Qualitative traits

10-8 M

1

None

No response

10-7 M

1

None

No response

–

2

vSERCA = 20→3, vPMCA = 8→10, vIP3R = 15→60

Small amplitude, fast decay transient response

–

3

vSERCA = 20→10, vIP3R = 15→35, vCa = 7→15

Small amplitude, slow decay transient response with a hump

10-6 M

1

vPMCA = 8→13, vIP3R = 15→12

Medium amplitude, medium decay oscillatory response

–

2

vSERCA = 20→3.5, vCa = 7→25

Small amplitude, fast decay transient response

–

3

vSERCA = 20→8, vPMCA = 8→7.5, vIP3R = 15→100, vCa = 7→35

Large amplitude, fast decay transient response with hump

–

4

vSERCA = 20→11, vPMCA = 8→5.5

Medium amplitude, slow decay oscillatory response

10-5 M

1

vSERCA = 20→5

Medium amplitude, fast decay transient response

–

2

vSERCA = 20→12, vPMCA = 8→9, vCa = 7→30

Medium amplitude, fast decay transient response with hump

–

3

vSERCA = 20→1.8, vPMCA = 8→2.8

Large amplitude, slow decay transient response

–

4

vSERCA = 20→35, vPMCA = 8→12, vIP3R = 15→14, vCa = 7→22

Large amplitude, slow decay oscillatory response

10-4 M

1

vSERCA = 20→2.25, vPMCA = 8→4

Medium amplitude, medium decay transient response

–

2

vSERCA = 20→2.25, vPMCA = 8→3.75, vIP3R = 15→25

Large amplitude, slow decay transient response

10-3 M

1

vSERCA = 20→5, vIP3R = 15→35

Large amplitude, fast decay transient response

–

2

vSERCA = 20→2.25, vPMCA = 8→6

Large amplitude, slow decay transient response

10-2 M

1

vSERCA = 20→3.75, vPMCA = 8→7.85

Medium amplitude, fast decay transient response

–

2

vSERCA = 20→2.75, vPMCA = 8→7, vIP3R = 15→35

Large amplitude, slow decay transient response

a Arrows point to modifications made to default parameter values stated to the left of each arrow. These modifications may not be unique

6 Discussion

Fast oscillations (flickers) are hypothesized to be caused by the presence of Ca2+ microdomains near the PM of osteoblastic cells. Rather than viewing Ca2+ signalling as a “common pool” event where Ca2+ rises and falls uniformly, it can better be understood as the spatial averaging of localized, short-lived Ca2+ microdomains that exist near the membrane of the cell (shell compartment), which eventually diffuse to the center of the cell (bulk compartment). Ca2+ transients exhibit faster dynamics with a greater rising rate and briefer duration in the submembrane layer than in the slow bulk cytosol due to the presence of cell membrane receptors and channels, allowing for timescale separation to exist between the shell and the bulk compartments of the cell. Ca2+ transients immediately beneath the cell membrane (flickers) are critically involved in processes such as excitation-contraction coupling in muscle cells, hormone secretion in (neuro)endocrine cells [17], and neurotransmitter release in neurons, which are thought to be driven by Ca2+ concentration 10 to 100 times higher than those measured in the bulk cytosol [9]. This suggests that similar important physiological processes may also occur in bone cells.

Previous studies have explored how to account for the spatiotemporal dynamics of Ca2+ signals using partial differential equations (PDEs) [17, 18]. To avoid using such complex models, we have divided the cell into shell and bulk compartments to describe osteoblastic Ca2+ signals. Mathematically, the fast dynamics in the shell is achieved by the numerous PM fluxes in the shell compared to the bulk compartment. Such model formalism aptly captures the slow and fast dynamics of the signal, including [Ca2+] flickers and spikes. The parameter regime which produces these flickers relies on the interplay of a few key mechanisms involving PMCA, SERCA, IP3R and Ca2+ exchange. The rate constants of these fluxes are vital in reproducing the superimposed flickers, while Ca2+ exchange needs to be sufficiently slow so as to not immediately homogenize the shell and bulk compartments and lose the timescale separation between the bulk and shell compartments needed to generate these fast oscillations.

Understanding the potential impact of experimental noise on Ca2+ signals, especially when differentiating flickers from white noise, is imperative. Firstly, some light smoothing has been applied to reduce the noise, while maintaining the fast dynamics superimposed on the Ca2+ signals. Since many signals still have these superimposed oscillations after being smoothed (albeit with varying amplitude due to noise), and because many signals do not have any superimposed oscillations to begin with, there is no evidence to suggest that these flickers should be immediately discarded as extraneous noise. Rather, we argue that recognizing these dynamic processes is vital in appropriately understanding the revised dogma of Ca2+ flicker dynamics, while providing an apt framework on how signal-transduction functions in osteoblasts. With that said, it’s crucial to note that our model is inherently deterministic, which, in turn, has hindered our ability to accurately capture the observed variations in flicker amplitudes in few experiments.

Model simulations for the clusters obtained from the data do not produce any flickers. This showcases the sensitivity of the model and its corresponding parameter regime in producing this unique dynamic behaviour. While the raw Ca2+ signal clusters themselves are smoothed to remove any fast oscillations for the sake of a reasonable clustering analysis, most of the signals in each cluster are already sufficiently homogenized and do not have fast oscillations to begin with. This implies that visualizing these fast oscillations is a relatively rare event.

The clustering analysis highlights the heterogeneity of the data and the signatures of signals at various [ATP]. While the Ca2+ signals generally transition from transient to oscillatory to transient responses as [ATP] goes from low (10-8,10-7) M to medium (10-6,10-5) M and then to high (10-4,10-2) M levels, the clustering analysis illustrates that this principle is not strict. For instance, transient signals are observed at medium [ATP] as well. To elucidate between the various clusters at the different [ATP], we have perturbed the parameters of the flux-balance model to provide an appropriate fit to one characteristic signal of the cluster, and then inferred what the parametrization implies physiologically. We have concluded that the key mechanisms responsible for controlling the main dynamics of the signals are fluxes due to PMCA pump, SERCA pump, IP3R and Ca2+ exchange. Our model simulations suggest that PMCA is primarily responsible for controlling the amplitude of a signal and its decay phase, SERCA is primarily responsible for controlling the large-scale oscillatory behavior of the Ca2+ signal, IP3Rs is responsible for controlling the amplitude of the signal, and Ca2+ exchange is instrumental in controlling the rate of spatial Ca2+ homogenization, a factor that aids in eliciting a transient response rather than an oscillatory one, and in eliminating the flickers by reducing the timescale separation between the shell and bulk compartments. Physiologically speaking, the effects of these fluxes are linked to either increasing the likelihood of proliferative or apoptotic events within the cell, as they determine if the Ca2+ signal reaches a sufficient magnitude to trigger one event or the other [19].

In conclusion, in this study, we have successfully developed a deterministic model that can capture the variability of the osteoblastic Ca2+ responses and explain their underlying dynamics. We have focused mainly on the temporal dynamics and accounted for the spatial inhomogeneity by considering a shell-bulk formalism.

References

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Yavropoulou, M.P., Yovos, J.G. The molecular basis of bone mechanotransduction. J. Musculoskelet. Neuronal Interact. 16(3), 221 (2016)

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Lazarowski, E.R., Rochelle, L.G., O’Neal, W.K., Ribeiro, C.M.P., Grubb, B.R., Zhang, V., Harden, T.K., Boucher, R.C.: Cloning and functional characterization of two murine uridine nucleotide receptors reveal a potential target for correcting ion transport deficiency in cystic fibrosis gallbladder. J. Pharmacol. Exp. Therapeut. 297(1), 43–49 (2001)

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Alan North, R.: Molecular physiology of p2x receptors. Physiol. Rev. (2002)

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von Kügelgen, I.: Pharmacological profiles of cloned mammalian p2y-receptor subtypes. Pharmacol. Therapeut. 110(3), 415–432 (2006)

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Orriss, I.R., Key, M.L., Brandao-Burch, A., Patel, J.J., Burnstock, G., Arnett, T.R.: The regulation of osteoblast function and bone mineralisation by extracellular nucleotides: the role of p2x receptors. Bone 51(3), 389–400 (2012)

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Jeffrey Dixon, S., Sims, S.M.: P2 purinergic receptors on osteoblasts and osteoclasts: potential targets for drug development. Drug Dev. Res. 49(3), 187–200 (2000)

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Xing, S., Grol, M.W., Grutter, P.H., Jeffrey Dixon, S., Komarova, S.V.: Modeling interactions among individual p2 receptors to explain complex response patterns over a wide range of atp concentrations. Front. Physiol. 7 (2016)

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Mikolajewicz, N., Smith, D., Komarova, S.V., Khadra, A.: High-affinity p2y2 and low-affinity p2x7 receptor interaction modulates atp-mediated calcium signaling in murine osteoblasts. PLoS Comput. Biol. 17(6), e1008872 (2021)

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Wang, S.-Q., Wei, C., Zhao, G., Brochet, D.X.P., Shen, J., Song, L.-S., Wang, W., Yang, D., Cheng, H.: Imaging microdomain ca2+ in muscle cells. Circul. Res. 94(8), 1011–1022 (2004)

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Li, Y.-X., Rinzel, J.: Equations for insp3 receptor-mediated [ca2+] i oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. J. Theoret. Biol. 166(4), 461–473 (1994)

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Khadra, A., Tomić, M., Yan, Z., Zemkova, H., Sherman, A., Stojilkovic, S.S.: Dual gating mechanism and function of p2x7 receptor channels. Biophys. J. 104(12), 2612–2621 (2013)

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Grygorczyk, C., Grygorczyk, R., Ferrier, J.: Osteoblastic cells have l-type calcium channels. Bone Miner. 7(2), 137–148 (1989)

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Zeng, S., Li, B., Zeng, S., Chen, S.: Simulation of spontaneous ca2+ oscillations in astrocytes mediated by voltage-gated calcium channels. Biophys. J. 97(9), 2429–2437 (2009)

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Croisier, H., Tan, X., Perez-Zoghbi, J.F., Sanderson, M.J., Sneyd, J., Brook, B.S.: Activation of store-operated calcium entry in airway smooth muscle cells: insight from a mathematical model. PLoS One 8(7), e69598 (2013)

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Putney Jr., J.W.: Store-operated calcium channels. In: Handbook of Cell Signaling, pp. 911–914 (2010)

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Cao, P., Tan, X., Donovan, G., Sanderson, M.J., Sneyd, J.: A deterministic model predicts the properties of stochastic calcium oscillations in airway smooth muscle cells. PLoS Comput. Biol. 10(8), e1003783 (2014)

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Fletcher, P.A., Li, Y.-X.: An integrated model of electrical spiking, bursting, and calcium oscillations in gnrh neurons. Biophys. J. 96(11), 4514–4524 (2009)

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Oprea, L., Desjardins, N., Jiang, X., Sareen, K., Zheng, J.Q., Khadra, A.: Characterizing spontaneous ca2+ local transients in opcs using computational modeling. Biophys. J. 121(23), 4419–4432 (2022)

19.

Liu, L., Li, H., Cui, Y., Li, R., Meng, F., Ye, Zi., Zhang, X.: Calcium channel opening rather than the release of atp causes the apoptosis of osteoblasts induced by overloaded mechanical stimulation. Cell. Physiol. Biochem. 42(2), 441–454 (2017)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_28

Parameter Estimation of Hodgkin-Huxley Model: A Comparative Study of Genetic Algorithm and Artificial Neural Network Approach

Pengfei Yue1  , Allan R. Willms1   and Monica G. Cojocaru1  

(1)

Department of Mathematics and Statistics, University of Guelph, Guelph, Canada

Pengfei Yue (Corresponding author)

Email: yuep@uoguelph.ca

Allan R. Willms

Email: awillms@uoguelph.ca

Monica G. Cojocaru

Email: mcojocar@uoguelph.ca

Abstract

The Hodgkin-Huxley model is a well-known mathematical model for the generation and propagation of action potentials in neurons. However, determining certain parameters of the model involves a complex combination of experiments and data tuning. Parameter estimation of the Hogkin-Huxley model has been an active research area for several decades, and various optimization techniques have been used including Genetic Algorithms. The Genetic Algorithms have been successful in finding parameter values that fit experimental data well, but they can be computationally expensive and may require significant expertise to tune. More recently, Artificial Neural Networks have been applied to parameter estimation of mathematical models, it can often provide faster and more accurate results than traditional and global optimization methods. In this research, we propose an dense neural network approach for parameter estimation in the Hodgkin-Huxley model, which is trained on a large voltage-clamp type simulations. Our findings showcase its superiority over Genetic Algorithms, in terms of computational efficiency and accuracy.

Keywords

Parameter estimationMachine learningGenetical algorithmInverse problemArtificial neural networksEvolutionary algorithms

1 Introduction

The Hodgkin-Huxley (HH) model is a mathematical model that describes the generation and propagation of action potentials in neurons, developed by Alan Hodgkin and Andrew Huxley in 1952 [8]. It is based on the flow of current through ion-selective channels in the neural membrane and contains probabilistic representations of the activation and inactivation of ionic conductances [1, 14]. The model is a dynamical system whose state variables are the activation and inactivation variables for the Sodium and Potassium channels. Hodgkin and Huxley themselves estimated some the the parameters in the model using a combination of experiments and data tuning, which can be extensively complicated and time-consuming. In this study, we use current recordings from voltage clamp simulations of isolated ionic currents. The voltage clamp holds the membrane potential at a constant level thus facilitating an analytic solution of the differential equations.

Studies have been carried out concerning the parameter estimation of the HH model using optimization methods, including the Levenberg-Marquardt algorithm [10] and gradient descent methods [3]. However, these optimization algorithms may encounter challenges, including convergence to local minima. A good way to avoid such local solutions is to use metaheuristics. Nevertheless, these global search algorithms require a large computational effort for estimation as emphasized in [2]. On the other hand, artificial neural networks (ANN) have proven effective in parameter estimation for the FitzHugh-Nagumo model, a simplified version of the HH model [12]. However, there have not been studies utilizing and comparing both approaches to solve the parameter problem in the HH model. In this paper, we develop a dense neural network model to estimate parameters involved in the Potassium channel of the HH model, and compare it with the Genetic Algorithm, revealing the dense neural network model to be more accurate and reliable.

2 Hodgkin-Huxley Formalism and the Voltage Clamp Simulations

2.1 Hodgkin-Huxley Ion Channel Model Formulation

A voltage-gated ion channel is a channel where the opening and closing of the pore is in response to the voltage difference across the membrane. The Hodgkin-Huxley formalism can be employed to model the current of a general voltage-gated ion channel, expressed as:

I=g¯mphq(V-Erev),

(1)

where g¯ is the maximal conductance, m is the activation variable, h is the inactivation variable, V is the voltage across the membrane, Erev is the reversal potential for this ionic current, p and q are non-negative integers that represent the number of activation and inactivation gates in a single channel, respectively.

The voltage gating is achieved through the dependence of the activation and inactivation variables on V. The HH formalism for these kinetics is

dxdt=αx(V)(1-x)-βx(V)x,x∈{m,h},

(2)

where the positive functions αx(V) and βx(V) are the rates of opening and closing of the gates, respectively, for x∈{m,h}. Mathematically, Eq. (2) is equivalent to

dxdt=x∞(V)-xτx(V),

(3)

where

x∞=αx(V)αx(V)+βx(V),τx(V)=1αx(V)+βx(V),x∈{m,h}.

(4)

Since at a fixed voltage V, x(t) will approach to x∞(V) as t increases, x∞(V) is the steady state. Moreover, τx(V) is the time constant, governing the rate at which x approaches to x∞(V).

To avoid having to introduce time constants at every voltage level as in [17] and to prevent time constants from being too small, we formulate opening and closing rate of activation and inactivation gate by a hyperbolic tangent function:

αx(V)=Mxα1+expV-V2xαsxα,βx(V)=Mxβ1+expV-V2xβsxβ,x∈{m,h}.

(5)

The positive parameters Mxα and Mxβ represent the maximum opening and closing rates. The parameters V2xα and V2xβ represent the voltage levels at which the rates are half maximal, while the parameters sxα and sxβ govern the sensitivity of the rates to voltage changes. It is necessary that smα<0 and smβ>0 so that αm is increasing with V and βm is decreasing. Similarly, we require shα>0 and shβ<0.

If the voltage is held constant (as with the voltage-clamp situation we are considering) the solution to Eq. (3) with initial condition x0 can be expressed as

x=x∞(V)+(x0-x∞(V))exp(-t/τx(V)),x∈{m,h}.

(6)

Thus, in a voltage-clamp situation, a voltage-gated ion channel, either Potassium or Sodium, can be described as a a system composed of Eqs. (1), (6), (4), and (5).

In this study, we focus on the Potassium channel, where q is zero due to the absence inactivation gates and p is three. There are eight parameters to estimate, which can be expressed in vector form: X=[g¯,Erev,Mmα,Mmβ,V2mα,V2mβ,smα,smβ]. However, the full parameter estimation problem consists of parameters associated with Potassium, Sodium, and the leakage channel.

2.2 Voltage Clamp Simulations

Voltage clamp studies are generally performed by first holding the voltage at a fixed level, Vprestep, for a sufficiently long time so that the activation and inactivation variables have essentially reached their equilibrium values. Then the clock is set to t=0 and the voltage is instantaneously changed to a new fixed level, Vstep. We simulated two sets of voltage clamp experiments. In the first set, the prestep voltage was set to -100 mV, and the step potential was one of a sequence of nine equidistant step voltages ranging from 0 to 80 mV. The purpose of these experiments is to reveal the activation parameters at high voltage levels. The second set of voltage clamp experiments had a prestep level of either -80 mV, -50 mV, or -20 mV, and a step voltage of 80 mV. This latter set of experiments is intended to reveal the activation parameters at low voltage levels. These experiments are depicted in Fig. 1.

Fig. 1

Voltage clamp simulations. Left: varying step potential. Right: varying prestep potential

Assuming that the prestep potential is held long enough for the variables to reach their steady state, Eq. (6) may be expressed as follows for the activation of the Potassium channel:

m=m∞(Vstep)+(m∞(Vprestep)-m∞(Vstep))exp(-t/τm(Vstep)).

(7)

3 Methodology

3.1 Genetic Algorithm (GA)

Genetic algorithms are a type of metaheuristic inspired by the process of natural selection. They rely on biologically inspired operators such as mutation, crossover, and selection. The following steps outline how a Genetic Algorithm for the problem of identifying parameters for the Hodgkin-Huxley model works:

1.

Initialization: The process begins with a set of randomly selected individuals, which is called a population. Each individual represents a set of parameters, X, for the HH model, encoded as a binary string of size 400 (individual size). Each chromosome representing a parameter is encoded as a sub-string of length 50, and the binary unit for the chromosome indicates a precision defined by the upper and lower bound of that parameter.

2.

Fitness evaluation: A fitness function is defined to quantify how close a population member is to the target parameters, Xtarget, based on their simulated current traces using the voltage clamp experiments as illustrated in Sect. 2.2. Since the information of parameters are embedded in the rising phase of the simulated current traces, we applied a threshold to cut off all the traces:

|Iij-Ii+1j|ΔIj≥threshold,

(8)

where i indexes the time point, Ij denotes j-th current trace from the twelve voltage clamp simulations, and ΔIj is defined as the disparity between the largest and smallest current recordings in the trace j. This effectively stops the current trace at the place where the trace is close to saturated. Additionally, a set of N representative points from the target trace are selected to encapsulate the entirety of that trace. These points are chosen by first selecting N values evenly spaced between the initial (time zero) target current recording, and the value of the target current when threshold is reached. Using linear interpolation, the time points corresponding to these values are approximated from the computed target trace. The target current is then re-evaluated at these N time points to yield the N points that will represent the entire trace. With this setup, the fitness function is defined as

F(X)=∑t=1N∑j=112(Itj(X)-Itj(Xtarget))2,

(9)

3.

Selection: The fittest individuals are selected from the current population to be the parents, producing the children for the next generation.

4.

Crossover: It combines two parents to form children for the next generation. This is done by exchanging parts of the binary string between parents to produce offspring.

5.

Mutation: Mutation rules apply random changes to individual parents to form children. This introduces genetic diversity into the population and prevents the algorithm from converging to a local optimum.

These steps are repeated over successive generations, and the population “evolves” toward an optimal solution. We provide details on the GA we used in Sect. 4.1. In Sect. 4.2, we investigate different number of generations for the Genetic Algorithms, ranging from 500 to 6000.

3.2 Artificial Neural Network (ANN)

Our ANN architectures consist of a sequence of dense, or fully connected, layers. Each dense layer l consists of nu(l) nodes, and each node of a layer is connected to all other nodes of a neighboring layer. A dense layer is composed of an affine mapping and a nonlinear function, Fy(yl-1)=ϕ(Wlyl-1+bl), where the weight matrix Wl∈Rnu(l)×nu(l-1), and bl∈Rnu(l) is the bias vector of layer l [6]. The non-linear activation function ϕ is commonly chose as the rectified linear unit (ReLU) function [13]. Here, we employ a smoother activation function known as Swish, which shares similarities with ReLU but has been demonstrated to mitigate the vanishing gradient problems, particularly in the training of deeper neural networks [4, 7]. In Sect. 4, we use ANN models with 4–7 dense hidden layers and 32–512 nodes in each layer to cover a number of suitable architectures. The total number of trainable parameters of the neural network models ranges from 17,384 to 769,544 and these are optimized by mini-batch stochastic gradient descent using the Adam optimizer [9]. The mean squared error (MSE) loss function is calculated on the target parameters and their corresponding estimates.

The ANN models take the twelve current traces from the voltage clamp simulations as inputs. To reduce the dimensionality, we applied the same techniques to reduce the trace to N points as we did when we defined the fitness function in Sect. 3.1 for the GA. Consequently, each trace can be represented by N time values and N current values. Each input sample is described as a one-dimensional vector with a length of 2×12×N, structured as follows: [t11t21⋯tN1I11I21⋯IN1t12t22⋯IN12]. Due to inconsistent scales of input features, normalization is applied to enhance the convergence and training stability of the models. The final layer of the neural network consists of eight nodes, each representing one of the estimated parameters.

4 Results

4.1 Experimental Setup

To evaluate the fitness function for the GA and to generate the data set for ANN training, we executed simulations as in Sect. 2.2 for a duration of 6 ms with a sampling period of Δt=0.01 ms. Although the simulation period in our study is noticeably shorter compared to a previous investigation utilizing metaheuristics [2], we observe that the current traces in the majority of simulations reach a plateau within the first 6 ms, as modelled by our HH formalism. Through extensive experiments, we found it sufficient to take a set of 20 points (i.e. N=20) to represent each current trace, so that we can minimize the computational cost of the GA and reduce the input dimension to the ANN models. Tables 1 and 2 provide summaries of experimental settings for the genetic algorithm and ANN models, respectively. Table 3 works as the search space of the GA and the parameter bounds to generate the data set for the ANN training. Additionally, we implemented the GA using the DEAP package [5] and the ANN using Pytorch [11].

Table 1

Settings for the GA

GA setting

Value

Individual size

400

Crossover probability

0.7

Mutation probability

0.2

Selection mechanism

Tournament

Number of participants

10

Table 2

Settings for ANN training

Dataset and optimization setting

Value

Number of training samples

2,641,302

Number of validation samples

146,739

Sample parameter distribution

Uniform

Loss function

MSE

Optimizer

Adam

Learning rate

0.001

Batch size

1024

Number of epochs

300

Table 3

The search space in GA or the parameter bounds for the data set applied to ANN models, which are chosen to be sufficiently large to cover the biologically realistic values

Parameters

g¯

Erev

Mmα

Mmβ

V2mα

V2mβ

smα

smβ

(mS/cm2)

(mV)

(ms-1)

(ms-1)

(mV)

(mV)

(mV)

(mV)

Minimum

100.0

-100.0

0.0

0.0

-60.0

-60.0

-50.0

0.5

Maximum

140.0

-80.0

50.0

50.0

60.0

60.0

-0.5

50.0

4.2 Exploration of Genetic Algorithms and ANN Architectures

In this subsection, we investigate various population parameters for the Genetic Algorithm, along with exploring different architectures for dense neural network models. To evaluate the Genetic Algorithm, we inspect the Mean Absolute Error (MAE) and the Mean Absolute Percentage Error (MAPE) on 1000 estimations with reference parameters sampled within the bounds specified in Table 3. We observe a consistent improvement with the increase in population size, as shown in Table 4. On the other hand, despite the reduction in the standard deviations (std.) of the errors with the increasing population size, they persist as relatively large values, which may be due to the stochastic nature of the Genetic Algorithm.

Table 4

Results of the Genetic Algorithm with different population sizes

Population size

MAE (mean)

MAE (std.)

MAPE (mean)

MAPE (std.)

500

7.98

10.66

0.65

8.46

1000

7.42

10.35

0.55

5.24

2000

6.58

9.52

0.52

5.63

3000

5.77

8.51

0.40

3.39

4000

5.56

8.26

0.37

3.03

5000

5.20

8.08

0.36

3.64

6000

5.07

7.53

0.35

3.01

Bold values indicate the lowest errors for each error type (each column)

Additionally, we investigate the influence of different neural network architectures from Sect. 3.2 on the parameter estimations. The ANN models are trained with settings shown in Table 2, the prediction capability of ANNs are evaluated with validation loss shown in Table 5, and we can deduce that the network with the 4 layers and 512 nodes in each layer leads to the lowest prediction error.

Table 5

Validation losses of ANNs with different layers and nodes per layer nu

nu

4 Layers

5 Layers

6 Layers

7 Layers

32

25.49

21.61

18.71

18.16

64

15.32

12.89

11.44

11.02

128

10.55

9.21

8.57

8.21

256

8.29

7.66

7.47

7.46

512

7.15

7.32

7.32

8.04

Bold values highlight the lowest validation loss across all tested layer and node configurations

4.3 Comparison

In this comparison, we assess the Genetic Algorithm using the optimal population size and the Artificial Neural Network model with the identified optimal architecture. The evaluation is conducted on the same set of 1000 samples obtained from voltage clamp simulations, as utilized for the GA in Sect. 4.2. When implemented in Python with 2 x AMD Rome 7532 @ 2.40 GHz 256M cache L3 CPUs from the Narval high-performance computing cluster operated by the Digital Research Alliance of Canada, the GA requires approximately seven hours to perform 1000 estimations. In contrast, the trained ANN model exhibits remarkable efficiency, predicting 1000 samples in a mere 22.60 ms, with a training time of about 6 h. The GA generates a set of current traces from each sample, and use it as a reference in the fitness function to search for the optimal parameters. On the other hand, the time and current traces from the same set of samples are fed into the ANN model, which outputs the estimated parameters. Table 6 summarizes the estimation performances of both the GA and the ANN on each parameter. We observe that the ANN model excels in estimating all parameters, with only one exception of slightly higher MAPE for Mmα. This discrepancy might arise from the model’s challenge in accurately predicting small values of Mmα. The distributions of the individual MAPEs of the GA estimations and the ANN model estimations are compared in Fig. 2 on a logarithmic scale, due to the substantial variance in estimation errors. Notably, the ANN model exhibits a lower and narrower 95% confidence interval in comparison to the GA estimations. This observation suggests that the ANN model is a relatively more reliable method for parameter estimation in this context.

Table 6

Comparison of estimation errors of the GA and ANN model

g¯

Erev

Mmα

Mmβ

V2mα

V2mα

smα

smβ

Overall

MAE (GA)

5.9945

5.1835

0.6378

6.3447

4.9670

11.5225

2.8099

3.0934

5.0692

MAPE (GA)

0.0504

0.0576

0.0326

0.5401

0.5351

1.2579

0.1708

0.1779

0.3528

MAE (ANN)

2.0445

2.1822

0.5177

1.5369

1.0354

1.9922

0.7094

0.8122

1.3538

MAPE (ANN)

0.0174

0.0243

0.0524

0.3906

0.0848

0.1768

0.0751

0.0723

0.1117

Bold values indicate the smaller MAE or MAPE values between the GA and ANN for each parameter (column)

Fig. 2

Comparison between the distributions of the MAPEs from the GA and ANN model estimations on the 1000 samples. Each MAPE is the average of the MAPEs over all parameters for a given sample. The dotted lines represent the 95% confidence interval of the estimation errors

5 Conclusion and Future Work

The presented study focuses on the parameter estimation of the Potassium channel in the Hodgkin-Huxley model, employing both Genetic Algorithm and Artificial Neural Network approaches. A comparative analysis is conducted, and the optimal configurations for GA and ANN are identified. The results suggest that a larger population size leads to an improved prediction Mean Absolute Percentage Error (MAPE) of 0.3528 for the GA. In the case of the ANN, a densely connected neural network with four layers, each containing 512 nodes, achieves the lowest MAPE of 0.1117 on all the parameters. Comparison between GA and ANN on the same set of 1000 voltage clamp simulations reveals that the ANN model exhibits a more reliable performance, as evidenced by a lower and narrower 95% confidence interval for the MAPEs. In conclusion, the ANN model, configured with the identified architecture, demonstrates superior performance in parameter estimation for the Hodgkin-Huxley model’s Potassium channel compared to the GA approach. Future work may involve further refinements of the ANN architectures, exploring other hyperparameter settings of the GA [18]. Additionally, the study could be extended to incorporate the Sodium and leackage channels, possibly using noisy or real-life datasets to improve the model’s robustness and the potential for real-world deployment.

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Willms, A.R.: NEUROFIT: software for fitting Hodgkin-Huxley models to voltage-clamp data. J. Neurosci. Methods 121, 139–150 (2002)

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Willms, A.R., Baro, D.J., HarrisWarrick, R.M., Guckenheimer, J.: An improved parameter estimation method for Hodgkin Huxley models. J. Comput. Neurosci. 6, 145–168 (1999)

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Yue, P.: Parameter estimation of the potassium channel in the Hodgkin-Huxley model. Master’s Thesis, University of Guelph (2024)

Part VMathematics and Computation in Finance, Economics, and Social Sciences

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_29

Credit Risk Modelling with Occupation Times Under Nonlinear Local Volatility Models

Giuseppe Campolieti1  , Hiromichi Kato1   and Roman N. Makarov1  

(1)

Wilfrid Laurier University, Waterloo, ON, Canada

Giuseppe Campolieti (Corresponding author)

Email: gcampolieti@wlu.ca

Hiromichi Kato

Email: kato4650@mylaurier.ca

Roman N. Makarov

Email: rmakarov@wlu.ca

Abstract

We study new structural credit risk models that provide flexible alternatives to the well-known Black-Cox model. The defaults within our models are characterized in accordance with Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process) of the U.S. Bankruptcy Code. The risk-neutral default probabilities hence involve joint probability distributions of the underlying firm value with imposed killing at the liquidation barrier and its occupation time with respect to the reorganization barrier. Occupation times are powerful tools in credit risk modelling as they correctly measure the cumulative time spent in each regime. The proposed model bases on a non-linear state-dependent volatility model and the application of the spectral methodology where the analytical closed-form pricing formulas for credit-related covenants can be expressed explicitly as infinite series that converge rapidly. We also calibrate our models to market credit default swap (CDS) spreads. Our calibration results show that the non-linear state-dependent volatility model outperforms the geometric Brownian motion (GBM) counterpart.

Keywords

Credit risk modelsOccupation timesSpectral expansionsCredit default spreadNon-linear local volatility diffusions

1 Introduction

There are two main branches of mathematical models in credit risks: structural and reduced-form models. Structural models assume a credit event triggers based on the movement of the current firm’s value and some event-triggering barriers [2, 12]. Reduced-form models are intensity-based models where the likelihood of a default is measured by its intensity [13].

The presence of occupation times distinguishes the bankruptcy and liquidation of the underlying firm. The occupation time structural models presume an insolvency of the firm as soon as the cumulative amount of time the firm’s value stay below a reorganization barrier exceeds some time allowance [10, 11]. Occupation time hazard rate models consider stochastic extensions to the (deterministic) hazard rate model, where the valuation of the firm’s credit-related covenants boil down to some expressions involving joint probability distributions of a geometric Brownian motion (GBM) process and its occupation time [1, 6]. These models have gained popularity in the CDS market as they provide a near-perfect fit to typical market CDS data. The more general framework of stochastic hazard rates can be found in [8], where the distributions may not necessarily involve occupation times.

This paper depicts a first extension to the model proposed in [6] with a non-linear diffusion, namely, the Bessel-K diffusions, which arises from the squared Bessel (SQB) process and a measure change via the Doob-h transform. The approach, named the “diffusion canonical transform” methodology, is described in [5]. We employ the spectral expansion approach (e.g., [9]) to obtain the no-arbitrage values of credit-related covenants. Laplace transforms of the joint probability distributions of a drifted BM or GBM (with or without imposed killings) and its occupation time can be found in some references such as [3, 7]. The main spectral theorems for the transition probability density functions with instantaneous killing rate at a proportional occupation time (with or without imposed killings) are derived in [7]. In the past literature, the spectral methodology was carried out to derive the no-arbitrage values of “step options” [4].

2 Firm’s Value Processes in the Risk-Neutral Measure

We consider a risk-neutral filtered probability space (Ω,F,P~,F) where F={Ft}t≥0 is a filtration for standard Brownian motion {W~t}t≥0 satisfying right-continuity and completeness. Let {Ft}t≥0 be an almost surely (a.s.) positive F-adapted time-homogeneous diffusion process, representing the firm’s value process. A typical example of the firm’s value {Ft}t≥0 is a GBM with stochastic differential equation (SDE)

dFt/Ft=rdt+σdW~t;F0>0,t≥0,

(1)

where r≥0 is a constant risk-free rate, σ>0 is a constant volatility. Another example is the so-called drifted Bessel K process [5], with an affine (linear) drift, governed by SDE

dFt/Ft=rdt+σloc(Ft)dW~t;F0>0,t≥0,

(2)

where σloc(F) is the local volatility function defined by

σloc(F)=2ρKμ+1(2ρx)Kμ(2ρx)+2(ρ+r)Iμ+1(2(ρ+r)x)Iμ(2(ρ+r)x),

(3)

with x=X(F) and X=F-1 being the unique inverse mapping of F given by

F(x)=aKIμ(2(ρ+r)x)Kμ(2ρx),

(4)

where aK,ρ,μ are positive parameters. The 3-parameter family of Bessel K processes generate local volatility profiles with pronounced skews (see Fig. 1). In the GBM or Bessel K model, the discounted firm value process {e-rtFt}t≥0, is a (P~,F)-martingale.

Fig. 1

Bessel-K local volatility curves with ρ=0.00001, r=0.02. Other parameters are calibrated such that σloc(F0)=0.25 (left) and σloc(F0)=0.4 (right) at F0=100

The Bessel-K diffusion is constructed as follows. Suppose that the underlying diffusion X is an SQB process on I=(0,∞). It has generator

Gf(x):=12σ2(x)f′′(x)+ν(x)f′(x)=1m(x)f′(x)s(x)′,

(5)

where ν(x)=2(μ+1) and σ2(x)=4x. The speed m(x) and scale s(x) densities, where s′(x) and m(x) are continuous and positive for x∈I (see [3]), are defined via the drift and diffusion coefficient functions as follows:

s(x):=exp-∫x2ν(z)σ2(z)dz=x-1-μ,m(x):=2σ2(x)s(x)=12xμ.

(6)

The Bessel-K diffusion is given by Ft=F(Xt(ρ)) with monotonic mapping defined in (4). The X(ρ)-diffusion is constructed from the X-diffusion via a Doob-h transform where h(x)=u^ρ(x):=x-μ/2Kμ(2ρx) is the ρ-excessive function. Let X(a,b)(ρ) be the diffusion X(ρ) with imposed killing at a and b (where a<b), defined by X(a,b),t(ρ):=Xt(ρ) if t<T(a,b) and X(a,b),t(ρ):=∂† if t≥T(a,b), where T(a,b):=inf{t≥0:Xt(ρ)∉(a,b)} is the first exit time from the interval (a, b), and ∂† denotes the cemetery (killed) state.¹ The transition density has the closed-form spectral expansion:

P~xX(a,b),t(ρ)∈dy/dy=u^ρ(y)u^ρ(x)e-ρtm(y)∑n=1∞e-λntϕn(x)ϕn(y),

(7)

for any x,y∈(a,b), t≥0, and zero otherwise. The normalized eigenfunctions on (a, b) are given by the cylinder functions

ϕn(x)=Nnx-μ/2Jμ(2λna)Yμ(2λnx)-Yμ(2λna)Jμ(2λnx)

(8)

with normalization constant Nn=πλn[(Jμ(2λna)/Jμ(2λnb))2-1]-12. Eigenvalues {λn}n≥1 are the set of increasing roots solving

Jμ(2λna)Yμ(2λnb)-Yμ(2λna)Jμ(2λnb)=0.

(9)

The transition density of the Bessel-K F(A,B),t=F(X(a,b),t) starting at F0=F(x), with imposed killing at A=F(a) and B=F(b), is related to that of the X(ρ)-diffusion and the unique differentiable inverse X=F-1:

P~F0F(A,B),t∈dy/dy=X′(y)·u^ρ(z)u^ρ(x)e-ρtm(z)∑n=1∞e-λntϕn(x)ϕn(z),

(10)

for any F0,y∈(A,B), t≥0, where z=X(y), x=X(F0), and zero otherwise. In our applications, we send the upper killing level B↗∞ since our models consider the firm’s value process with one imposed killing at lower level A, which we assume is a liquidation barrier level.

3 Occupation Time Hazard Rate Model

An occupation time At(F),L,+ (or At(F),L,-) is defined as the cumulative time the firm value stays above (or below) the occupation level L:

At(F),L,+:=∫0tI[L,∞)(Fs)ds;t≥0,

(11)

At(F),L,-:=∫0tI[0,L](Fs)ds;t≥0.

(12)

Suppose now that (Ω,H,P~,H) is a filtered probability space where H={Ht}t≥0 with Ht=Ft∨σ(ξ), describing the complete information up to time t. We clearly see that F⊂H as F only contains information about the firm’s value. Define an F-adapted hazard process:

Γt:=∫0tλ(Fs)ds;t≥0.

(13)

For now, let us assume that the instantaneous killing rate has the form

λ(Fs)=α1I{Fs≥L}+α2I{Fs≤L};0≤α1≤α2,

(14)

where L is the reorganization barrier. We define an H-stopping time τ(α1,α2):=min{τLα1,α2,τA}, where

τLα1,α2:=inf{t≥0:α1At(F),L,++α2At(F),L,->ξ},

(15)

ξ∼Exp(1) is an F-independent exponential random variable with unit rate, and τA is the first hitting time to the liquidation barrier. The default time τ(α1,α2) characterizes both Chapters 7 and 11 type defaults.

It is enough to consider only the occupation time below L (i.e., α1=0) thanks to a simple identity At(F),L,++At(F),L,-=t. The hazard process in (13) can be simplified to

Γt=α1t+(α2-α1)At(F),L,-.

(16)

In the remainder of this section, we shall assume α1=0 and α2=α≥0. The model specification with α=0 convenes an idle reorganization barrier and simplifies to a first passage time model. The hazard process is given by Γt=αAt(Xρ),ℓ,-, where At(Xρ),ℓ,- is an occupation time of the underlying X(ρ) diffusion below the occupation level ℓ∈(a,b). We have a simple relationship between the occupation times of the two diffusions X(ρ) and F by

At(F),L,-=AtX(ρ),ℓ,-:=∫0tI[0,ℓ]X(ρ)ds,

(17)

where X0(ρ)=X(F0)=x and ℓ=X(L).

Define the following diffusion with the instantaneous killing rate as: X~(a,b),t(ρ):=X(a,b),t(ρ) if Γt<ξ, and X~(a,b),t(ρ):=∂† if Γt≥ξ. By the Doob-u^ρ transform, the transition density of the underlying diffusion X(a,b)(ρ) with the instantaneous killing rate is given by

P~xX~(a,b),t(ρ)∈dy/dy=u^ρ(y)u^ρ(x)e-ρtm(y)∑n=1∞e-λ~ntϕ~n,αℓ,-(x)ϕ~n,αℓ,-(y),

(18)

for t>0,x,y∈(a,b), and zero otherwise, where {ϕ~n,αℓ,-}n≥1 are eigenfunctions with eigenvalues {λ~n}n≥1 as the set of increasing roots of the equation provided in the (24). Interested readers may refer to [4, 7] for more details. The survival probability can be obtained via term-by-term integrations of the infinite series (18):

P~(τ(α1,α2)>t)=limb↗∞∫abu^ρ(y)u^ρ(x)e-ρtm(y)∑n=1∞e-λ~ntϕ~n,αℓ,-(x)ϕ~n,αℓ,-(y)dy.

(19)

4 Calibration of CDS Spreads: Bessel K Case

We consider CDS spreads with quarterly payments on the time grid {T1,T2,⋯,Tp} with 0<T1<⋯<Tp=T until the default event and one last payment at the liquidation time. The explicit valuation formulas can be found in our related paper [6]. We calibrate our models to some market data where the underlying firm’s value is assumed to be the Bessel-K process. We extract the market CDS spreads data for the Total Energies company (TTE) with spot time on June 1, 2022. The market data contain eight sample market data points at the following tenor values: T=0.5,1,2,3,4,5,7,10 years. Our calibration approach is find optimal parameter values that (locally) minimizes the gap between the market and model prices through a loss function. Here, we use the (unweighted) root mean squared error (RMSE) as the loss function (where Θ is the set of parameters in a given model):

L(Θ)=∑i=18RTiMdl-RTiMkt28

(20)

where RTiMkt is the ith observed market CDS spread and RTiMdl is the CDS spread implied by the model at Ti. We compare the following models (1)–(3):

Model (1)   The first passage time model (i.e., α1=α2=0) ;

Model (2)   The single hazard rate model (i.e., α1=0, α2≥0) ;

Model (3)   The double hazard rate model (i.e., 0≤α1≤α2).

Table 1

Calibration results for model parameters (the underlying process is Bessel-K), with F0=55.59, LGD=0.6, r=0.05. LGD is the (constant) loss given default. NA means not applicable. ρ=10-5 is assumed only in the first passage time model (1) since ρ is not a robust parameter (i.e. the results are sensitive to the initial guess of the parameter)

Variable

Description

Model (1)

Model (2)

Model (3)

μ

Skewness

0.1532

0.3558

0.3417

ρ

1×10-5

7.0216×10-6

1.7153×10-5

α2

Killing rate (below L)

0 (fixed)

0.0116

0.0207

α1

Killing rate (above L)

0 (fixed)

0 (fixed)

0.0018

A

Liquidation barrier

0.2648

2.0926×10-8

3.1880×10-7

aK

Coefficient

212.4356

232.4076

175.7550

L

Reorganization barrier

NA

48.4362

37.2913

σloc(F0)

Spot volatility

0.2558

0.3082

0.3065

L(Θ)

Loss function value

4.9571×10-6

1.5695×10-8

2.6777×10-10

Fig. 2

Local volatility profile for each model: (1) the first passage time model (no haz), (2) the single hazard rate model (single haz), (3) the double hazard rate model (double haz)

5 Conclusion

The main objective was to calibrate the hybrid model to CDS market data where the firm’s value is driven by a non-linear diffusion process. We see that, in terms of the loss function values in Table 1, the occupation time hazard rate model with the Bessel-K process replicated the CDS values better than the occupation time hazard rate model with the GBM process (the corresponding value for GBM was 5.7714×10-10). This may be due to the flexibility of non-linear diffusion processes. As suggested in Fig. 2, the parameter α2 plays a crucial role in calibrations since the first passage time and the hazard rate models exhibit different local volatility profiles. The single and double hazard rate models have relatively similar profiles, making α1 a less significant parameter.

Acknowledgements

This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grant numbers 2018-06176 and 2020-04782.

Appendix

In this section, we provide explicit formulas for the transition density of the X(a,b)(ρ)-diffusion in (18). Here, we conveniently define the cylinder function

Sμ,λ(x,y)=(xy)-μ/2[Iμ(2λx)Kμ(2λy)-Kμ(2λx)Iμ(2λy)],λ>0,-π2(xy)-μ/2[Jμ(2|λ|x)Yμ(2|λ|y)-Yμ(2|λ|x)Jμ(2|λ|y)],λ<0,

and its derivative (w.r.t. x)

Cμ,λ(x,y)=2λ2x(xy)-μ/2[Iμ+1(2λx)Kμ(2λy)+Kμ+1(2λx)Iμ(2λy)],λ>0,π22|λ|2x(xy)-μ/2[Jμ+1(2|λ|x)Yμ(2|λ|y)-Yμ+1(2|λ|x)Jμ(2|λ|y)],λ<0.

The normalized eigenfunctions in (18) are given in terms of the cylinder functions:

ϕ~n,αℓ,-(x)ϕ~n,αℓ,-(y)=(Cn,αℓ,-)-1Sμ,-λ~n+α(x,a)Sμ,-λ~n+α(y,a)(1/2)Sμ,-λ~n+α(b,a)W-λ~n+α,-λ~n2,2(ℓ),x≤ℓ,y≤ℓ,(Cn,αℓ,-)-1ℓ-1-μSμ,-λ~n+α(x,a)Sμ,-λ~n(b,y),x≤ℓ,y≥ℓ,(Cn,αℓ,-)-1ℓ-1-μSμ,-λ~n(b,x)Sμ,-λ~n+α(y,a),x≥ℓ,y≤ℓ,(Cn,αℓ,-)-1Sμ,-λ~n(b,x)Sμ,-λ~n(b,y)(1/2)Sμ,-λ~n(b,a)W-λ~n+α,-λ~n1,1(ℓ),x≥ℓ,y≥ℓ,

(21)

for x,y,ℓ∈(a,b), and zero otherwise, where

Cn,αℓ,-=Cμ,-λ~n+α(ℓ,a)-λ~n(μSμ,-λ~n(b,ℓ)+bCμ,-λ~n(b,ℓ)-ℓCμ,-λ~n(ℓ,b))+Cμ,-λ~n(ℓ,b)-λ~n+α(μSμ,-λ~n+α(ℓ,a)+ℓCμ,-λ~n+α(ℓ,a)-aCμ,-λ~n+α(a,ℓ))+12(bSμ+1,-λ~n(b,ℓ)Sμ,-λ~n+α(ℓ,a)-aSμ+1,-λ~n+α(ℓ,a)Sμ,-λ~n(b,ℓ)),

(22)

and

W-λ~n+α,-λ~n1,1(ℓ)=Sμ,-λ~n+α(ℓ,a)Cμ,-λ~n(ℓ,a)-Sμ,-λ~n(ℓ,a)Cμ,-λ~n+α(ℓ,a),W-λ~n+α,-λ~n2,2(ℓ)=Sμ,-λ~n(b,ℓ)Cμ,-λ~n+α(ℓ,b)-Sμ,-λ~n+α(b,ℓ)Cμ,-λ~n(ℓ,b),

(23)

are the Wronskian factors. Eigenvalues {λ~n}n≥1 are the set of increasing roots solving

Sμ,-λ~n(b,ℓ)Cμ,-λ~n+α(ℓ,a)+Sμ,-λ~n+α(ℓ,a)Cμ,-λ~n(ℓ,b)=0.

(24)

References

1.

Alfonsi, A., Lelong, J.: A closed-form extension to the Black-Cox model. Int. J. Theoret. Appl. Financ. 15(08), 1250053 (2012)MathSciNet

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Black, F., Cox, J.C.: Valuing corporate securities: some effects of bond indenture provisions. J. Financ. 31(2), 351–367 (1976)

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Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae. Springer Science & Business Media (2015)

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Campolieti, G., Makarov, R., Wouterloot, K.: Pricing step options under the CEV and other solvable diffusion models. Int. J. Theoret. Appl. Financ. 16(05), 1350027 (2013)

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Campolieti, G., Makarov, R.N.: Solvable diffusion models with linear and mean-reverting nonlinear drifts. SIAM J. Financ. Math. 8(1), 146–170 (2017)MathSciNet

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Campolieti, G., Kato, H., Makarov, R.N.: Spectral expansions for credit risk modelling with occupation times. Risks 10(12), 228 (2022)

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Campolieti, G.: Solvable Diffusions in Financial Mathematics. Springer, manuscript in preparation, to be published

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Carr, P., Linetsky, V.: A jump to default extended CEV model: an application of Bessel processes. Financ. Stochast. 10, 303–330 (2006)MathSciNet

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Linetsky, V.: The spectral decomposition of the option value. Int. J. Theoret. Appl. Financ. 7(03), 337–384 (2004)MathSciNet

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Makarov, R., Metzler, A., Ni, Z.: Modelling default risk with occupation times. Financ. Res. Lett. 13, 54–65 (2015)

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Makarov, R.N.: Modeling liquidation risk with occupation times. Int. J. Financ. Eng. 3(04), 1650028 (2016)MathSciNet

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Merton, R.C.: On the pricing of corporate debt: the risk structure of interest rates. J. Financ. 29(2), 449–470 (1974)

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Rutkowski, M., Bielecki, T.: Credit Risk: Modelling, Valuation and Hedging. Springer (2002)

Footnotes

1

The cemetery state ∂† is not included in I. When the process is killed and immediately sent to the cemetery state, it stays there indefinitely.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_30

Financial News Headlines Sentiment Analysis Enhances Stock Market Prediction

Wenjing Fan1  , Roman N. Makarov1   and Xu Wang1  

(1)

Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

Wenjing Fan

Email: fanx5821@mylaurier.ca

Roman N. Makarov (Corresponding author)

Email: rmakarov@wlu.ca

Xu Wang

Email: xwang@wlu.ca

Abstract

The information within financial news headlines plays a crucial role in predicting stock market movements and providing valuable insights for investors. This study focuses on assessing the impact of financial news headlines sourced from Thomson Reuters on the stock returns of Apple (AAPL) from December 30th, 2011, to May 31st, 2019, with the goal of developing a profitable trading strategy. In the data preparation phase, we meticulously clean, tokenize, and extract features from the headlines. By employing sentiment analysis through the Financial Bidirectional Encoder Representations from Transformers (BERT) model, we establish a link between the sentiment of Apple’s news and its corresponding stock prices. Subsequently, we utilize various statistical machine learning (SML) models, such as Decision Trees (Breiman et al., Classification and regression trees, 1984), Random Forests (Breiman in Machine learning. Springer, pp 5–32, 2001), and Neural Networks (Lawrence in Introduction to neural networks. California Scientific Software, 1993), with stratified cross-validation for performance evaluation and hyperparameter tuning. Additionally, the study explores trading strategies that utilize European call and put options, leveraging predictions generated by the SML models. By exploring the relationship between financial news headlines and stock market fluctuations, our research provides valuable insights. Investors can optimize their trading strategies and make informed decisions by understanding this intricate relationship.

Keywords

Sentiment analysis of financial newsFinBERTStatistical machine learning modelsStratified cross-validationStock returns predictionMulti-class classificationOption trading strategies

1 Introduction

For individuals actively involved in the stock market seeking reliable predictions, daily financial news emerges as a pivotal source of information on recent market trends. Beyond mere data dissemination, financial news headlines encapsulate sentiments or attitudes toward the financial status of the associated businesses. These sentiments can be categorized as positive, negative, or neutral, potentially exerting corresponding influences on the stock market.

Sentiment analysis, also referred to as opinion mining, has evolved into a powerful tool for understanding the sentiments and emotions embedded in textual data. From conventional rule-based methods like Valence Aware Dictionary and sEntiment Reasoner (VADER) [1], which rely on predefined sentiment dictionaries, the field has progressed to sophisticated machine learning models leveraging transformers [4] and Bidirectional Encoder Representations from Transformer (BERT) [3]. This evolution has given rise to specialized tools like FinBERT [2] designed for context-specific financial sentiment analysis.

Building upon the insights from [5, 6] regarding the impact of financial news sentiment analysis on stock market prediction, our study delves into the same dataset, employing distinct sentiment analysis techniques and diverse statistical machine learning (SML) models to identify the most effective approach for market prediction. Key focal points include utilizing FinBERT for sentiment scoring, incorporating new performance score metrics for hyperparameter optimization in SML models, and implementing a European option-based trading strategy to assess model accuracy. The study presents four major contributions:

Application of FinBERT [2] as a sentiment analysis tool;

Introduction of two novel performance metrics to enhance model evaluation;

Training and optimization of three SML models: Decision Trees, Random Forests and Neural Networks;

Implementing an option-based trading strategy to evaluate prediction accuracy under the three SML models.

In our study, sentiment scores are extracted from headlines using Natural Language Processing (NLP) techniques. Section 2 compares VADER and FinBERT, and prepares the final dataset, while Sect. 3 presents three model performance evaluation metrics and their comparison. Sections 4 and 5 apply three multi-class classification SML models and construct trading strategies to maximize returns, with Sect. 6 summarizing our findings and proposing future work.

2 Data Preparation with NLP Techniques

To ensure fair and consistent comparisons with the results from [5, 6], we use the same data and text processing approach. Firstly, we remove irrelevant stop words like “a”, “to”, and “in”, as well as punctuation marks. Furthermore, word substitution is conducted, particularly for the synonyms and homology words for Apple products or shares. Next, tokenization and lemmatization techniques are used to break down sentences into individual words, and then convert these words to their base forms. Finally, word bigrams and positions are also considered to identify additional features. The final feature list comprises normalized words, bigram groups, and words with positions.

VADER is a lexicon and rule-based sentiment analysis tool designed for social media sentiment expression [1]. It is suited for analyzing short, informal texts typical for social platforms. In VADER, positivity (pos_sum) and negativity (neg_sum) scores are calculated from lexicon scores for each word in a sentence. If the score is 0, the neutral count (neu_count) is incremented. In addition to these three scores, VADER provides a compound score that represents the overall sentiment assessment of a text. The compound score can range from -1 (most negative) to 1 (most positive).

Financial BERT (FinBERT) is a specialized, state-of-the-art sentiment analysis tool designed for financial texts [2]. As an extension of the original Bidirectional Encoder Representations from Transformers (BERT) model, FinBERT is pre-trained on a large corpus of financial news articles and reports. Unlike traditional rule-based approaches, FinBERT leverages a special feature of BERT, which is a transformer-based architecture. For example, in our case, the transformer allows FinBERT to capture the contextual information of each token by considering its surrounding tokens. After this, the model employs a multi-layer neural network to make sentiment predictions across three categories: positive, negative, and neutral.

This study uses a dataset encompassing Apple’s stock prices and related news headlines from December 30, 2011, to May 31, 2019, with around 1863 trading days. The preprocessed dataset is reduced to 1502 rows due to missing headlines on some days, containing a total of 7197 headlines, averaging 4 to 5 per day. The dataset features 1196 columns, including extracted features, stock-related information, and sentiment scores from two different sentiment analysis tools. The stock-related data are collected from Yahoo Finance, focusing on daily log returns calculated using closing prices. Additionally, sentiment scores are calculated as the average score of all the headlines on a given day.

To label financial news headlines as positive, negative, or neutral based on market returns, we use log returns and predefined thresholds. A headline’s impact is classified as negative if the corresponding log return is below the lower threshold, positive if the log return is above the upper threshold, and neutral if it is within the bounds. We employ Python’s quantile function to set these thresholds, using 95% or 90% quantiles for the upper bound and 5% or 10% quantiles for the lower bound as in [6]. This results in four threshold combinations: 5% and 95%, 10% and 95%, 5% and 90%, and 10% and 90% used to classify the sentiments of the headlines. However, in this study, we consider only one combination with the lower and upper thresholds equal to 5% and 95%, respectively.

3 Performance Metrics

This section introduces three Performance Score (PS) metrics to evaluate SML models, guiding the selection of optimal hyperparameters and determining the final parameter configuration based on the highest achieved performance score.

Table 1

Confusion matrix for three-group classification

Predict positive

Predict neutral

Predict negative

Actual positive

True Predict1

False Predict12

False Predict13

Actual neutral

False Predict21

True Predict2

False Predict23

Actual negative

False Predict31

False Predict32

True Predict3

After the trained model is applied to test data to yield predictions, a confusion matrix, as given in Table 1, is generated. Key metrics for evaluating model accuracy include Precision and Sensitivity matrices given in (1). Precision is calculated by dividing each entry from Table 1 by its column total, reflecting the accuracy of predictions. Sensitivity is determined by dividing each entry by its row sum, indicating the proportion of actual cases correctly identified by the models.

Precision=TP1PPosFP12PNeuFP13PNeg[10pt]FP21PPosTP2PNeuFP23PNeg[10pt]FP31PPosFP32PNeuTP3PNeg,Sensitivity=TP1APosFP12APosFP13APos[10pt]FP21ANeuTP2ANeuFP23ANeu[10pt]FP31ANegFP32ANegTP3ANeg,

(1)

In Eq. (1), TP refers to True Predict, FP denotes False Predict; PPos, PNeu, and PNeg represent the sum of predictions in the positive, neutral, and negative columns in Table 1, respectively; APos, ANeu, and ANeg represent the sum of actual cases in the positive, neutral, and negative rows, respectively.

The performance score PS is calculated using the dot product of matrices M and A, as described in (2):

PS=M·A=∑i,jmijaij,

(2)

where the predefined 3×3 matrix M=(mij)i,j=1,2,3 represents weights of entries in Table 1, and the elements of A=(aij)i,j=1,2,3 are calculated from Table 1 based on the selected metric as described below.

The first method of calculating the performance score is by Precision. In this method, we define

M=10-1010-101,

(3)

and matrix A equals Precision given in (1). In the product M·A, the top left (a11) and bottom right (a33) entries of A, signifying accurate predictions, are assigned a weight of 1. Conversely, the top right (a13) and bottom left (a31) elements, indicating inaccurate opposite class predictions, are weighted with -1. The limitation of this method is its focus only on accuracy within predictions, neglecting the actual prediction accuracy, leading to the introduction of two new performance metrics described below.

The geometric performance score is calculated as a dot product of M defined in (3) and A derived from the element-wise geometric average of the Precision and Sensitivity matrices:

A=Sensitivity∘Precision=TP1PPos·APosFP12PNeu·APosFP13PNeg·APosFP21PPos·ANeuTP2PNeu·ANeuFP23PNeg·ANeuFP31PPos·ANegFP32PNeu·ANegTP3PNeg·ANeg.

That is, matrix A is the positive square root of the dot product of Sensitivity and Precision.

The second new performance score metric introduced in this study is calculated by only considering Positive and Negative related elements. The formula for the Pos-Neg Performance Score is given by

Pos-NegPS=M0·A0,withM0=10-1000-101,

(4)

where the entries in matrix A 0 are calculated by dividing the entries of the confusion matrix in Table 1 by the sum of the elements except for the second row. For example, a11=TP1APos+ANeg. The second row and column of matrix M 0 are composed entirely of zeros, indicating that when both predicted and actual sentiments are neutral, it leads to no action and thus no impact on profit, as detailed in Sect. 5.

4 Multi-class Classification

This section discusses multi-class classification SML models, assigning data to Positive, Negative, and Neutral classes based on financial news sentiment. We enhance model robustness using stratified cross-validation (CV), and explore three SML algorithms: Decision Trees (DT), Random Forests (RF), and Neural Networks (NN), for this multi-class classification challenge.

For each model, we want to find the optimal set of hyperparameters for each performance score metric. The tuning method involves running various parameter combinations and calculating three distinct performance scores, as detailed in Sect. 3, selecting the combination with the highest score as the final configuration for the corresponding performance score metric. Furthermore, we focus on the thresholds 5% and 95% to compare with the results from [5, 6].

Our study comprises 1502 observations, divided into a training set of 1302 observations (from 2011-12-30 to 2018-05-31) and a test set of the last 200 observations (from 2018-06-01 to 2019-05-31). Each set includes a response variable (Y), representing sentiment classes assigned to headlines, and predictor variables (X1,…,X1177) encompassing three FinBERT sentiment scores and features from the headlines.

To assess the performance of three SML models, we use a stratified k-fold cross-validation (CV) method, dividing the dataset into k equally sized subsets. For instance, if there are x positive, y negative, and z neutral responses in the training set, each fold should contain about xk positive, yk negative, and zk neutral responses. In our study, k=5. To achieve accurate results and enhance reliability, we repeat this process 10 times in DT and RF, and twice in NN due to computation intensity.

Decision Trees (DT) use a greedy algorithm to select the best feature for splitting a node based on some specific criterion. Table 2 lists the hyperparameters of DT to be fine-tuned and all the optional values used in this study.

Table 2

Tuning hyperparameters in DT

Name

Values

Max depth

2, 3, 4, 5, 10, 15

Min samples split

2, 3, 4, 5, 10

Class weight

Balanced

Criterion

Gini, entropy

Min samples leaf

1, 2, 4, 8

Table 3

Tuning hyperparameters in RF

Name

Values

N estimators

200, 300, 500

Max depth

2, 4, 8, 16, 32

Min samples split

2, 8, 16, 32

Class weight

Balanced

Max features

sqrt, log2

Criterion

Gini, entropy

Min samples leaf

1, 4, 16

After processing cross-validation for 240 hyperparameter combinations of DT (Table 3), we compile Table 4 as an overview of the best hyperparameter combination within the DT model that results in the maximum PS for the respective metric.

Table 4

Best combinations of hyperparameters in DT

Precision PS

Geometric PS

Pos-Neg PS

Max depth

2

15

2

Min samples split

5

2

2

Class weight

Balanced

Balanced

Balanced

Criterion

Gini

Entropy

Entropy

Min samples leaf

1

2

8

Random Forests (RF) are an ensemble learning technique. Table 3 lists the hyperparameters of RF to be fine-tuned and all the optional values used in this study. Table 5 provides an overview of the best combination of hyperparameters for the RF model that results in the maximum PS for the respective metric.

Table 5

Best combinations of hyperparameters in RF

Precision PS

Geometric PS

Pos-Neg PS

N estimators

500

500

500

Max depth

16

32

2

Min samples split

16

8

16

Class weight

Balanced

Balanced

Balanced

Max features

log2

sqrt

sqrt

Criterion

Entropy

Gini

Gini

Min samples leaf

1

4

1

Neural Networks (NN) are a type of machine learning process that uses interconnected nodes or neurons in a layered structure. For NN, the stratified five-fold cross-validation is conducted twice due to computing capacity. Table 6 lists the hyperparameters of NN to be fine-tuned and all the optional values used in this study. Table 7 provides an overview of the best combination of hyperparameters for NN that results in the maximum PS for the respective metric.

Table 6

Values used to tune hyperparameters in NN

Name

Values

Hidden layer sizes

(64, ), (128, ), (256, ), (128, 128)

Activation

Sigmoid, Relu

Solver

SGD, ADAM

Max iteration

1000

Alpha

0.001, 0.01, 0.05

Early stopping

False, True

Initial learning rate

0.01, 0.001, 0.0001

Table 7

Best combination of hyperparameters in NN

Precision PS

Geometric PS

Pos-Neg PS

Hidden layer sizes

(128, 128)

(128, 128)

(256, )

Activation

Relu

Relu

Relu

Solver

SGD

SGD

ADAM

Max iteration

1000

1000

1000

Alpha

0.001

0.001

0.01

Early stopping

False

False

False

Inital learning rate

0.001

0.001

0.01

5 Testing Method and Trading Strategy

Upon determining the best hyperparameters, we develop a trading strategy based on our classification models’ predictions, incorporating an unrealistic assumption: assume that we know the sentiment of day i on the previous day i-1, allowing the purchase of put or call options accordingly. This assumption, while impractical in reality as news headline content cannot be known beforehand, is necessary for our study to test the predictive power of our models and compare with the results in [5, 6]. Our objective is to verify if using other machine learning methods and new performance metrics enables the construction of trading strategies that outperform those based on multi-stage regression methods. Thus, we consider the case with unrealistic strategies to make a comparison with the results from [5, 6].

Our approach can be described in the following step-by-step manner.

1.

First, train the model(s) as described in Sects. 2–4. We need a preprocessed dataset with keywords and FinBERT scores for each headline and respective daily log returns (to label each headline as having a positive, negative, or neutral impact on the stock return). If several models with the same or different performance metrics are trained, we implement the plurality-vote approach.

2.

Once the model is trained, we start using it to predict if each new headline has a positive, negative, or neutral impact on the stock price. If several headlines are posted on the same day, we combine them into a single one.

3.

Depending on the outcome, we take one of the following actions:

If the positive impact is predicted, we purchase an at-the-money (ATM) call option since we expect that the stock price will go up.

If the negative impact is predicted, we purchase an ATM put option since we expect that the stock price will go down.

Otherwise, we purchase no option. Alternatively, one can purchase a so-called butterfly spread that includes two long and two short call options.

In our strategy, we use at-the-money (ATM), one-day-to-expiry options, setting the strike price K equal to the stock’s closing price St on the purchase day t. The profit from a call option, expiring on day t+1, is max(St+1-St,0)-Ct, and for a put option, it equals max(St-St+1,0)-Pt, where Ct and Pt are the respective premiums of the options bought on day t, calculated by the Black–Scholes option pricing formulas.

In our analysis, we introduce a method called the plurality vote. This method involves repeating the testing process m times for each metric, yielding m sets of predictions (Pred_Y) per model. We then apply plurality vote to these sets, where the most common prediction among the m models is chosen for each day. For instance, if 100 retrained models on a given day predict more negative sentiments than positive or neutral, the day’s sentiment is classified as negative. Additionally, the plurality vote aggregates these predictions across all three performance scores for each model, finalizing the prediction for each day based on the plurality sentiment across the metrics. For example, if the plurality sentiment on day t is positive in two out of three metrics, the final plurality vote prediction for that day would be positive.

As mentioned in Sect. 4, we focus on the results under the thresholds 5% and 95%. Table 8 shows the features we obtain from trading.

Table 8

Features in trading strategy

Name

Description

Purchased

Total number of purchased options

Exercised

Total number of exercised options

Expenses

Total cost of purchased options

Payoff

Total payoff of the exercised options

Profit

Total payoff minus the total cost

Percentage of return

ProfitExpenses·100%

First, we conduct a comparative analysis of the trading performance across the two data split approaches. Table 9 presents the Return derived from applying the unrealistic trading strategy, with the first 1302 observations as the training set and the last 200 observations as the test set.

Table 9

Return of the trading strategy

Name

Decision trees (%)

Random forests (%)

Neural networks (%)

Precision PS

1.445

159.657

32.685

Geometric PS

-1.242

83.427

32.685

Pos-Neg PS

1.535

-4.368

33.313

Final plurality Vote

-2.804

83.427

32.685

Then, we compare the performance of VADER and FinBERT by extracting the highest return value in Table 9 for a detailed analysis against VADER-related results [5], as presented in Table 10. LM, and SVM stand for Logistic Regression Model and Support-Vector Machine, respectively.

Table 10

Comparison between VADER and FinBERT

Name

Purchased

Exercised

Expense

Payoff

Profit

Return (%)

FinBERT-RF

7

6

3.719

9.66

5.937

159.66

VADER-LM [5]

35

22

43.786

86.32

42.535

97.14

VADER-SVM [5]

48

28

60.517

116.32

55.806

92.22

Table 11

Comparison for SML models with and without FinBERT Sentiment Score

Name of score

Purchased

Exercised

Expense

Payoff

Profit

Return (%)

DT-No Sentiment

63

28

34.677

31.866

-2.811

-8.11

DT-With FinBERT

140

67

75.605

76.697

1.092

1.45

RF-No Sentiment

5

4

2.790

4.601

1.811

64.90

RF-With FinBERT

7

6

3.719

9.656

5.937

159.66

NN-No Sentiment

6

3

3.373

3.814

0.441

13.07

NN-With FinBERT

5

3

2.875

3.814

0.940

32.69

Using FinBERT with the random forest model yields a return of about 159.657%, exceeding VADER’s performance by roughly 60%. One key observation of FinBERT is its conservativeness in its purchasing approach, resulting in only 7 options being bought, leading to low premium expenses, while maintaining high accuracy with 6 out of 7 options exercised successfully. Next, to evaluate the impact of sentiment analysis, we retrain a model excluding sentiment scores, using the same hyperparameters for the corresponding classification model and trading strategy. We then compare the results, as shown in Table 11, to analyze the differences in outcomes between scenarios with and without the use of FinBERT sentiment scores in Decision Trees, Random Forests, and Neural Networks models. The results demonstrate that sentiment scores do have a significant impact on the trading strategy. By incorporating sentiment analysis, more accurate predictions and better trading strategies can be achieved.

6 Conclusion

This study confirms our speculation that the FinBERT sentiment scores significantly improve option returns. Here are the three main results we derive after utilizing the unrealistic trading strategy with Decision Trees, Random Forests, and Neural Networks models. (1) Sentiment analysis is critical when text data are used to predict market trends. (2) FinBERT outperforms VADER in analyzing financial news headlines. (3) The study demonstrates the power of diverse machine learning models and their potential to enhance trading strategies.

Acknowledgements

The authors would like to thank the National Science and Engineering Research Council of Canada (NSERC) for its support. The research was funded by the NSERC Discovery Grants RGPIN-2019-05917 and RGPIN-2020-04782.

References

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Hutto, C., Gilbert, E.: VADER: a parsimonious rule-based model for sentiment analysis of social media text. In: Proceedings of the International AAAI Conference on Web and Social Media, vol. 8, no. 1, pp. 216–225 (2014)

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Araci, D.: FinBERT: Financial sentiment analysis with pre-trained language models (2019). arXiv preprint arXiv:​1908.​10063

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Devlin, J., Chang, M.-W., Lee, K., Toutanova, K.: BERT: pre-training of deep bidirectional transformers for language understanding (2018). arXiv preprint arXiv:​1810.​04805

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Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I.: Attention is all you need. In: Advances in Neural Information Processing Systems, vol. 30 (2017)

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He, J.: Application of sentiment analysis in stock market prediction. Theses and Dissertations (Comprehensive), no. 2544 (2023)

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He, J., Makarov, R.N., Tuero, J., Wang, Z.: Performance evaluation metric for statistical learning trading strategies. Data Sci. Finance Econ. 4(4), 570–600 (2024). https://​doi.​org/​10.​3934/​DSFE.​2024024

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_31

Self-Exciting Point Processes in Real Estate

Ian Fraser1  , Devan Becker1  , Yang Liu2   and Xu Wang1  

(1)

Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

(2)

Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada

Ian Fraser (Corresponding author)

Email: fras1890@mylaurier.ca

Devan Becker

Email: dbecker@wlu.ca

Yang Liu

Email: yangliu@wlu.ca

Xu Wang

Email: xwang@wlu.ca

Abstract

This study pioneers a stochastic process-based approach for analyzing and predicting residential property sales, treating the number of daily residential sales as a point process. Utilizing self-exciting point process and sinusoidal-based regression models, the results demonstrate the temporal Hawkes process’s effectiveness in predicting home sales, particularly addressing seasonality with only one known data cycle. Compared to linear regression, Poisson regression, and ARIMA models, our point process models with self-exciting features exhibit superior predictive accuracy without relying on external data. The study’s findings underscore the potential of point processes in refining temporal predictions in real estate, encouraging a new direction for future analytics.

Keywords

Real estateSelf-exciting point processSinusoidal-based regressionTemporal Hawkes processesARIMA modelsPoisson regression

1 Introduction

The real estate market, a complex and dynamic entity, exerts a substantial influence on financial well-being for individuals, cities, and countries [1]. This study employs point processes to analyze residential property sales, treating them as interconnected events. Despite advancements in computational capabilities, the application of self-exciting point processes, like the Hawkes Process, remains unexplored in real estate. This research aims to fill this gap, evaluating the effectiveness of a variety of point process models in predicting real estate transactions.

Real estate transactions, each with unique temporal and spatial traits, present a compelling case for Spatio-Temporal Point Process (STPP) analysis. The real estate industry has yet to consider temporal point processes, although they constitute a leading method in the fields of criminology, finance, and information propagation  [2, 3]. Our research, building on methods developed for modeling terrorism attacks  [4, 5], adopts a novel approach using binned point process models with self-excitation for predicting home sales, diverging from traditional real estate analysis.

Our research’s primary contribution is the successful integration of a sinusoidal background with the Hawkes process, addressing seasonality in data with only one observed cycle. This method contrasts traditional models that rely on kernel density estimation, underscoring the utility of point processes in real estate market analysis and forecasting. The proposed methodology outperforms leading time series techniques to address this type of data.

Traditional analysis of the real estate market, focused primarily on property prices, often utilizes hedonic and kriging models encompassing a range of factors like property attributes and economic variables  [6, 7]. These models assume that the variation in sales is part of a general trend (either spatial, temporal, or both), but do not contain a mechanism by which one home sale may affect future home sales.

The minimal existing literature utilizing point processes on real estate market analysis provides valuable insights to the effectiveness of the point process framework  [8, 9] and this study applies novel methodology to address features of the data. However these existing methods do not consider self-exciting properties of point processes, and nor do they compare their methodologies with existing time series methods in terms of predictive accuracy, our proposed method will address these isssues.

2 Methodology

This section starts by describing the chosen data and required cleaning and transformations, followed by the methodology used in this study, and concludes with the performance measures selected for evaluating each model’s predictive capabilities.

2.1 Data

We apply our methodology to a data set of real estate transactions for King County, Washington, US, which contains 21,613 residential property sales between May 2 2014 and May 27 2016  [10]. The variables of interest are longitude, latitude, and date representing the spatial and temporal fields of the point pattern. Boundaries of municipal districts  [11] are used to create two subsets of varying size to represent alternative spatial observation windows (Fig. 1). Weekends and holidays are removed from the time series of counts per day of transactions, resulting in 251 total days of transactions. The training test split for each of the three regions chosen for analysis is 80%:20%, resulting parameter estimation using the first 200 days, and testing over the last 51 days. Note that the date of a transaction is the closing date, not the date on which the home buyers take possession of the house. We are assuming that the closing of a house (when the “sold” sign goes up) may influence future buyers in the area.

Fig. 1

Spatial plots of transaction in King County (right) and the two selected subsets defined by political boundaries (left and middle). There are notable rivers running through Bellevue upon which no houses can be sold

2.2 Model Building

In order to predict daily transaction counts across different geographic regions. We employ several approaches for comparison with the proposed methodology. This includes ARIMA, linear and Poisson regression, random jittering for simple point processes, and binned point process modeling. We highlight three of the methods below: linear regression, random jittering for simple point processes, and binned point process modeling.

Linear Regression

To account for the known cyclicity in real estate markets, seasonal decomposition is commonly used, however, this can only be done with at least two full seasons of data. Due to the limitation of having only one cycle in our observation window, a sinusoidal-based linear model is used and fitted using least squares regression. The model takes the form:

μsin(t)=μ+αsin2πt|T|+βcos2πt|T|,

(1)

where μ is a constant, T={1,2,…,t,⋯251} is the total number of intervals in the observation window, and α, β are the cosine and sine’s amplitude to the process. This function is used as the linear predictor in a linear model as well as a Poisson generalized linear model.

Random Jittering

A binned point process is one where events happen at continuous time and then are “binned” into discrete time points. For house sales, the exact time of sale is not recorded, only the day of the sale. This means that the point process is not simple; there are multiple observations at a given time point, and methods for dealing with the binning are required. One potential method is called “Random Jittering” and involves random perturbations of the points within a unit interval surrounding the binned time point. The Random Jittering method adds a uniform random distribution to each event which has another on the same time stamp to ensure a unique time for all events, simply constructed by the routine: for all ti=tj, let tj=tj+Uniform(-0.5Δt,0.5Δt), where Δt is the interval between each recorded time. In our scenario where each time step is one day, we have Δt=1.

Binned Hawkes Process

Another popular approach is the Binned Hawkes Process, which directly incorporates the binning. This model is based on a simplification of the Hawkes process with a parameter p∈[0,1] which represents the proportion of events within a bin that are attributable to the background intensity. This allows for direct evaluation of the degree of self-excitation. The model takes the form:

λ(t|Ht)=pμ(t)+(1-p)∑ti<tk(t),

(2)

where Ht={N1,N2,⋯,Nt} is the history of the events up to time t with Nt∈Z+ being the count of events per bin, μ(t) is the background intensity at time t, and k(t) is the kernel defining the decaying contribution that previous points contribute to the current time t. The most common and widely used exciting kernel is exponential, where k(t)=ωe-ωt-ti. In previous research, μ(t) has been a Gaussian kernel density function. We instead use μsin(t) as the background intensity for its superior performance in following the pattern of our data. Hence our proposed model takes the form:

λ(t|Ht)=pμsin(t)+(1-p)k0∑ti<tωe-ω(t-ti),

(3)

where k0 is a scalar representing the amount of excitation generated by the collection of prior events, and ω defines the rate of decay for the triggering density.

Expanding on this model, the addition of a reward term to the intensity function and the artificial ordering of events of each bin allows for the consideration of intra-bin excitation. Intra-bin excitation allows for events which happen in the same time period to contribute to the likelihood of future events. For an arbitrary bin tq with Q events, we define the sequence of events within the bin as {tq,x}x=1,2,⋯,Q, meaning tq|x can be regarded as tq+(x-1)Δt,Δt→0. This model is shown in [4] to take the general form:

λ(tq|x|Ht)=pμ(t)+(1-p)k0∑ti<tωe-ω(t-tq)+(x-1)ωe0.5.

(4)

Although unlikely that home sales which occur within the same day are influenced by each other, this model is evaluated for robustness as it is one of few alternative binned model designs.

2.3 Model Fitting and Evaluation

The baseline models for linear regression has parameters estimated using least squares, whereas Poisson regression and ARIMA models are fit using Maximum Likelihood Estimation (MLE) in R. This section focuses on the methodology of the binned Hawkes process and the evaluation metrics chosen for the study.

2.3.1 Maximum Likelihood Estimation (MLE)

The parameter estimation of the point process is achieved through MLE. For point processes, the likelihood function is the joint density function of all the points in (t1,⋯,tn)∈[0,T), and for the general Hawkes process, it is given by:

L(θ→|T)=f(t1,…,tn)=∏i=1nf(ti)=∏i=1nλ(ti)e-∫0Tλ(s)ds.

(5)

The model parameters are obtained by maximizing the likelihood function, or equivalently by minimizing the negative of the log-likelihood function:

ℓ(θ→|T)=∑i=1nlogpμ(t)+(1-p)k0∑ti<tωe-ω(t-ti)-pn+(1-p)k0∑i=1ne-ω(tn-ti)-1,

(6)

where T is our data set of counts per day.

Optimization of the above function is achieved using built-in optimizers in R, specifically the optim() function with method set to L-BFGS (Limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm).

2.4 Performance Metrics

For the temporal prediction of event counts per day, three metrics have been selected to evaluate model performance. Each relies on the actual count per day in the training days, Ni in comparison to their predicted values, Ni^.

Mean Absolute Error (MAE) =1n∑i=1n|Ni-Ni^| is a straightforward metric measuring average error magnitude, being less sensitive to outliers.

Root Mean Squared Error (RMSE) =1n∑i=1n(Ni-N^i)2 is useful for emphasizing larger errors, being more sensitive to outliers and highlighting significant deviations.

Number of Correct Predictions (NCP) =1n∑i=1nI{Ni=Ni^} counts the number of exactly correct predictions, which is most useful for count data with low values.

This collection of performance metrics is applicable for home sales prediction because MAE offers a direct error magnitude measure, RMSE prioritizes larger errors, and NCP directly assesses prediction precision. Note that a better model should have a lower MAE and RMSE, but a higher NCP.

The proposed models with estimated parameters were fit to the training data. For the non-deterministic models, 50 sets of predicted values were produced from each model. Each performance indicator was calculated for each set of predicted values, then the average was recorded (this explains the non-integer values of NCP in the results section).

3 Results

The region of King County and the municipalities of Bellevue City and Mercer Island are fit and evaluated on a variety of models to conduct a robust test of the daily prediction of home sales. The models are grouped into categories based on the assumptions on the dependence structure and the results are shown in Table 1. Note that the line of best fit found through the sinusoidal regression is utilized as the background rate where sin is present in the model name. The models in the lowest five positions of the table are the Hawkes-type processes which contain exponential self-excitation.

Table 1

Temporal prediction performance metrics on each region’s count per day of transactions. The best performance indicators are highlighted in grey. Second and third best performance indicators are highlighted in light grey

For King County, both the homogeneous and inhomogeneous (with sinusoidal background), point processes modeled after random jittering have superior performance. Whereas in Bellevue, the top two performers are the Binned Hawkes Process with and without intra-bin self-excitation. And for Mercer, the smallest data set, the simple linear predictions are most accurate. This is attributable to the small count per day for Mercer. Of the Point Process models, it is the sinusoidal backgrounds on the binned point processes outperforming the rest.

Table 2

Parameter estimates of the binned Hawkes process with intra-day self-excitation and a sinusoidal background intensity

p

k0

ω

King County

0.986

5.415

0.001

Bellevue

0.953

8.576

0.001

Mercer

0.878

7.144

0.021

The values for p in Table 2 indicate that the process is mostly attributable to the background, with modest self-exciting properties. This can be clearly seen in Fig. 2, wherein the calculated values for the fitted intensity closely follow the curve of the sinusoidal background. The values of ω indicate a very short self-excitation effect, with the increase in intensity wearing off in 1 or 2 days.

Fig. 2

Sinusoidal Background with Intra-Bin Excitation: Fitted background (Green Line), evaluated intensity function (Blue Line), example simulated values (Green Points), and actual test points (Red Crosses)

4 Discussion

Table 1 demonstrates some clear patterns in the data and reveals interesting facets of our models. While not surprising, it is worth noting that the data are clearly well-described by a sinusoidal pattern. Almost all of the models that were in the top three of any given performance indicator included a sinusoidal term. With multiple years of data we may wish to estimate a seasonal trend that differs slightly from a sinusoidal pattern, but it is a indication that we should use sinusoidal terms in this model.

It is surprising, however, that the simple linear model performed so well for these data. We suspect that this is because the simple linear model also called curve fitting is estimated as a nearly horizontal line, and the sinusoidal nature of the data leads to predictions that were essentially the estimates of the mean. It is coincidence that the test data were all close to the overall mean of the training data, which leads to higher than expected performance of a linear model.

The autoregressive models were never in the top three. This is likely because there is not sufficient data to estimate a seasonal effect.

Looking at the results of King County, we can see that the homogeneous point process and the inhomogeneous point process with a sinusoidal mean intensity were in the top three for all of the performance indicators. For the models that did include self-excitation, the p parameter was 0.986, indicating nearly all of the points are attributable to the background intensity rather than self-excitation. Together, these two facts indicate little evidence of self-excitation for King County overall.

Bellevue had a similarly large value of p, but the models with self-excitation, specifically sinusoidal mean Hawkes process with intra-bin effects, performed very well on the data. Mercer had a smaller value of p, yet the models with self-excitation were not the usually the top performers. These two locations were fairly well explained by models with self-excitation, which leads us to believe that further investigation into the nature of self-excitation for residential sales is warranted.

We also note that the value of the p parameter was larger with smaller sample sizes (King County is larger than Bellevue, which is larger than Mercer). It may be the case that points are attributed to self-excitation in order to explain more variance, which becomes less necessary as the sample size increases. Further work involves investigation into whether the inclusion of covariates will reduce the apparent effect of self-excitation.

5 Conclusion

In conclusion, this research has demonstrated the effectiveness of using self-exciting point processes with a sinusoidal background trend in real estate market modeling and prediction. The integration of a sinusoidal background with an exponentially triggered intensity function has captured seasonal variations and perceived randomness in the realization of home sales, outperforming traditional time series methods over the time scale of our data. Hence we have shown that the inherently stochastic nature of house sales is more accurately modelled by point processes compared to the leading methods used in the field. This study opens avenues for future research, including the application of temporal or spatial covariates to assess the impact of significant events on the short-term realization of real estate transactions.

The success of this novel application of temporal point processes offers new perspectives and concrete tools for modelling, analyzing, and predicting the dynamics of property sales. Furthermore, this shows that there may still exist untapped fields of research for which point processes may offer valuable insights and improved modeling capabilities.

Acknowledgements

The authors would like to thank the National Science and Engineering Research Council of Canada (NSERC) for its support. The research was funded by the NSERC Discovery Grants RGPIN-2019-05917, and RGPIN-2023-05471.

References

1.

Evangelou, N.: In: How do Home Sales Affect the Economy and Job Market in Your State? Economists Outlook. https://​www.​nar.​realtor/​blogs/​economists-outlook/​how-do-home-sales-affect-the-economy-and-the-job-market-in-your-state. Cited 1 Dec. 2023

2.

Egesdal, M., Fathauer, C., Louie, K., Neuman, J., Mohler, G., Lewis, E.: Statistical and Stochastic Modeling of Gang Rivalries in Los Angeles. SIAM (2010)

3.

Li, C., Song, Z., Wang, X.: Nonparametric Method for Modeling Clustering Phenomena in Emergency Calls Under Spatial-Temporal Self-Exciting Point Processes, vol. 7. IEEE Access (2010)

4.

Wang, S., Wang, X., Li, C.: Modeling terror attacks with self-exciting point processes and forecasting the number of terror events. Entropy 25.7 (2023)

5.

Brantingham, P., Lewis, E., Mohler, G., Bertozzi, A.: Self-Exciting point process models of civilian deaths in Iraq. Secur. J. 25.3 (2011)

6.

Salzman, D., Zwinkels, E.: Behavioural real estate. J. Real Estate Literature 25(1), 990–1000 (2017)

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Abidoye, R., Chan, A.: Critical review of hedonic pricing model application in property price appraisal: a case of Nigeria. Int. J. Sustain. Built Environ. 6(1), 250–259 (2017)

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Paci, L., et al.: Analysis of residential property sales using space-time point patterns. Spatial Stat. 21, 149–165 (2017)MathSciNet

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Eduardo, P.: Understanding the Spatial Statistical Properties of a Real Estate Listings Point Pattern in San Jose, Costa Rica. Urban Analytics and City Science, Environment and Planning B (2023)

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Harlfoxem. House Sales in King County, USA (2016). https://​www.​kaggle.​com/​datasets/​harlfoxem/​housesalespredic​tion/​data

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King county. King County GIS Open Data (2023). https://​gis-kingcounty.​opendata.​arcgis.​com/​

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_32

Algebraic Structure and Complexity of Games

Zixuan Gao1  , Chrystopher L. Nehaniv1   and Attila Egri-Nagy2  

(1)

University of Waterloo, Waterloo, Canada

(2)

Akita International University, Akita, Japan

Zixuan Gao (Corresponding author)

Email: z72gao@uwaterloo.ca

Chrystopher L. Nehaniv

Email: cnehaniv@uwaterloo.ca

Attila Egri-Nagy

Email: egri-nagy@aiu.ac.jp

Abstract

In this study, we investigate the algebraic structures and complexities of several two-player perfect-information games using Krohn-Rhodes algebraic automata theory and computer algebra. We examine Nim, Subtraction Game, Tic-Tac-Toe, and the 3×3 Hex game, taking advantage of the small state space. Our approach involves a precise formulation of state-space representation and a detailed algebraic analysis of best-play automata generated using minimax tree searches and backtracking algorithms. Notably, our findings do not contradict established theoretical frameworks but offer new insights, particularly in the case of a specific Nim game setup that refutes an existing conjecture. Although our study found a reliable upper bound for the complexity of Tic-Tac-Toe and 3 × 3 Hex, the existence of some lower-complexity best-play automata remains an open question. Overall, this research attempts to bridge the gap between algebraic automata theory and classical game theory.

Keywords

Two-player complete information gamesOptimal game-play automataTransformation semigroupsKrohn-Rhodes complexityEssential dependencies and lower bounds

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée en partie par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669. This project was also funded by the Kakenhi grant 22K00015 by the Japan Society for the Promotion of Science (JSPS), titled ‘On progressing human understanding in the shadow of superhuman deep learning artificial intelligence entities’ (Grant-in-Aid for Scientific Research type C, https://​kaken.​nii.​ac.​jp/​grant/​KAKENHI-PROJECT-22K00015/​).

1 Introduction and Background

Scholars have delved into game theory and explored game strategies for centuries. However, research on the underlying structures of gameplay remains scarce, particularly on the algebraic structure of gameplay automata. This gap in the literature is somewhat addressed in Rhodes’ game complexity study [6, Part IV], which employs modern algebraic theory. This paper draws inspiration from Rhodes’ work to further explore these concepts.

Reusing Rhodes’ definition [6, Definition 6.106], a gameplay machine is a tuple C=(A,B,Q,λ,δ) where A is the set of moves that the opponent can make, B is the set of moves that I, the machine, can make, the state set Q is the set of possible opponent views (i.e., the board positions at the opponent’s turn), and two transition functions

λ:Q×A→Q with (q,a)↦(q·a)·b and b:=δ(q,a). Note that q is the opponent’s view, and a is the opponent’s move. Essentially, λ maps a current opponent view (and move) to a new opponent view after the move is made and I responded.

δ:Q×A→B provides my move given my view, i.e. q·a, after the opponent’s move. In short, δ describes my strategy.

A few additional notes regarding our construction:

1.

λ has a special case as Q does not include endgame positions; it loops back to the starting position q0 and thus resets the game. The game may be over after an opponent move (i.e., (q·a)) or after my response (i.e., (q·a)·b).

2.

Initial state q0∈Q. This implies that the opponent always goes first regardless of who won the preceding game.

3.

Making an illegal move results in a permanent garbage state ∅. That is, for every q∈Q,a∈A such that q·a is illegal, we have q·a=∅. Additionally, (∀a∈A)∅·a=∅. While Rhodes discusses several possible approaches to illegal moves, this particular rule is enforced here to eliminate any effect illegal moves may have on the algebraic structure of the game.

A best-play machine is a gameplay machine that guarantees the best outcome, i.e., the highest utility for myself.

We will now introduce the concept of Krohn-Rhodes Theory.¹ While we will not delve deeply into its intricacies, it is essential to understand that this theory allows for decomposing finite semigroups and automata. Such decomposition is achieved using elementary components such as finite aperiodic semigroups (like flip-flops) and finite simple groups (such as the modulo two counter C2, the alternating group A5). These components are combined in wreath products, creating a cascade that emulates the original system. In essence, the Krohn-Rhodes complexity refers to the least number of layers containing nontrivial permutation groups in all such cascades emulating the system. Furthermore, we define game complexity θ(G) as the minimum complexity of any best-play automaton that plays G.

2 Objective

For our study, we aim to analyze the algebraic structure of several two-player perfect-information games. These games, namely Nim, Subtraction Game, Tic-Tac-Toe, and 3×3 Hex game have small state space, enabling total minimax tree searches. Firstly, we shall find suitable state-space formulations, considering board symmetry. Secondly, we will construct the best-play automata. We will define optimality (utility function), perform minimax, and sample best strategies. Thirdly, we will utilize the computer algebra GAP package SgpDec [2, 3] to decompose the best-play automata and perform complexity analysis. We hope to obtain some results on the complexity of particular games and verify a few conjectured assertions made by Rhodes [6, Part IV, pp. 247–248 and p. 255 ], in particular, related to his theorem:

Theorem 1

(Rhodes (Theorem 6.108 [6])) If the game G has an n×m board where pieces continue to fill the board, then θ(G)≤nm/2.

Assertion/Conjecture 1

Tic-Tac-Toe has a complexity of 1.

Assertion/Conjecture 2

Nim has a complexity of 1.

Assertion/Conjecture 3

7×7 Hex has complexity ≤24 and is approximately 18.

The theorem gives complexity upper bound of 4 for Tic-Tac-Toe and 24 for 7×7 Hex, but no proof is given for further reduction of these bounds.

3 Methodology

We should develop suitable state-space representations for each game to begin the experiment.

3.1 Tic-Tac-Toe

Consider a machine C=(A,B,Q,λ,δ) that plays Tic-Tac-Toe. Let A=B={0,1,⋯,8} where action x∈{0,1,⋯,8} represents placing a piece at position x. (See Fig. 1.) Placing a piece in a position that is already occupied is illegal.

Fig. 1

Tic-Tac-Toe board representation

Q is the set of reachable board positions where we take symmetry into account. Given any position, rotating and mirroring the board will yield an equivalent position. Therefore, when constructing the gameplay machine, we pick a fixed representative for each set of equivalent positions. This will reduce the total number of states by up to a constant factor.

Fig. 2

Sample best-play machine of Tic-Tac-Toe

In a sample best-play automaton construction shown in Fig. 2, each state is a board position in the opponent’s view, and each arrow with a label (0–8) represents a possible move from the opponent. The arrow leads to a new state where our player has already responded optimally to the opponent’s move. An illegal move goes to a garbage state permanently.

3.2 Nim and Subtraction Game

Nim and Subtraction Games are both impartial games. For Nim, we set up the game with several piles of chips; players take turns removing any amount of chips from one of the piles until all of the chips run out. The player who removes the last chip wins. In our model, states Q describes the piles of chips. We use notation

x1:x2:⋯:xk

with xi∈N to represent k piles of chips where pile i has xi chips. Permuting the piles yields an equivalent state, so we choose sorted piles as the symmetry representation in this case. For player action, we write i←x to indicate the removal of x chips in pile index i. Removing more chips than are present in a pile is illegal.

Subtraction Game is a slightly lesser-known game. It is set up with a single pile of chips, and two players successively remove 1, 2 or 3 chips until the pile is emptied. The player who removes the last chip wins. In this case, the state is simply an x∈N, i.e., the number of chips in the pile; and each player’s action set is {1,2,3}. Removing more chips than are present in the pile is illegal.

3.3 Hex

Hex is a 2-player perfect information game where the red and blue players place colored pieces on a hexagonally tesselated Rhombus grid, one at a time. The first player that connects the two sides (e.g., top to bottom or side to side) that they are assigned wins the game. The states are board positions similar to Tic-Tac-Toe. However, the symmetry here is more restrictive, as only rotating 180∘ yields an equivalent state since the sides matter, and the rhombus shape prohibits 90∘ rotation. We use the same notation system as Tic-Tac-Toe for the actions.

3.4 Decomposition and Complexity

We may construct a best-play automaton with appropriate state-space formulation and convert it to a transformation semigroup. The utility function dictates the best-play automata. By only caring about winning (utility of 1), losing (utility of −1), or getting a draw (utility of 0), we can obtain the complete set of best-play automata. However, this leads to an enormous amount of automata, even in the simplest games like Tic-Tac-Toe, and absurd situations where a player purposefully drags the game on even though they could win in one move. Consequently, it is wise to take the duration of the game into account when calculating utility. For example, when I win, the utility will be 1/(number of turns). Similarly, if I am losing a game, I will try to survive for as long as possible, hoping my opponent will make a mistake. In other words, a victory with fewer turns should get a higher utility, and a loss with more turns should get a higher utility, too. As Nim and Subtraction Games are impartial games, we can use the Sprague-Grundy theorem to find the nimber of the game [7] and therefore determine winning moves directly. We would only need to apply minimax for losing games.

Ideally, we would like to construct all of the best-play automata by picking our optimal strategies one at a time since the complexity of the game is minimal complexity of all of these. However, this is unrealistic due to the sheer number of automata generated by all possible optimal strategies. The process is also highly nontrivial. One needs to recursively perform breadth-first search game tree construction, resulting in trillions of trees for even a simple game like Tic-Tac-Toe. Therefore, we only attempt to obtain the tightest possible complexity upper bounds of the game by collecting only a small sample of the best-play automata. We apply a compound uniform random sample (with possible replacement). We will convert each best-play automaton in the sample to a transformation semigroup to study its complexity.

The SgpDec package written in GAP [4] provides the holonomy decomposition, which provides a complexity upper bound for a system. We additionally implement a function to verify maximum chain of essential dependency [5], which guarantees a lower bound [5, 6]. We store each automaton sample’s complexity bounds and the maximum chain of essential dependency in a database.

4 Analysis

4.1 Complexity of Tic-Tac-Toe and Hex

As described earlier, we sample 500 random best-play automata for Tic-Tac-Toe and Hex and perform the complexity analysis on each one. The 500-sample size was chosen because it is large enough for an exploratory analysis and can be conducted within a reasonable time frame. Their complexity characteristics are summarized as pair plots in Fig. 3.

Fig. 3

Pair plots of complexity lower bound, upper bound, and number of states for a sample of N=500 best-play automata for a Tic-Tac-Toe and b 3×3 Hex

The pair plots comprise three scatter plots and three histograms for the complexity lower bound, upper bound, and the number of states for the sampled best-play automata; they provide many insights into their complexity characteristics. Firstly, there are no correlations between the complexity and the number of states in the automata. In both cases, the sampled best-play machines with the fewest states do not have the lowest complexity (of 2). Secondly, the histograms show that only a tiny proportion of sampled best-play automata achieved a complexity of 2 (for both the lower and the upper bounds). This indicates that we will need a considerably larger sample size to discover an automata with a lower complexity potentially. Thirdly, in both of the complexity lower bound vs. upper bound scatter plots, there exist automata whose complexity upper bound and lower bound are both 2, implying that some machines have an exact complexity of 2.

For Tic-Tac-Toe, one automaton with an exact complexity of 2 has the holonomy decomposition shown in Fig. 4a; this particular holonomy decomposition shows 27 total levels, but only 2 of them (levels 18 and 24) have nontrivial groups. There is also an essential dependency between these two levels with C2 structure. Since we only narrow the game complexity upper bound to 2, we cannot confirm Conjecture 1 that Tic-Tac-Toe has a complexity of 1 yet.

Fig. 4

Holonomy decompositions of a a Tic-Tac-Toe best-play machine with complexity 2 and b a Hex best-play machine with complexity 2

For the 3×3 Hex game, one automaton with an exact complexity of 2 has the holonomy decomposition shown in Fig. 4b; we see level 13 with C3 and level 16 with C2. Exploring a regular Hex game is currently unrealistic. Nonetheless, it does not contradict Theorem 1, as θ(G)≤2≤3·32.

4.2 Complexity of Subtraction Game and Nim

We start with a subtraction game since we can search all of the best-play machines for a game small enough, e.g., five chips only. Searching among game sizes of 5 up to 24 yields the following minimal bounds.

Fig. 5

Lower and Upper Bounds on Complexity of Subtraction Game versus Game Size

From Fig. 5, we obtain Proposition 1. Note that when Nmod4≡2, we do not know the exact complexity of the game, only that it is between 1 and ⌊N/4⌋.

Proposition 1

Let N be the size of the Subtraction Game, i.e., the number of chips in a pile at the start of the game. For N∈{5,6,⋯,24}:

If Nmod4≡0 or Nmod4≡3, then θ(G)=1.

If Nmod4≡1, then 1≤θ(G)≤⌊N/4⌋.

If Nmod4≡2, then θ(G)=⌊N/4⌋.

We would conjecture that this finding extends to any N≥5, and we can easily prove the following,

Theorem 2

For any Subtraction Game G with size Nmod4≡0, θ(G)=1.

Proof

Observe that when Nmod4≡0, the game is in a losing state (for the opponent), so whenever the opponent removes x∈{1,2,3} chips, our gameplay machine has no choice but to remove 4-x chips, and the resulting best-play machine always have the cyclic group structure Ci where i=N/4, as illustrated in Fig. 6. This implies a Krohn-Rhodes complexity of 1.    □

Fig. 6

Best-play automata for selected Subtraction Games (N=8,12,16,20,24). Nodes show the number of chips (in opponent view); transitions are labeled by opponent moves and lead to opponent view after the best-play automaton’s response

As for Nim, the best-play machine can vary wildly depending on how the game is set up. For simple configuration, 2 : 2 : 3, there are 12 best-play automata with an unknown game complexity that can be 1 or 2. However, for a simple configuration 1 : 2 : 3, there is only one unique best-play machine, and it has exact complexity 2; its holonomy decomposition has two levels with nontrivial groups and a maximum chain of essential dependency of length 2 (Fig. 7). Therefore, we disproved the Conjecture 2 that Nim has a complexity of 1.

Fig. 7

a The unique best-play automaton for the Nim Game 1 : 2 : 3. b Skeleton for the unique best-play automaton for Nim Game 1 : 2 : 3, having two levels with nontrivial groups and an essential dependency between them showing complexity 2

5 Conclusion

In this paper, we have undertaken a detailed analysis of the algebraic structures underlying several two-player games with perfect information, explicitly focusing on Nim, Subtraction Game, Tic-Tac-Toe, and the 3 × 3 Hex game. The relatively small state spaces of these games allowed for comprehensive minimax tree searches. Our approach involved crafting precise and compact state-space formulations for each game, considering factors such as board symmetry. Then, we backtrack our minimax algorithm to generate all possible best-play automata for smaller Nim and Subtraction Games, where we perform holonomy decomposition and find the maximum chain of essential dependency to confirm each one’s exact complexity. However, due to the practical limit and the absurd number of best-play automata for Tic-Tac-Toe and 3×3 Hex, we only sampled 500 best-play automata to estimate the tightest complexity bound (Fig. 3). We do not know the exact number of best-play automata for these games. Nonetheless, computational exploration over several days shows it is at least in the trillions and possibly orders of magnitude bigger. Our sample constitutes much less than 0.00000005% of this space for Tic-Tac-Toe, so it may be a more fruitful strategy to attempt to construct complexity 1 best-play automata directly, in parallel to further brute-force computational exploration.

To summarize, we have made several significant strides that further our understanding of the algebraic structure of games. Firstly, none of our findings contradicted Theorem 1, reaffirming its broad applicability. Secondly, a notable discovery was a Nim game setup with a complexity of 2, which refutes Conjecture 2. Thirdly, our analysis of Tic-Tac-Toe gives a complexity upper bound of 2. However, the possibility of a best-play machine with an exact complexity of 1 remains open.

References

1.

Dömösi, P., Nehaniv, C.L.: Algebraic Theory of Automata Networks, volume 11 of series Monographs in Discrete Mathematics. SIAM (2005)

2.

Egri-Nagy, A., Mitchell, J.D., Nehaniv, C.L.: SgpDec: cascade (de)compositions of finite transformation semigroups and permutation groups. In: Mathematical Software (ICMS 2014), pp. 75–82. Springer (2015)

3.

Egri-Nagy, A., Nehaniv, Mitchell, J.D.: SgpDec—software package for Hierarchical Composition and Decomposition of Permutation Groups and Transformation Semigroups, Version 0.9.7 (2023). https://​gap-packages.​github.​io/​sgpdec/​

4.

The GAP Group. GAP—Groups, Algorithms, and Programming, Version 4.13.0 (2024). https://​www.​gap-system.​org

5.

Nehaniv, J.R., Egri-Nagy, A., Dini, P., Rothstein Morris, E., Horváth, G., Karimi, F., Schreckling, D., Schilstra, M.: Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions. In: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 373, 20140223 (2015)

6.

Rhodes, J.: In: Nehaniv, C.L. (eds.), Applications of Automata Theory and Algebra: Via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games. Foreword by Morris W. Hirsch World Scientific Press (2010)

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Siegel, A.N.: Combinatorial Game Theory. American Mathematical Society, Providence, RI (2013)

Footnotes

1

See [6] or [1] for a detailed introduction.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_33

State-Complexity Relations in Evolved Players of the Iterated Prisoners’ Dilemma

Jing Lu1   and Chrystopher L. Nehaniv1  

(1)

University of Waterloo, Waterloo, ON, Canada

Jing Lu (Corresponding author)

Email: j289lu@uwaterloo.ca

Email: amylu2009@gmail.com

Chrystopher L. Nehaniv

Email: chrystopher.nehaniv@uwaterloo.ca

Abstract

We examine the complexity of players in evolved populations of finite automata that play the iterated prisoners’ dilemma. We track the complexity of players measured bounding their Krohn-Rhodes complexity, a rigorous integer-valued mathematical measure from algebraic automata theory. In particular, we test Rhodes’ hypothesis that evolved organisms will attain a complexity approximately equal to their number of possible states. Results confirm that number of states in evolved populations are close to the complexity of the evolved automata if there is a cost to having more states and that the difference between complexity and number states is smaller when costs for states are higher.

Keywords

Evolution-Complexity relationsEvolutionary computationKrohn-Rhodes complexityPrisoner’s DilemmaAlgebraic Automata Theory

1 Introduction

The relationship between the complexity and physical aspects of an organism has been difficult to study on the typical timescale of evolution in biological systems. By evolving digital organisms, which can be generated much faster, we can investigate how these factors might interact.

Digital organisms, specifically evolutionary algorithms, which represent individuals as finite state machines (FSM), have been used to investigate optimal player strategies to the iterated prisoners’ dilemma (IPD) [2, 6, 7]. The IPD is a classic study of decision-making and social behaviour, where player interactions show how populations might use cooperative strategies even when a rational actor would always choose to betray others. In other words, the IPD represents a suitable environment that drives evolutionary forces shaping player complexity and structure, which can then be analyzed through rigorous mathematical measures.

In line with the ‘Evolution-Complexity Relations’ Conjecture of mathematician John Rhodes [9, Sects. 6.59 and 6.118], we hypothesize that the complexity of the evolved automata players will be close to the maximum possible given their number of states. In particular, the complexity, which must be strictly less than the number N of reachable states [9], is expected to attain approximately that number in evolved automata. A tacit assumption of Rhodes’ argument is that it is more costly to have more states, leading to the conclusion the number of states will be optimized by evolution to be the minimum possible subject to achieving optimal play. Using artificial life techniques, we can study what happens in evolving populations of players with and without costs for more states to test this.

2 Background

Prisoners’ Dilemma

The Prisoners’ Dilemma serves as a model for decision making. It illustrates why two rational players might not cooperate, even when it is in their overall best interest to do so [6]. Two prisoners, or players, are given the choice to either cooperate (C) with or defect (D) against each other under interrogation. When they cooperate they have the best overall result, with both receiving a shorter punishment. However if Player 1 cooperates but is betrayed by Player 2, then Player 1 has the worst individual result (sucker payoff). Conversely, if Player 2 betrays the cooperating Player 1, then Player 2 has the best individual result (temptation payoff) [6]. Finally, if they both betray each other, have the worst possible overall result but better individual results than the sucker payoff. We represent this using points as shown in the payoff table in Table 1, where the more points the better [6]. This later translates into the fitness of the players.

Table 1

Payoff table showing the score associated with a move made in the Prisoners’ Dilemma for Two Players P1 and P2. Entries show the payoff for P1 / payoff for P2 for each combination of moves where they chose independently whether to cooperate or defect

P2

Cooperate

Defect

P1

Cooperate

3/3

0/5

Defect

5/0

1/1

Nash equilibrium describes a scenario where a player receives no benefits from changing their strategy while other players do not change theirs. A play of defect-defect (DD) would be a Nash equilibrium for the Prisoners’ Dilemma [6]. In other words, as rational actors, it would always make sense to defect in this game regardless of what choice the other player makes [6]. In comparison, Table 1 shows that mutual cooperation (CC) would triple both players’ scores. Acting in self-interest does not always produce the optimal outcome.

The most successful strategies in existing literature tend to have two traits: they can cooperate with fellow cooperators, and retaliate against defection [1, 2, 6, 7]. The iterated prisoners’ dilemma presents a good scenario for evolving organisms with the complexity required to compete and cooperate with each other to achieve a goal, a best pay-off over the course of repeated interactions.

Genetic and Evolutionary Algorithms

The use of genetic algorithms and evolutionary algorithms (evolving finite state automata) to encode players of the prisoners’ dilemma is well documented in existing literature. The idea of evolving finite state automata for a given task goes back to the seminal work of artificial life and computational intelligence pioneer Lawrence J. Fogel and his colleagues [5]. Evolutionary methods are an effective technique for solving optimization problems and model many processes of natural evolution as proposed by Darwin, including heredity, variation in traits from parents to offspring, differing fitness in the population, and limited resources through artificially imposed constraints or “death”.

Just like in biological evolution, the players must have some measure of fitness which determines their likelihood of reproducing and passing on their traits to the next generation of players. For the prisoners’ dilemma, this is usually the summed score of all games played by the player. Genetic recombination and mutation introduces variation in the population. After a number of generations, meaningful strategies are observed to emerge [1, 2, 6, 7].

To take advantage of playing multiple games, players must remember previously made moves and outcomes. Axelrod and Golbeck create players using a 64-bit string to encode all possible histories of a player’s last three games to determine the next move [1, 6]. This type of encoding will always yield Krohn-Rhodes complexity (see below) at most 7.¹ This structure constrains the long term dependencies in the behaviours of Axelrod and Golbeck’s players.

Dropping this constraint, evolved automata can not only have non-trivial complexity but their states and structure serve as a form of memory with less constrained hierarchical organization. Hourigan and Collins encode their players this way [2, 7]. They can be understood as finite state machines, and thus are characterized by associated finite semigroups. A finite state machine (FSM) is a mathematical model that can be in one of a finite number of states at any time. It transitions from one state to another depending on its current state and inputs received. They will be represented as Moore machines, i.e., FSMs whose current state determines their output move. As an example, Fig. 1 shows the tit-for-tat strategy as an FSM [1]. It always cooperates for its first move, and then copies the opposing player’s last move, effectively cooperating with cooperators and punishing defectors.

Fig. 1

Tit-for-tat player modelled as a Moore FSM. Its output is the same as the opponent’s previous move, allowing it to cooperate with cooperators and punish defectors. Transitions show the opponent’s current move, while nodes are labelled by the player’s response in the next round

Krohn-Rhodes Complexity

Every FSM can be represented as a finite semigroup together with its set of basic events (inputs) operating on states [4]. They are essentially equivalent. A semigroup is a set with any associative operation that takes two elements of the set and returns a third. The operation does not have to be commutative. For groups, the binary operation must be both associative and invertible. For full mathematical definitions and details see [4]. A finite semigroup can be decomposed into finite aperiodic semigroups and finite groups. The Krohn-Rhodes complexity of a finite semigroup S is the least number of groups in a wreath product of finite groups and finite aperiodic semigroups of which S is a divisor [4]. In other words, semigroups can be decomposed into simpler building blocks. To represent the semigroup, these building blocks are composed hierarchically—they are in levels. The levels deeper in the hierarchy receive information from the levels above [4]. Each level may also have a ‘reset’ state component. Krohn and Rhodes proved that this was enough to represent every semigroup [4]. The smallest number of levels with groups required to compose a semigroup S is the complexity of S. Determining the exact complexity is still an open problem, but there are good methods for getting the upper and lower bounds [4]. Holonomy decomposition is one way of finding the building blocks required for composing semigroups, and provides an upper bound on complexity [4]. It has been implemented in GAP with the SgpDec library [3], which we will be using for complexity analysis in this paper.

3 Methodology

Our simulations evolve players of the iterated prisoners’ dilemma, encoded as finite state automata. The player FSMs are represented in the code by two arrays, next_state and move, which determine the player’s next state and move by using the current state and opponent’s move as indices. See Table 2 for an example. The players always start in state zero. The max possible states for a player has been set to 15, but the actual number of reachable states will be equal to or less than 15 as the initial population members, crossover process, and mutation process are random. For example, if column 0 of next_state says to go to state 0 regardless of if the opponent cooperates or defects, the FSM would effectively only have one state. The unreachable states are functionally equivalent to unexpressed genetic information. They may be passed down to the player’s children, but do not affect the current player’s performance. When we discuss the number of states a player has in regards to the evolution-complexity relation, we always mean the number of actually reachable states.

Table 2

Tit-for-tat player represented in array form. The top array, next_state shows which state will be transitioned to given the player’s current state and the opponent’s move. The bottom array move represents each state’s corresponding move. For example, if the player is in state 0 it will output a Cooperate move, and if the opponent Defects, it transitions to state 1 and will defect in the next move

next_state

Player’s current state

0

1

Opponent’s prev move

COOP

0

0

DEFECT

1

1

move

Player’s current state

0

1

Player’s move

COOP

DEFECT

Every simulation run begins with 100 randomly generated players, which will reproduce until there have been 100 generations of players. To clarify the terminology used, a game of the Prisoners’ Dilemma is when both players make a move, and are rewarded points based on the payoff table in Fig. 1. A round consists of 30 games between two players. A cycle is complete when each player has played a round against every other player in the population [6]. The sum of all the points a player earns during a cycle becomes their score—a value which is used to calculate their fitness.

There are three parts to reproduction: the selection strategy, crossover, and mutation. The selection strategy determines which players reproduce.

The roulette wheel selection method was used in this experiment, where the likelihood of a player being chosen for reproduction was proportional to their fitness. The overall fitness of the entire generation is tallied, and used to normalize the player fitness, giving us the percentage likelihood a player will be selected as a parent, i.e., Player Selection Likelihood is PlayerFitnessTotalGenerationFitness.

This selection strategy gives the best strategies a high chance of reproducing, allowing for a relatively quick convergence, while maintaining genetic variation in the population by still giving less fit players a chance of reproducing [6].

This simulation also uses elitism in its selection strategy. To preserve the best strategies, the 30% of players with the highest fitness are copied directly to the next generation.

Crossover describes the process in which the genetic material of the two selected parents are combined. A random index representing the crossover point is generated. Before the crossover point, the next_state and move array of the child is a copy of the first parent. After the crossover point, it is a copy of parent 2. Since the crossover point could be at the precise beginning or end of the array, it is possible that the child ends up being an exact copy of one parent.

A mutation rate of 7.5% was used—each element in a next_state or move array has a chance of being randomly replaced with a new, randomly generated value. There are a few peculiarities to consider on how mutation works in this implementation. Since the newly generated value is random, it could be exactly the same as the original value. For the move array there are only two possible values, C and D, so this is not uncommon.

In addition, a state or move that is not reachable for the current player could be the one changed. This is similar to mutating an unexpressed gene. It does not affect the performance of the current player, but the mutation is hereditary and may later influence the player’s descendants. Any of a player’s 15 states have an equal chance of being mutated regardless of whether or not they are reachable. When a penalty function for reachable state count is applied players will tend to have less reachable states. This means it is less likely a mutation will actually affect their performance. However, when one of their few reachable states are mutated, the change is more likely to have a great effect on their strategy.

There were four groups being compared in the experiment, with a different fitness calculation for each:

(1)

no evolutionary pressure and no penalty function, where the fitness for every player is always equal (constant fitness group): Fitness=1.

(2)

evolutionary pressure and no penalty function, where fitness is determined only by score (no penalty group): Fitness is (Score100)2.

(3)

evolutionary pressure and subtractive penalty function, where fitness is determined by score and number of reachable states (exponential penalty group): Fitness is (Score-PopulationSize∗ReachableStateSize2100)2.

(4)

evolutionary pressure and divisor penalty function, where fitness is determined by score and number of reachable states (divisor penalty group): Player Fitness is (ScoreReachableStateSize+5).

Consider the example tit-for-tat player in Fig. 1. It has 2 states. Let’s say it scored 9000 points in one cycle with 100 players. In the first scenario, it would have a fitness of 1. In the second, it would have a fitness of 902=8100. In the third, it would have a fitness of (9000-100∗22100)2=7396. Fourth, a fitness of 90002+5=1286. It is possible that (Score-PopulationSize∗ReachableStateSize2) will be a negative value. To avoid negative fitness, the player’s fitness is instead set to zero. The penalty is applied before normalization to determine the percentage likelihood of being selected.

The purpose of adding a penalty is to model an environment with more evolutionary trade-offs or constraints, for example the energy cost of larger mass or more features. Constants were selected to avoid arithmetic overflow in the case of exponents or to tune the fitness functions appropriately, e.g., in scenario (4), the divisor is ReachableStateSize+5 to smooth the fitness landscape for very low numbers of states, such that the population does not almost always converge on the easy 1-state best solution (always defect). An overly harsh penalty for states without an appropriate reward for cooperating would make it extremely difficult to evolve any non-trivial behaviour, and the extreme case of just one state is not biologically plausible for organisms.

Each scenario was run ten times. Once a simulation run has been completed, every player is converted to GAP’s semigroup format so holonomy decomposition can be run. The upper bound for complexity is calculated and recorded, as well as the number of reachable states. Code is available at https://​github.​com/​mensshi/​GA-Prisoners-Dilemma.

Fig. 2

Graphs comparing the percentage of cooperative moves (red) to average fitness (green) of the population over 100 generations. When there is no evolutionary pressure (constant fitness function) the population does not stabilize. Although the fitness used for breeding was set to a constant 1 for this group, what is displayed on this graph is based on their actual game score, for analysis purposes. For the other three scenarios, the population begins randomly and then tends to converge to a stable population of majority cooperators or defectors. The graphs in this figure all show the majority cooperator scenario

4 Results and Discussion

For the figures in this section, cooperative move percentage per generation (COOP) is calculated by games where both players cooperate (C-C), divided by all games played. Fitness percentage is calculated by the fitness of a player divided by their theoretical maximum fitness, i.e., if a player scored 5 points in every game they played. It is impossible to have an average fitness of 100% in a generation because when a player scores 5 points, they have defected and taken the temptation payoff, so the opposing player has a score of 0. The other set of graphs compare Krohn-Rhodes complexity (an upper bound estimated through holonomy decomposition) and the number of reachable states a player has.

As seen in Fig. 2 on cooperative moves and fitness and Fig. 3 on reachablability states and complexity, when there is no evolutionary pressure fitness, cooperative move percentage, number of states, and complexity do not change very much from their initial values. Across all 100 generations, the number of states tends to be around 12 and the complexity upper bound tends to be around 6.

When evolutionary pressure exists, fitness and cooperative percentage tended to stabilize quite high or low, depending on whether the population converged to majority cooperators or defectors. In a few cases the population did not stabilize within 100 generations. For both scenarios, but especially with the exponential penalty, the number of states and complexity tends to become smaller over the generations. The difference between the two shrinks.

Fig. 3

Graphs comparing the average complexity upper bound and number of states of the same populations in Fig. 2, over 100 generations. In the groups with evolutionary pressure, complexity and state size tend to trend downward and change together, with the exponential and divided penalty groups dropping the fastest

Table 3

One-way ANOVA analysis of the mean difference in Krohn-Rhodes complexity and reachable state count for the constant fitness, no penalty, exponential penalty, and divisor penalty groups. Run with a significance level of 0.05, it indicates there is significant difference between the groups

Source of variation

SS

df

MS

F

P-value

F-crit

Between groups

78.29

3

26.1

94.36

4.05E-17

2.87

Within groups

9.96

36

0.28

Total

88.25

39

Table 4

Tukey HSD run as a post-hoc analysis after the omnibus ANOVA test. Here are the results when q-critical is 3.85. Other than exponential versus divisor penalty, the q-score between each pair of groups is larger than q-critical, so there is a significant difference between them

Group pairs

q score

Null versus no penalty

8.39

Null versus exponential

21.43

Null versus divided

18.09

No penalty versus exponential

13.05

No penalty versus divided

18.09

Exponential versus divided

3.34

As seen in Table 3 a standard significance level of 0.05 was used for the ANOVA. The ANOVA results show that there is a significant difference in the means, with a F-value of 94.36, which is larger than the F-critical value of 2.87. The p-value of 4.05×10-17 is less than 0.01 (highly significant).

Furthermore, the Tukey results in Table 4 show there is a significant difference between groups. Our Tukey q-critical value is 3.85. Except for the exponential versus divisor penalty, each of the q values calculated between the groups is larger. The groups with evolutionary pressure differ from the one without, and the groups with a penalty differ from the no penalty scenario. Then it appears that evolutionary pressure from the environment of the Prisoners’ Dilemma does cause the players’ complexity to become close to their number of states.

It also appears that complexity and number of states did not evolve to be maximal with or without a penalty function, i.e., maximum possible state number is 15, so the maximum possible Krohn-Rhodes complexity would be 14 [9]. The reason for this would require further investigation. One possible explanation is that it is easier to reduce the number of states than it is to evolve more complex strategies, particularly in the environment of the Prisoners’ Dilemma. Tit-for-tat is one of the most successful strategies in the literature and it has complexity zero in our encoding and only two states. We see that in the exponential penalty function group, complexity in the final generation often averaged 1. Since it is possible to be very successful even with a low complexity and small number of states, there may not be enough environmental pressure to evolve higher complexity and more states. This may change if the environment changes to a more complex and less abstracted model, e.g., by introducing local spatial structure constraining play rather than having each player play all others (cf. [8]).

5 Conclusion

Our results show that automata under evolutionary pressure have complexity close to their number of states, in line with Rhodes’ Conjecture [9]. We also see that different methods of determining fitness may lead to different results in how close the complexity is to the number of states, with a closer match between complexity and number of states as the cost for more states increases. One limitation to note is that we are estimating the Krohn-Rhodes complexity by using an upper bound.

However, although unexpected from our initial hypothesis, the number of states and complexity does not evolve maximally to use the number of states possible with these genetics. In particular, with an exponential or divisor penalty function that penalizes players for having more states, the number of states and complexity of the final generation is often very low. It is unclear why this happens, but it may be that the problem just does not require high complexity. Alternatively, it may be the genetic variability is so great in the system that potential optimal high-complexity strategies are disrupted before they can even evolve; however our use of elitism (always maintaining a best-so-far player) suggests this is not the case. It may be worth investigating with evolving organisms in environments more complex than the Prisoners’ Dilemma or in spatially structured ones.

Acknowledgements

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2019-04669. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), numéro de référence RGPIN-2019-04669. We thank Prof. Hiroki Sayama for useful discussions and an anonymous referee for help in improving this paper.

References

1.

Axelrod, R.: The Evolution of Strategies in the Iterated Prisoner’s Dilemma. In: Davis, L.D. (ed.) Genetic Algorithms and Simulated Annealing, pp. 32–41. Pitman, London (1987)

2.

Collins, D.R.: Review: a genetic algorithm approach to the iterated Prisoner’s Dilemma. Undergrad. Res. Nat. Clin. Sci. Technol. J. 3(8), (2019)

3.

Egri-Nagy, A., Nehaniv, C.L., Mitchell, J.D.: SgpDec: Cascade (de)compositions of finite transformation semigroups and permutation groups. In: Mathematical Software-ICMS 2014: 4th International Congress, Seoul, Korea, 5–9 Aug. 2014, pp. 75–82. Springer (2015)

4.

Egri-Nagy, A., Nehaniv, C.L.: A Beastiary of transformation semigroups for the Holonomy decomposition. In: Recent Developments in Mathematical, Statistical and Computational Sciences, pp. 37–46. Springer Nature (2021)

5.

Fogel, L.J., Owens, A.J., Walsh, M.J.: Artificial Intelligence Through Simulated Evolution. Wiley, New York (1966)

6.

Golbeck, J.: Evolving Strategies for the Prisoner’s Dilemma. Ph.D. thesis, University of Maryland (2002)

7.

Hourigan, E., Eck, D.: The genetic algorithm, the evolution of cooperation, and “Niceness" in the Iterated Prisoner’s Dilemma. J. Young Investig. (2004). https://​www.​jyi.​org/​2004-september/​2004/​9/​14/​the-genetic-algorithm-the-evolution-of-cooperation-and-niceness-in-the-iterated-prisoners-dilemma. Accessed 13 Mar. 2023

8.

Nowak, M.A., May, R.M.: Evolutionary games and spatial chaos. Nature 359, 826–829 (1992)

9.

Rhodes, J.: Applications of Automata Theory and Algebra: Via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games. editor: Nehaniv, C.L., Foreword by Hirsch, M.W.: World Scientific Publishing Co., Singapore (2010)

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Yang, X.-S.: Chapter 6—Genetic algorithms. In: Nature-Inspired Optimization Algorithms, 2nd edn., pp. 91–100. Academic Press (2021)

Footnotes

1

More generally, with states encoding length n histories, any long enough sequence w of plays by the opponent determines the last n opponent’s moves and there are 2n possibilities for the remaining player’s moves. It follows that any element s with s2=s in the FSM’s semigroup has image of size no than more 2n states, hence complexity is at most 2n-1 as follows from Theorem 6.107 of [9].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_34

The Sparse Grid Combination Method for Multidimensional Black-Scholes Partial Differential Equations

Ruining Wu1   and Christina C. Christara1  

(1)

Department of Computer Science, University of Toronto, Toronto, Canada

Ruining Wu (Corresponding author)

Email: rwu@cs.toronto.edu

Christina C. Christara

Email: ccc@cs.toronto.edu

Abstract

A fundamental challenge in the numerical solution of multidimensional partial differential equations (PDEs) is the exponential increase of the number of unknowns as the dimensions increase, known as the curse of dimensionality. This creates difficulties in obtaining accurate solutions for even moderate-dimension problems. Researchers have used the sparse grid combination method to reduce the computational workload for multidimensional PDEs. To resolve the unstable order of convergence that arises from the nonsmooth initial conditions, typical of finance problems, researchers applied smoothing (in Fourier space) or, in the case of a Basket option, a clever coordinate transformation that aligns the line of discontinuity with a coordinate axis. We view the transformation through the lens of quantization error, which allows us to determine the minimum order of smoothing required for restoring the order of convergence, and to highlight certain features of smoothing techniques. Additionally, we present numerical results for American options and for options with payoffs that cannot be transformed to align their nonsmoothness regions with a coordinate axis.

Keywords

Computational financeCurse of dimensionalitySparse gridsSmoothingsQuantization error

1 Introduction

There are several option pricing problems described by multidimensional PDEs. The most basic form of the Black-Scholes equation [1], that models the price of a contingent claim on a single underlying asset, is one-dimensional (involves one spatial variable). One source of additional dimensions comes from multiple assets. There are options written on arbitrarily many assets, such as basket options, geometric average options, and more.

To mitigate the curse of dimensionality that arises from numerically solving multidimensional PDEs, researchers have used the sparse grid combination method [5], which uses substantially fewer degrees of freedom compared to the full grid method, and hence is more computationally efficient, at the cost of a slightly deteriorated order of convergence from O(hβ) to O(hβ(logh)d-1), for β-order methods. Sparse grid methods have originally been developed for multidimensional numerical integration [14]. The success of the sparse grid combination method relies heavily on smoothness requirements, which are not normally satisfied in computational finance, as the typical initial conditions (ICs) exhibit discontinuities. Some recent works on the pricing of European options use high-order methods [6] with a grid transformation [9], which aligns the plane of discontinuity for basket options with a coordinate axis, reducing the smoothness requirements to one dimension, and simplifying smoothing of the ICs when required. Additionally, in [7], no coordinate transformation is applied, and a more general multidimensional smoothing [8] is applied to restore the order of convergence. In our work, we

compare various smoothing techniques with or without the coordinate transformation, and highlight certain features of each,

demonstrate, through numerical examples, that the theory of quantization error [2] for the one-dimensional problem can be applied to certain multidimensional cases to explain smoothness requirements of the ICs,

apply the sparse grid combination method to a nonlinear PDE arising from an American option pricing problem with payoff depending on both dimensions, and

provide examples where discontinuities in the payoff cannot be aligned with a coordinate axis, and show how averaging [11] restores the order of convergence.

2 Problem Formulation

The financial options we consider in this paper are under the Black-Scholes model. For European options, the PDE that governs the price of the option is given by

Vτ=LV≡12σ2S2VSS+rSVS-rV,

(1)

where V denotes the option price that we are solving for, S the price of the underlying asset, τ the reverse time counted from expiry T (τ=T-t, t is forward or real time), σ the volatility, r the interest rate, and subscripts denote partial derivatives.

In d dimensions, with d assets, the Black-Scholes PDE is given by

Vτ=LV≡12∑i,j=1dρi,jσiσjSiSjVSi,Sj+∑i=1drSVSi-rV,

(2)

where Si and σi denote the price and volatility of the i-th asset, and ρi,j denotes the correlation between Si and Sj.

Payoff functions correspond to ICs. There are many different payoffs in d dimensions; we use the Basket Put, given by V(0,Si)=max(K-∑wiSi,0), the Min-Put, given by max(K-min(Si),0), and the Digital (or binary) Basket Call, given by 1(wiSi-K>0), where K is known as the strike price, wi are the weights, conventionally taken to sum to 1, and 1 is the indicator function. Note that Basket Put and Min-Put payoffs are C0, while Digital is C-1. We are interested in the option values at τ=T and Si≈K, as this is considered the interesting region to practitioners.

2.1 American Options

In contrast to European options, American options can be exercised at any time up to expiry. This early exercise right gives rise to a linear complementarity problem, which we solve by replacing with the nonlinear PDE [4]

Vτ=LV+ρmax(V∗-V,0),

(3)

where ρ is the reciprocal of the desired accuracy ϵ for solving with a penalty iteration algorithm. Solving the American PDE (3) using second-order numerical methods results in a deteriorated order of convergence, as shown in [4]. The authors propose to resolve this issue using adaptive timesteps. Further research [13] has shown that a quadratic transformation of the uniform timesteps also restores the order of convergence. We prefer to use the latter in our work due to its simplicity.

3 Sparse Grid Combination Method

The sparse grid combination method [5] (henceforth, combination method) is a discretization method for mitigating the curse of dimensionality. It makes use of existing PDE solvers, each based on a different, anisotropic tensor product discretization. By combining the numerical solutions of the subproblems appropriately, error terms from dimensions which have coarser discretizations are cancelled out, resulting in an accurate approximation to the numerical solution computed by the full grid method. Additionally, since the subproblems have no interdependence on each other, parallelization comes easily.

Consider the case in two dimensions. Let ui,j denote the numerical solution computed by the two-dimensional grid with level i and j in the first and second variables respectively. Each increase by one in level doubles the number of gridpoints in that dimension.

In two dimensions, the combination method solution at grid level q is computed by

uqc=∑i+j=q+1ui,j-∑i+j=qui,j,

(4)

and if the method for solving the subproblems admits an error expansion of the form

u-ui,j=C1hi2+C2hj2+Dhi2hj2

(5)

with hi and hj the stepsizes in the first and second dimensions, respectively, and where C1, C2 and D are bounded above by a constant independent of hi or hj, then, with asymptotic error expansions, it can be shown [5] that the error terms on the coarse grids in both dimensions cancel out, and only q-level errors remain. The combination formulas can be generalized to d dimensions, resulting in O((logN)d-1) subproblems, each of size O(N) for a total complexity of O(N(logN)d-1). The combination method solution for the general d-dimensional case is given by

uqc=∑p=0d-1(-1)pd-1p∑∑li=q+(d-1)-pul1,l2,…,ld.

(6)

In other words, it is a sum of d sums, and the coefficient associated with each term is given by binomial coefficients with alternating signs.

Due to the slightly deteriorated order of convergence, typical orders expected for 2D problems vary from ∼1.4 to ∼1.8, while for 3D problems from ∼1.2 to ∼1.7, assuming about 7 grid refinements.

The combination method was originally developed for elliptic PDEs. In the application to parabolic PDEs, we apply the appropriate combination formula to the solution computed at the final timestep at the respective levels. The timesteps are chosen to be the same size as that for a full grid method on the same grid level.

4 Numerical Methods

In our work, we use second-order accurate methods. Specifically, we use Crank-Nicolson with Rannacher smoothing [12] (CNR) for timestepping and three-point finite difference (FD) formulas on nonuniform grids generated by smooth mappings [3] for spatial discretization. Note that while the nonuniform three-point FD approximation to the second derivative is technically first-order, because we apply it on a smooth nonuniform grid, it can be shown to be second-order accurate. Multidimensional grids are created from tensor products of one-dimensional grids.

For each spatial variable Si, we truncate the semi-infinite domain [0,∞) to [0,Si,max] where Si,max is chosen sufficiently large such that the accuracy of the solution near the strike is affected minimally. For 2D problems, we set Si,max=8K, and for 3D problems, we set Si,max=6K.

In a multidimensional discretization, the straightforward application of CNR timestepping in the combination method may lead to spurious oscillations, so we use smoothing and other techniques described subsequently to remove the nonsmoothness of ICs. While the nonsmoothness of the ICs may also affect the convergence of the full grid method, the effect is less visible, especially on C0 payoffs.

To have a unique solution to the PDE problem, it is not always necessary to have boundary conditions. For computational purposes, we impose some equations on the boundary. All boundary conditions we impose are of Dirichlet type, and derived by time-discounted payoffs (i.e. V(τ,∂Ω)=V(0,∂Ω)exp(-rτ)).

4.1 Smoothing Operators and Quantization Error

For the smoothing of functions, we turn to their representation in Fourier space. The well-known paper [8] gives smoothing operators for various orders of smoothness; relevant here are the first- and second-order smoothings, denoted by Φ1 and Φ2, respectively. While these operators are presented in [8] for uniform grids, we consider the generalization of Φ1 that appears in [11] and is known as averaging, defined as

us(xj)=1xj+1/2-xj-1/2∫xj-1/2xj+1/2u(x)dx,

(7)

where u is the IC, us(xj) is the smoothed function sampled at the gridpoints xj, xj-1/2 is halfway between the points xj and xj-1, and xj+1/2 likewise. It can be seen that us is the average value of u in [xj-1/2,xj+1/2], hence the name. The averaging technique can also be seen as a simple approximate extension of Φ1 of [8] to nonuniform grids. It also generalizes easily to higher dimensions. For example, in two dimensions,

us(xj,yi)=1(xj+1/2-xj-1/2)(yi+1/2-yi-1/2)∫xj-1/2xj+1/2∫yi-1/2yi+1/2u(x,y)dxdy,

(8)

and higher dimensions are analogous.

Quantization error is loosely defined as the error arising from the placement of the point of nonsmoothness on the discrete grid. In [2], for one-dimensional problems, explicit formulae are developed for the quantization error with various ICs of varying degrees of nonsmoothness. We briefly review some facts directly relevant to this paper. Let α∈(0,1] be the relative placement of the nonsmooth point on the grid. As shown in [2], for Heaviside (digital), the quantization error without smoothing has an O(h) term with coefficient α-0.5, and an O(h2) term with coefficient dependent on α. Thus, aligning the discontinuity with a midpoint in all grid refinements results in stable O(h2) convergence. Equivalently, the Φ1-smoothed Heaviside gives stable O(h2) convergence, if α remains constant in all refinements, and the Φ2-smoothed Heaviside also gives stable O(h2) convergence, regardless of α. For ramp (call, put, etc.) ICs, the quantization error without smoothing has an O(h2) term with coefficient dependent on α. Thus, aligning the discontinuity with a gridpoint (easy to maintain in all grid refinements in 1D) results in stable O(h2) convergence. Equivalently, the Φ1-smoothed ramp function also gives stable O(h2) convergence, regardless of α. Note that, in multiple dimensions, maintaining α constant is not easy, if not impossible.

5 Numerical Experiments

To demonstrate the improved efficiency of the methods along with requirements on smoothness, we present numerical experiments for the two-dimensional American Basket Put, two-dimensional Digital Basket Call, and three-dimensional Min-Put options.

In all numerical experiments, the solution of all subproblems takes place in parallel, with enough workers, such that it is possible to distribute only one subproblem to each worker.

Throughout the section of numerical results, q denotes the grid level, with the sparse grid solution for a certain grid index qk intended to approximate the full grid solution at the same index qk, Np denotes the number of subproblems, NS denotes the number of spatial gridpoints used, Nτ denotes the number of timesteps taken (for the combination method, we count the gridpoints and timesteps from all subproblems), and convergence order is calculated with successive differences; that is,

order=log2(uq-uq-1)/(uq+1-uq)

(9)

where uq is the solution of the q-th grid level, measured pointwise at (K, K) in the two-dimensional case, or (K, K, K) in the three-dimensional case.

5.1 American Basket Put

Our first problem is the American Basket Put, with results shown in Table 1. In all tables and relevant discussion, “Sparse Grid method” refers to the combination method. The Full Grid method is not as sensitive to the unsmooth C0 payoff, and even with unsmoothed ICs, exhibits second-order convergence. The Sparse Grid method on the original coordinates with 2D-averaging gives compatible (if not slightly better) results. The Sparse Grid method on the transformed coordinates as given in [9] is also comparable. We point out that, since the coordinate transformation aligns the discontinuity with one coordinate axis, the Fourier transform of the IC reduces to the one-dimensional case, which is studied extensively in [2]. When using the coordinate transformation, we ensure that our one-dimensional grid in the direction orthogonal to the plane of nonsmoothness has a gridpoint (1D-alignment with gridpoint) in the location of the nonsmoothness. This allows the combination method to exhibit a stable order of convergence. In the terminology of [2], the quantization error is always present and is second order, with a constant coefficient (Table 1).

Table 1

Two-dimensional American basket put option, with parameters σi=0.4, ρ=0.2, r=0.1, T=1, and K=10, solved with penalty method, and quadratic transformation of time domain. In the Sparse Grid method, all subproblems are solved in parallel

q

Np

Nτ

NS

Iters

Value

Difference

Order

Time (s)

Full Grid method, no smoothing

1

1

34

1089

63

0.874869332

–

–

2.79e-01

2

1

66

4225

129

0.877385013

2.52e-03

–

2.48e+00

3

1

130

16641

258

0.878047559

6.63e-04

1.92

2.36e+01

4

1

258

66049

515

0.878225544

1.78e-04

1.90

2.58e+02

5

1

514

263169

1027

0.878265165

3.96e-05

2.17

2.46e+03

Sparse Grid method, original coordinates, 2D-averaging

2

3

198

5379

365

0.877620427

–

–

9.30e-01

3

5

650

17029

1203

0.878118233

4.98e-04

–

2.97e+00

4

7

1806

46471

3373

0.878241795

1.24e-04

2.01

1.28e+01

5

9

4626

117641

8687

0.878276333

3.45e-05

1.85

5.18e+01

6

11

11286

284555

21153

0.878284761

8.43e-06

2.03

3.11e+02

7

13

26650

667533

50235

0.878287086

2.33e-06

1.86

1.44e+03

Sparse Grid method, transformed coordinates, 1D-alignment with gridpoint

2

3

198

5379

382

0.876834538

–

–

2.57e+00

3

5

650

17029

1231

0.877947454

1.11e-03

–

4.73e+00

4

7

1806

46471

3429

0.878204302

2.57e-04

2.12

1.26e+01

5

9

4626

117641

8797

0.878267177

6.29e-05

2.03

5.72e+01

6

11

11286

284555

21583

0.878282833

1.57e-05

2.01

3.03e+02

7

13

26650

667533

50819

0.878286697

3.86e-06

2.02

1.51e+03

Table 2

Two-dimensional Digital Basket Call Option, with parameters σx=0.4, σy=0.2, ρ=0.2, r=0.1, T=1, and K=10. In the Sparse Grid method, all subproblems are solved in parallel

q

Np

Nτ

NS

Iters

Value

Difference

Order

Time (s)

Full Grid method, original coordinates, no smoothing

1

1

34

1089

157

0.5335624882

–

–

3.58e-02

2

1

66

4225

401

0.5413892716

7.83e-03

–

1.76e-01

3

1

130

16641

981

0.5454967719

4.11e-03

0.93

1.49e+00

4

1

258

66049

2630

0.5475973997

2.10e-03

0.97

1.70e+01

5

1

514

263169

7058

0.5486592436

1.06e-03

0.98

1.76e+02

Full Grid method, transformed coordinates, no smoothing, 1D-alignment with gridpoint

1

1

34

1089

136

0.5316505597

–

–

3.80e-02

2

1

66

4225

301

0.5402955722

8.65e-03

–

1.53e-01

3

1

130

16641

657

0.5449069062

4.61e-03

0.91

1.09e+00

4

1

258

66049

1555

0.5472907262

2.38e-03

0.95

1.11e+01

5

1

514

263169

3623

0.5485028562

1.21e-03

0.98

9.54e+01

Full Grid method, original coordinates, 2D-averaging

1

1

34

1089

170

0.5506961685

–

–

5.36e-02

2

1

66

4225

398

0.5499706420

−7.26e-04

–

1.95e-01

3

1

130

16641

1038

0.5497891960

−1.81e-04

2.00

1.73e+00

4

1

258

66049

2632

0.5497438271

−4.54e-05

2.00

1.74e+01

5

1

514

263169

7065

0.5497323344

−1.15e-05

1.98

1.77e+02

Full Grid method, transformed coordinates, 1D-alignment with midpoint

1

1

34

1089

138

0.5520491862

–

–

1.12e-01

2

1

66

4225

322

0.5502341477

−1.82e-03

–

1.70e-01

3

1

130

16641

658

0.5498469489

−3.87e-04

2.23

1.08e+00

4

1

258

66049

1555

0.5497573230

−8.96e-05

2.11

1.05e+01

5

1

514

263169

3624

0.5497357514

−2.16e-05

2.05

9.32e+01

Sparse Grid method, transformed coordinates, 1D-alignment with midpoint

2

3

198

5379

809

0.5502339155

–

–

8.77e-01

3

5

650

17029

2565

0.5498468301

−3.87e-04

–

1.14e+00

4

7

1806

46471

6796

0.5497572770

−8.96e-05

2.11

1.76e+00

5

9

4626

117641

17033

0.5497357335

−2.15e-05

2.06

5.06e+00

6

11

11286

284555

41471

0.5497304484

−5.29e-06

2.03

1.95e+01

7

13

26650

667533

92999

0.5497291349

−1.31e-06

2.01

7.35e+01

8

15

61470

1531791

206345

0.5497287984

−3.37e-07

1.96

3.33e+02

9

17

139298

3456913

450930

0.5497286947

−1.04e-07

1.70

1.59e+03

Sparse Grid method, original coordinates, 2D-Φ2 smoothing

Gamma (Vxx)

Value (V)

Time (s)

2

1.729238e-04

–

–

0.5501696945

–

–

2.06e-01

3

1.726895e-04

−2.34e-07

–

0.5498427171

−3.27e-04

–

3.82e-01

4

1.726071e-04

−8.24e-08

1.51

0.5497595027

−8.32e-05

1.97

1.18e+00

5

1.725810e-04

−2.61e-08

1.66

0.5497369071

−2.26e-05

1.88

3.88e+00

6

1.725732e-04

−7.83e-09

1.74

0.5497308590

−6.05e-06

1.90

1.48e+01

7

1.725709e-04

−2.31e-09

1.76

0.5497292769

−1.58e-06

1.93

6.26e+01

8

1.725701e-04

−7.79e-10

1.57

0.5497288599

−4.17e-07

1.92

2.76e+02

9

1.725699e-04

−2.09e-10

1.90

0.5497287516

−1.08e-07

1.94

1.28e+03

5.2 Digital Basket Call

We consider the Digital Basket Call option to demonstrate our numerical method on C-1 ICs, show that the Greeks have stable convergence also, and to further show that the coordinate transformation in [9] allows us to analyze basket options using the one-dimensional case. Although the coordinate transformation method allows us, with a simple 1D-alignment with a gridpoint, to avoid smoothing for C0 ICs, as we have seen in the previous example, this is a feature of the nonsmooth but continuous ICs and not entirely of the transformation itself. When we have discontinuous ICs, such as this problem, we cannot avoid smoothing even on full grids. This is apparent from the first sub-table of Table 2, where the Full Grid method is first-order. In the second sub-table, the Full Grid method on transformed coordinates and neither smoothing, nor proper 1D-alignment with a midpoint is also first-order; see also first-order results in Table 8 of [9]. But, in the third sub-table, 2D-averaging (second sub-table) restores second-order for the Full Grid method. We note that, in theory, 2D-averaging (approximation to 2D-Φ1) may not be enough for stable second-order, but the Full Grid method is not as sensitive to nonsmoothness as the Sparse Grid method. Also, in the fourth sub-table, transformed coordinates and 1D-alignment with a midpoint give second-order errors.

Due to lack of space, we present Sparse Grid method results exclusively with smoothing, since smoothing is necessary even for Full Grid. To get second-order with the Sparse Grid method, we use transformed coordinates, and 1D-alignment with a midpoint in all grid refinements (fifth sub-table), which is easy to apply in one dimension, namely the dimension orthogonal to the plane of discontinuity. The second-order error can be explained mathematically, by applying the multidimensional Fourier transform to the IC. Since the IC in the new coordinate system depends on only one variable, the Fourier transform is the same as the one-dimensional transform in that variable. Therefore, the theory in [2] applies. In the original coordinate system, we cannot apply the theory of grid alignment, since the anisotropic grids from the sparse grid combination method cause gridpoints to have varying alignments with the plane of discontinuity. If we wish to keep the original coordinates with the Sparse Grid method, we need to apply 2D-Φ2 smoothing (sixth sub-table).

The last sub-table has a different structure, because we also want to show convergence of the Greeks; we indicate the Gamma (Vxx). Note that Np, Nτ and NS are the same as the previous sub-table. The number of iterations would be very similar, hence is omitted. Convergence is stable and monotonic.

Table 3

Three-dimensional European Min-Put Option, with parameters σi=0.4, ρi,j=0.2 for i≠j, r=0.1, T=1, and K=10. In the Sparse Grid method, all subproblems are solved in parallel

q

Np

Nτ

NS

Iters

Value

Difference

Order

Time (s)

Full Grid method, no smoothing

1

1

14

2197

42

1.631619998

–

–

2.66e-01

2

1

26

15625

78

2.082438148

4.51e-01

–

2.25e-01

3

1

50

117649

201

2.177623887

9.52e-02

2.24

3.78e+00

4

1

98

912673

588

2.200021333

2.24e-02

2.09

9.13e+01

5

1

194

7189057

1769

2.205548944

5.53e-03

2.02

1.60e+03

Sparse Grid method, no smoothing (minN=12)

3

10

500

64090

1082

2.210251041

–

–

1.94e+00

4

19

1862

222247

3907

2.245322399

3.51e-02

–

1.42e+00

5

31

6014

682087

12028

2.212317996

−3.30e-02

0.09

2.74e+00

6

46

17756

1937098

34874

2.218271375

5.95e-03

2.47

1.10e+01

7

64

49280

5214448

89798

2.208918361

−9.35e-03

−0.65

6.84e+01

8

85

130730

13497241

219145

2.210013624

1.10e-03

3.09

3.57e+02

Sparse Grid method, 3D-aver., minN=12

Sparse Grid method, 3D-aver., minN=24

3

2.226693054

–

–

–

–

–

–

4

2.215203163

−1.15e-02

–

2.210620071

–

–

1.37e+01

5

2.209757568

−5.45e-03

1.08

2.208408052

−2.21e-03

–

2.38e+01

6

2.208213011

−1.54e-03

1.82

2.207671564

−7.36e-04

1.59

7.16e+01

7

2.207626025

−5.87e-04

1.40

2.207493028

−1.79e-04

2.04

3.11e+02

8

2.207437289

−1.89e-04

1.64

2.207433828

−5.92e-05

1.59

1.46e+03

5.3 Three-dimensional Min-Put

Unlike basket options, the Min-Put, a put option written on the minimum of several assets, has ICs with multiple regions of discontinuity, that make alignment with a coordinate axis difficult, if not impossible. Hence, these experiments demonstrate the more general nature of the averaging technique, compared to applying transformations. In Table 3, we present three sub-tables, one showing second-order convergence of the Full Grid method, and two showing steady and erratic convergence of the Sparse Grid method with and without smoothing, respectively. In the last sub-table, we include results with minimum number of subintervals in all grids and subproblems minN=12, and minN=24. A larger minN improves the convergence rate, at the cost of increased runtime [10]. For minN=12, Nτ and NS are the same as no smoothing, with similar runtime. For minN=24, Nτ is doubled and NS octupled. The efficiency of the Sparse Grid method is emphasized in the 3D problem, with the highest grid index (q=5) of the Full Grid method using similar time as the highest grid index (q=8) of the Sparse Grid method for 100 times the error. For the Full Grid method, we estimate that it would take about three months for q=8.

6 Conclusions

The sparse grid combination method allows numerical solutions of multidimensional PDEs to be computed on grids that have finer resolution than is feasible for standard full grid methods. We focused on problems from computational finance, which are characterized by nonsmooth ICs, that are shown to cause trouble if the sparse grid combination method is blithely applied. Like other researchers [6, 7, 9], we turned to smoothing techniques to remedy this. We highlighted particular features. The coordinate transformation [9] eliminates discontinuities in all but one dimension, and allows the application of grid-alignment theory from [2]. Averaging [11] allows the treatment of more general ICs with discontinuities that cannot be aligned to a coordinate axis, and gives rise to errors equivalent to coordinate transformation, for C0 ICs. We obtained second-order accurate solutions and Greeks (without visible deterioration) for 2D problems, including American options. For 3D problems and complicated payoffs, we obtained the expected (slightly reduced) order of convergence.

Subsequent work involves the application of sparse grids to other multidimensional nonlinear PDEs, their smoothing requirements, the study of the quantization error in multiple dimensions, dimension-adaptive techniques for transformed variables, and closed-form formulae for the smoothing of common payoffs.

References

1.

Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)MathSciNet

2.

Christara, C., Leung, N.: Analysis of quantization error in financial pricing via finite difference methods. SIAM J. Numer. Anal. 56(3), 1731–1757 (2018)MathSciNet

3.

Clarke, N., Parrott, K.: Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance 6(3), 177–195 (1999)

4.

Forsyth, P., Vetzal, K.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput. 23(6), 2095–2122 (2002)MathSciNet

5.

Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, pp. 263–292 (1992)

6.

Hendricks, C.: High-order methods for parabolic equations in multiple space dimensions for option pricing problems. Ph.D. thesis, Bergische Universität Wuppertal (2016)

7.

Hendricks, C., Heuer, C., Ehrhardt, M., Günther, M.: High-order ADI finite difference schemes for parabolic equations in the combination technique with application in finance. J. Comput. Appl. Math. 316, 175–194 (2017)MathSciNet

8.

Kreiss, H.O., Thomée, V., Widlund, O.: Smoothing of initial data and rates of convergence for parabolic difference equations. Commun. Pure Appl. Math. 23(2), 241–259 (1970)MathSciNet

9.

Leentvaar, C., Oosterlee, C.W.: On coordinate transformation and grid stretching for sparse grid pricing of basket options. J. Comput. Appl. Math. 222(1), 193–209 (2008)MathSciNet

10.

López-Salas, J.G., Pérez-Rodríguez, S., Vazquez, C.: AMFR-W numerical methods for solving high-dimensional SABR/LIBOR PDE models. SIAM J. Sci. Comput. 43(1), B30–B54 (2021)MathSciNet

11.

Pooley, D., Vetzal, K., Forsyth, P.: Convergence remedies for non-smooth payoffs in option pricing. J. Comput. Finance 6(4), 25–40 (2003)

12.

Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numerische Mathematik 43(2), 309–327 (1984)MathSciNet

13.

Reisinger, C., Whitley, A.: The impact of a natural time change on the convergence of the Crank-Nicolson scheme. IMA J. Numer. Anal. 34(3), 1156–1192 (2014)MathSciNet

14.

Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii Nauk, Rus. Acad. Sci. 148(5), 1042–1045 (1963)

Part VITheory and Computational Methods for Differential Equations

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_35

Input-to-State Stability for Cascaded Impulsive Systems and H∞ Control

Mohamad S. Alwan1  

(1)

University of Saskatchewan, Saskatoon, Canada

Mohamad S. Alwan

Email: m.alwan@math.usask.ca

Abstract

This paper deals with a class of nonlinear cascaded systems undergoing impulsive actions and an external input disturbance signal with bounded amplitude. Our main interests are to establish sufficient conditions to guarantee the input-to-state stability (ISS) and design a feedback H∞ control. To do so, the system is decomposed into two cascaded (or interconnected) impulsive subsystems, namely the leader subsystem with the external input signal and follower subsystem with input being the output of the leader subsystem. We develop the ISS for each isolated impulsive subsystem and then for the cascaded system. We observed that the cascade of two ISS impulsive subsystems implies the ISS of the cascaded system. Later, these results are applied to establish another characterization of ISS where the controlled measured output of the leader is cascaded to the follower. Besides, the continuous leader subsystem is unstable but stabilized by an H∞ controller. The Lyapunov method is used to achieve ISS where each subsystem admits an ISS-Lyapunov function, then the cascaded system is ISS with a Lyapunov function being the composite of the former functions. A numerical example and simulations are presented to enhance the findings of this work.

Keywords

Cascaded systemsImpulsive effectsH∞Input-to-state stabilityLyapunov functionAlgebraic Riccati equation

1 Introduction

Many evolutionary processes such as those in engineering, sciences, ecosystems are subjected to abrupt changes in their states at certain instances during the continuous evolution. When compared to the total duration of the process, the duration of these changes are very small; hence, they can be reasonably approximated as instantaneous changes of the state or as impulses. The resulting system is called an impulsive system, a special class of hybrid systems. Mathematically, impulsive systems are described by a differential equation representing the continuous evolution of the process and a difference equation representing the impulses [1].

The notion of ISS is concerned with the system response to a bounded input signal when the unforced system is stable [2–4]. The problem of finding sufficient conditions to ensure the stability and/or ISS of nonlinear dynamical systems are often challenging. An avenue to cope with such a difficulty for certain complex coupled systems is to divide the system into leader-follower interconnected subsystems that are simpler to analyze. Moreover, the state or output of the leader is fed forward as an input disturbance to the follower. An interconnected system having such a structure is called a cascaded system [5–7]. These systems, as multiple-control-loop structures, are often used in manipulating (e.g. purifying or enriching) substances such as those used as fuels [9]. In [10], the cascade structure is used to synchronize the frequencies of two connected oscillating pendula (leader and follower) both perturbed by sinusoidal external forces where the first pendulum is to oscillate at the frequency of the second. In spacecraft formations [6], the leader spacecraft is controlled to track a given reference signal, and simultaneously the follower spacecraft is controlled to synchronize its motion with the leader’s.

Motivated by the above applications and others such as those in biochemical cascades, and by the fact that external disturbances and impulses in continuous and discrete dynamical systems are inevitable. In this work, the leader is disturbed by an external disturbing force, and both leader and follower undergo destabilizing impulsive effects. The contribution of this article is to address the notions of input-to-state stability and stabilization for a class of cascaded systems with impulsive effects. To do so, the Lyapunov method is used to write sufficient conditions that guarantee the ISS of the subsystems and then the cascaded system. In Sect. 3, the unforced continuous leader subsystem (i.e., zero input) is assumed to be stable and its full state is cascaded to the follower, while in Sect. 4 it is assumed to be unstable but stabilized by an H∞ controller and its available state is cascaded to the follower. The role of the H∞ controller is to input-to-state stabilize the unstable system in the presence of impulses and disturbance input, and to eliminate the undesirable peaking phenomenon that may occur in cascaded systems [5]. In fact, H∞ control is an effective tool to stabilize unstable systems undergoing external input disturbances. We believe that the proposed results and approach find applications in cascade-based dynamical systems with disturbances and impulsive effects.

2 Problem Formulation

Consider the following nonlinear cascaded subsystems with impulses

Σ1:x˙=f(t,x,z),t≠tk,Δx(t)=Ik(t-,x(t-)),t=tk,k∈N,x(t0)=x0

(1)

Σ2:z˙=g(t,z,u),t≠tk,Δz(t)=Jk(t-,z(t-)),t=tk,k∈N,z(t0)=z0

(2)

where Σ1 is the x-subsystem with input z and state x∈Rn1 and Σ2 is the z-subsystem with input u∈Rm and state z∈Rn2. The input signal u:[0,∞)→Rm is a piecewise continuous and belongs to class L2[t0,∞) functions; that is, ‖u(t)‖[t0,∞)L22=∫t0∞‖u(t)‖2dt with ‖u(t)‖=u(t)Tu(t), the Euclidean norm. The impulsive instants tk satisfy 0≤t0<t1<t2<⋯<tk<⋯ for k∈N. In the discrete equation, Δx(t)=x(t+)-x(t-) and Ik(t-,x(t-)) represents the pulse amount; similarly, we define Δz(t) and Jk(t-,z(t-)). The vector fields f:[tk-1,tk)×Rn1×Rn2→Rn1 and g:[tk-1,tk)×Rn2×Rm→Rn2 are piecewise continuous (PC) in t with f (and g) being locally Lipschitz in x and z (and z and u).

Definition 1

([3]) A continuous function ρ:[0,∞)→[0,∞) is said to be in class K if it is strictly increasing and ρ(0)=0.

Definition 2

System (2) is said to be input-to-state exponentially stable (ISS) if there exist ρ∈K and positive constants λ and λ¯ such that for any bounded signal u(t), the solution z(t)=z(t;t0,z0) exists and satisfies

‖z(t)‖2≤λ¯‖z0‖2e-λ(t-t0)+ρ(supt0≤τ≤t‖u(τ)‖),∀t≥t0.

The following impulsive assumptions are considered throughout this paper.

Assumption A1. There exist constants 0≤ρ1≤ρ such that, for all tk∈R+ and x∈PC(R≥0;D), with D⊂Rn, if ‖x(tk-)‖2<ρ1, then ‖x(tk)‖2<ρ.

Assumption A2. For any k∈N, we have the following two cases: tsup=sup{tk-tk-1}<∞ and tinf=inf{tk-tk-1}>0.

Remark 1

Assumption A1 is made to ensure that if the solution is bounded just before a pulse (i.e., at t=tk-) then it stays bounded just right after the pulse (i.e., at t=tk+). Assumption A2 guarantees that the interval between any two consecutive impulsive instances be neither infinity nor zero; that is, Assumption A2 is made to avoid chattering and trivialness, respectively.

3 Input-to-State Stability Result

In the following theorems, we establish the ISS property.

Theorem 1

(z-subsystem with input u) Assume that Assumptions A1 and A2 are satisfied, there exist positive constants c12,c22,α2,σ2, and a class-K function ρ2, and there exists a positive-definite, piecewise smooth, decrescent function V2(t,z(t)) for all (t,z)∈[tk-1,tk)×Rn2. Then, the z-subsystem is ISS with input u if the following conditions hold:

(i)

c12‖z‖2≤V2(t,z)≤c22‖z‖2;

(ii)

for all t≠tk (k∈N), V˙2(t,z,u)≤-σ2‖z‖2+ρ2(‖u‖[tk-1,tk)L2);

(iii)

at t=tk+(=tk), V2(tk+,z(tk-)+Jk(tk-,z(tk-)))≤d2kV2(tk-,z(tk-)) where d2k>1;

(iv)

for all k∈N, tk-tk-1<α2 and ln(d2k)≤σ2α2-σ2(tk-tk-1).

Remark 2

Conditions (i) and (ii) are to guarantee the ISS of the impulse-free z-system with input u. Particularly, in (i) the Lyapunov function V is positive-definite and decrescent, while in condition (ii) the time derivative of V can be re-written as: V˙≤-σ‖z‖2 whenever ‖z‖≥ρ(‖u‖) for some constant σ>0 and class K function ρ which means V is decreasing outside a ball having radius as function of u [3]. Condition (iii) shows jumping-up in V at tk+ due to the destabilizing impulses. The first part of condition (iv) guarantees that the time between two consecutive impulses be bounded by α2>0, and the second part is an algebraic condition to prove the inequality in (14).

Proof of Theorem 1

Let z(t)=z(t;t0,z0) be the solution state of the z-subsystem. To prove the result of this theorem, we consider the following claim.

Claim 1. For all t∈[tk-1,tk) with k∈N, there exists a constant M2>1 and a K function γ2 such that conditions (i)-(iv) imply that

V2(t,z(t))≤M2‖z0‖2e-σ2(t-t0)+γ2‖u‖[tk-1,tk)L2.

(3)

Proof of claim 1. Choose M2>1 such that

c22‖z0‖2<M2‖z0‖2e-σ2(t1-t0)+γ2‖u‖[t0,t1)L2

(4)

That is, the z-system is ISS at t0. For all t∈[tk-1,tk) and k∈N, the claim is proved by using the mathematical induction as follows. For k=1 (i.e. t∈[t0,t1)),

V2(t,z(t))≤M2‖z0‖2e-σ2(t1-t0)+γ2(‖u‖[t0,t)L2)

(5)

If this relation was not true, there would exist t¯=inftt∈(t0,t1) such that

V2(t¯,z(t¯))>M2‖z0‖2e-σ2(t1-t0)+γ2(‖u‖[t0,t1)L2)>c22‖z0‖2≥V2(t0,z(t0)),

(6)

Since V2(t,z(t)) is continuous over [t0,t1), there exists t∗∈(t0,t¯) such that

V2(t∗,z(t∗))=M2‖z0‖2e-σ2(t1-t0)+γ2(‖u‖[t0,t1)L2‖))

(7)

and for all t∈[t0,t∗], we have V2(t,z(t))≤M2‖z0‖2e-σ2(t1-t0)+γ2(‖u‖[t0,t1)L2). Also (due to the continuity of V2), there exists t∗∗∈[t0,t∗) such that

V2(t∗∗,z(t∗∗))=c22‖z0‖2

(8)

and for all t∈[t∗∗,t∗), V2(t,z(t))≥c22‖z0‖2. By condition (ii), D+V2(t,z(t))≤0 for t∈[t∗∗,t∗) which by integration leads to

V2(t∗∗,z(t∗∗))≥V2(t∗,z(t∗))

(9)

This inequality leads to c22‖z0‖2≥M2‖z0‖2e-σ2(t1-t0)+γ2(‖u‖[t0,t1)L2), which is a contradiction with (4); thus, the inequality in (3) is true when k=1.

Assume that (3) is true for k=1,2,…,m; that is, for all t∈[tk-1,tk),

V2(t,z(t))≤M2‖z0‖2e-σ2(tk-t0)+γ2(‖u‖[t0,tk)L2).

(10)

Now, we show that (10) is also true for l=m+1, i.e., for all t∈[tm,tm+1),

V2(t,z(t))≤M2‖z0‖2e-σ2(tm+1-t0)+γ2(‖u‖[t0,tm+1)L2).

(11)

If (11) was not true, define t¯=inftt∈[tm,tm+1) such that

V2(t¯,z(t¯))>eσ2αM2‖z0‖2e-σ(tm+1-t0)+γ2(‖u‖[t0,tm+1)L2)>M2‖z0‖2e-σ(tm+1-t0)+γ2(‖u‖[t0,tm+1)L2)

(12)

By the continuity of V2(t,z(t)) over [tm,tm+1), there exists t~∈[tm,t¯), such that

V2(t~,z(t~))=eσ2αM2‖z0‖2e-σ2(tm+1-t0)+γ2(‖u‖[t0,tm+1)L2)

(13)

and for t∈[tm,t~) V2(t,z(t))≤eσ2αM2‖z0‖2e-σ2(tm+1-t0)+γ2(‖u‖[t0,tm+1)L2). At t=tm+, with the aid of assumptions (iii) and (iv), we have

V2(tm,z(tm))≤d2kV2(tm-,z(tm-))<eσ2αe-σ2(tm+1-tm)V2(tm-,z(tm-))=eσ2αM2‖z0‖2e-σ2(tm+1-t0)+e-σ2(tm+1-tm)γ2(‖u‖[t0,tm)L2)≤eσ2αM2‖z0‖2e-σ2(tm+1-t0)+γ2(‖u‖[t0,tm+1)∞)<V2(t¯,z(t¯)).

(14)

So that, there exists t∗∈(tm,t¯) such that

V2(t∗,z(t∗))=eσ2αM2‖z0‖2e-σ2(tm+1-t0)+γ2‖u‖[t0,tm+1)L2

(15)

and hence D+V2(t∗,z(t∗))>0 which is a contradiction. Thus, (3) holds true for k=m+1 and hence is true for t∈[tk-1,tk). This completes the proof of Claim 1. Using assumption (i), we get ‖z‖2≤M2c12‖z0‖2e-σ2(t-tl-1)+1c12γ2‖u‖[tk-1,t)L2. This completes the proof of Theorem 1. Thus, the z-system (2) is ISS with input u.

Theorem 2

(x-subsystem with input z) Assume Assumptions A1 and A2 are satisfied, there exist positive constants c11,c21,α1,σ1, and a class-K function ρ1, and there exists a positive-definite, piecewise smooth, decrescent function V1(t,x) for all (t,x)∈[tk-1,tk)×Rn1. Then, system (1) is ISS with input z if the following hold:

(i) c11‖x‖2≤V1(t,x)≤c21‖x‖2;

(ii) for all t≠tk and k∈N, V˙1(t,x,z)≤-σ1‖x‖2+ρ1(‖z(t)‖[tk-1,tk)) (iii) at any t=tk (for k∈N), V1(tk+,x(tk-)+Ik(tk-,xtk-)))≤d1kV1(tk-,x(tk-)),d1k>1;

(iv) for all k∈N, tk-tk-1≤α1 and ln(d1k)≤σ1α1-σ1(tk-tk-1).

Proof of Theorem 2

The proof is similar to that of Theorem 1 except the input of the x-subsystem is z; so, it is omitted here due to page limit.

We are in position to prove the ISS of the cascaded system with input u.

Theorem 3

(Cascaded System with input u) Suppose that

(i) subsystems Σ2 and Σ1 are ISS in the sense of Theorems 1 and  2;

(ii) there exists a positive-definite, piecewise smooth, decrescent function V(t, x, z) for all (t,x,z)∈[tk-1,tk)×Rn1×Rn2 such that c1‖(x,z)T‖2≤V(t,x,z)≤c2‖(x,z)T‖2 with c1,c2>0 and the dissipation inequality V˙((t,x,z),u)≤-σ~1‖x(t)‖-12σ~2‖z‖+ρ~2(‖u‖[tk-1,tk)L2) where σ~1,σ~2>0 and ρ~2 is a class K function;

(iii) at t=tk+, V(tk+,x(tk+),z(tk+))≤dkV(tk-,x(tk-),z(tk-)) where dk=max{d1k,d2k};

(iv) for all k∈N, tk-tk-1<α and ln(dk)≤σα-σ(tk-tk-1) where α=max{α1,α2} and σ=min{σ~1,σ~2}.

Then, the cascaded system, Σ:u→xz, given in (1)–(2) is ISS with input u.

Proof of Theorem 3

Since both subsystems, Σ1 and Σ2, are ISS, then for all t≠tk the following dissipation inequalities hold:

V˙2(t,z(t))≤-σ2‖z0‖2+ρ2(‖u‖[t0,t)L2),V˙1(t,x(t))≤-σ1‖x0‖2+ρ1(‖z‖[t0,t))

For all (t,x,z)∈R≥0×Rn1×Rn2, define V(t,x,z)=V1(t,x)+V2(t,z) as a composite Lyapunov function candidate for the cascaded system. Then, for all (t,x,z,u)∈R≥t0×Rn1×Rn2×Rnm with t≠tk, the time derivative of V along the trajectories of the continuous cascaded system is given by

V˙((t,x,z),u)=V˙1((t,x),z)+V˙2((t,z),u)≤-σ1‖x(t)‖2+ρ1(‖z‖)-σ2‖z(t)‖2+ρ2(‖u‖L2)

(16)

Referring to [8], there exist a new supply pairs (σ~1,ρ~1) and (σ~1,ρ~1). In the present work, we have

σ~2=σ2,ρ~2(‖u‖L2)=2c222σ2ρ22(‖u‖L2),σ~1(‖x(t)‖2)=-σ1c112‖x(t)‖4,ρ~1(‖z(t)‖2)=12σ~2‖z(t)‖2,

Then, using the new supply pairs, inequality (16) becomes

V˙((t,x,z),u)≤-σ1c112‖x(t)‖4+12σ~2‖z(t)‖2-σ~2‖z(t)‖2+2c222σ2ρ22(‖u‖L2)=-σ1c112‖x(t)‖4-12σ~2‖z(t)‖2+2c222σ2ρ22(‖u‖L2)≤-(1-θ∗)σ1c112‖x(t)‖4-(1-θ∗)12σ~2‖z(t)‖2,0<θ∗<1

whenever xz≥max{‖x‖,‖z‖}≥1θ∗σ1~ρ~2(‖u‖L2)4,2θ∗σ2~ρ~2(‖u‖L2). Following the proof of Theorem 1, V is an ISS Lyapunov function for the interconnected system. Hence, the solution of cascaded system is ISS with input u.

4 Application to H∞ Control

Consider the cascaded subsystems:

Σ11:x˙=f(t,x(t),y),t≠tkΔx(t)=Ik(t-,x(t-)),t=tk,k∈N,x(t0)=x0,

(17)

Σ21:z˙=Az(t)+Bw(t)+Gu(t),t≠tl,y(t)=Cz(t)+Fw(t)Δz(t)=Jk(t-,z(t-)),t=tk,k∈N,z(t0)=z0,

(18)

where w∈Rq is the control input of the form Kz with K∈Rp×q being a control gain matrix, u∈Rm is a disturbance signal assumed to be in L2[0,∞), y∈Rr is the controlled measured output, and A∈Rn1×n1, B∈Rn1×p, G∈Rn1×m, C∈Rr×n1 and F∈Rr×q are real constant matrices. In this section, the unforced z-system (i.e., when w(t)≡0) is unstable, but stabilized by the feedback control w=Kz. Hence, the matrix pair (A, B) is assumed to be controllable or particularly stabilizable.

Definition 3

Given a constant ξ>0, the continuous z-system is said to be input-to-state stabilizable with an H∞-norm bound ξ if there exists a state feedback law w(t)=Kz(t) where K=-12εBTP2 for some a constant ε>0 and positive-definite matrix P2 such that the closed-loop system is ISS and the output y satisfies: ‖y‖L22:=∫0∞‖y(t)‖2dt≤ξ2‖u‖[0,∞)L22+m0,m0>0.

Theorem 4

Suppose that:

(i) the control gain K and disturbance level ξ>0 are given;

(ii) there is a constant σ2>0 and a positive-definite matrix P2 such that the following Riccati-like equation holds

(A+BK)TP2+P2(A+BK)+ξ-2P2GGTP+CTC+σ2P2=0;

(19)

(iii) at any t=tk+, V2(z(t-)+Jk(t-,z(t-)))≤d2kV2(z(t-)) where d2k>1;

(iv) ∑k=1∞Lk‖Jk(tk-,z(tk-))‖<∞ for some constant Lk>0, and (v) for all k∈N, tk-tk-1≤α2 and ln(d2k)≤σ2α2-σ2(tk-tk-1) where α2>0. Then, whenever V2(z)≥1θ∗ε2‖u‖[0,∞)L22 for some positive constants θ∗ and ε2, the z-subsystem is input-to-state stabilized by the feedback control law w=Kz with an H∞-norm bound ξ>0.

Proof of Theorem 4

For all t≠tk, let z(t)=z(t;t0,z0) be the solution of z-system and define V2(z)=zTP2z as a Lyapunov function candidate. Then

V˙2(z)≤zT[ATP2+P2A+2KTBTP2]z+2zTPGu≤zT[(A+BK)TP2+P2(A+BK)]z+ε2zTP2GGTP2z+1ε2uTu.

Making use of the Riccati-like equation in (19) with ε2=ξ-2, we obtain V˙2(z,u)<-σ2zTP2z+1ε2uTu=-(σ2-θ∗)zTP2z-θ∗zTP2z+1ε2uTu=-σ¯2V2(z) (with σ¯2=σ2-θ∗>0) provided that V2(z)≥1θ∗ε2‖u‖2 (or ‖z‖≥1λmax(P2)θ∗ε2‖u‖). Following the proof of Theorem 1, there exist a constant M2>1 and a function γ2∈K such that (for all t≥t0) V2(z)≤M2‖z0‖2e-α¯(t-t0)+γ2(‖u‖L2). Using λmin(P2)‖z‖2≤zTP2z, the last inequality implies ‖z(t)‖2≤M¯2‖z0‖2e-α¯(t-t0)+γ¯2(‖u‖L2) with M¯2=M2/λmin(P2) and γ¯2(·)=1/λmin(P2)γ2(·).

To prove the upper bound on the output magnitude, we consider the objective functional J=∫0∞(yTy-ξ2uTu)dt. It follows that

J=∫0∞(yTy-ξ2uTu)dt≤∫0∞(yTy-ξ2uTu)dt+∫0∞V˙2(z(t))dt-limt→∞V2(z(t))+V2(z(0))+∑k=1∞Lk‖Jk(tk-,z(tk-))‖≤∫0∞(yTy-ξ2uTu)dt+∫0∞[zT((A+BK)TP2+P2(A+BK)-ξ-2P2GGTP2+ξ-2P2GGTP2)z+2zTP2Gu]dt+∑k=1∞Lk‖Jk(tk-,z(tk-))‖+V2(z(0))=∫0∞zT[(A+BK)TP2+P2(A+BK)+ξ-2P2GGTP2+CcTCc]zdt-∫0∞(u-ξ-2GTP2z)Tξ2(u-ξ-2GTP2z)dt+∑k=1∞Lk‖Jk(tk-,z(tk-))‖+V2(z0)≤∑k=1∞Lk‖Jk(tk-,z(tk-))‖+V2(z0)

where we use the theorem assumption in (19) and the integrand of the second integral being strictly negative. Finally, the first integral and last inequality lead to ‖y‖L22≤ξ2‖u‖L22+∑k=1∞Lk‖Jk(tk-,z(tk-))‖+m0, with m0=V2(zt0).

Remark 3

The algebraic Riccati equation in (19) is solved for the positive-definite matrix P2 to ensure V˙2<-σ¯2V2(z). Lk>0 is a Lipschitz constant such that |V2(tk+,z(tk+))-V2(tk-,z(tk-))|≤Lk‖z(tk+)-z(tk-)‖=Lk‖Jk(tk-,z(tk-))‖.

Theorem 5

(Cascaded System) Suppose that: (i) Σ2 is ISS with input u and Σ1 is ISS with input y; (ii) there exists a positive-definite, piecewise smooth, decrescent function V(t, x, z) for all (t,x,z)∈[tk-1,tk)×Rn1×Rn2 such that c1‖(x,z)T‖2≤V(x,z)≤c2‖(x,z)T‖2 with c1,c2>0 and V˙((t,x,z),u)≤-σ~1‖x(t)‖-12σ~2‖z‖+ρ~2(‖u‖[tk-1,tk)L2) holds where σ~1,σ~2>0 and ρ~2 is a class K function; (iii) at t=tk+, V(tk+,x(tk+),z(tk+))≤dkV(tk-,x(tk-),z(tk-)),dk=max{d1k,d2k}, and (iv) for all k∈N, tk-tk-1≤α and ln(dk)≤σα-σ(tk-tk-1), σ=min{σ1,σ2}. Then, the cascaded system, Σ:u→xz, given in (17)–(18) is ISS with input u.

Proof of Theorem 5

Similar to the proof of Theorem 3. So that, it is omitted here.

Fig. 1

|(z1,z2)T| & γ(|u|) versis time, t

Fig. 2

γ(|z1|) & |x| versis time, t

Fig. 3

|(z1,z2,x)T| & γ(|u(t)|) versis time, t

Fig. 4

|(z1,z2,x)T| & γ(|u(t)|) versis time, t

Numerical Example. In the follower system (17), let x∈R such that x˙=-2x-3x3+(1+x2)sin2(z1) (when t≠tk) and discrete equation Δx(tk)=0.5x(tk-) (destabilizing impulses). In the leader system (18), let z=[z1,z2]T∈R2 where the system information are as follows: A=-3001, B=0.30.20-7, G=00.90.90, the external input disturbance u=sin2(t)sin2(t), C=I2×2 and F=0.2-0.2-10. In Theorem 5, take σ=0.2, α=2, θ∗=1.8, and ε=1/0.04, then we solve the Riccati equation in (19) to get the positive-definite matrix P2=0.1485-0.0468-0.04680.5646 with λmax(P2)=0.5698 and control gain matrix K=-0.00450.0014-0.03570.3962. For the leader, define the Lyapunov function V2(z)=zTP2z; then, one can show V˙2(z)≤-0.2V2(z) provided that ‖z‖≥0.1777sin2(t). For the follower, define V1(x)=12x2; then, one can show that V˙1(x)≤-2V1(x) provided that |x|≥sin2(z1). In condition (iv) of Theorem 5, the time between impulses satisfies tk-tk-1<2. The simulation results are presented in Figs. 1, 2, 3 and 4. Figure 1 illustrates the external force u and the input-to-state stabilization of the leader in the presence of destabilizing impulses; that is, the simulation verifies the outcome of Theorem 4. In Fig. 2, we show the ISS properties of the leader and follower where, as it can be seen, the output of the disturbed leader is cascaded into the follower. Figure 3 validates the conclusion of Theorem 5; namely, provided that both subsystems are ISS, the impulsive cascaded (or composite) system is ISS with input u. The fact that, when u≡0, the input-to-state stability reduces to exponential stability [2, 3] is clarified in the impulsive composite system by considering the decaying input u(t)=e-1.15t[sin2(t),sin2(t)]T, Fig. 4. In other words, the findings and approach of this paper can be adapted to prove the exponential stability for unforced systems with/without impulses.

References

1.

Lakshmiknatham, P.J., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

2.

Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34(4), 435–443 (1989)MathSciNet

3.

Khalil, H.K.: Nonlinear Systems, 3rd ed. Prentice-Hall (2002)

4.

Teel, A.R., Moreau, L., Nešic, D.: A note on the robustness of input-to-state stability. In: Proceeding of the 40th IEEE in Decision and Control, FL, vol. 1, pp. 875–880 (2001)

5.

Loria, A., Panteley, E.: Cascaded nonlinear time-varying systems: analysis and design, Advanced Topics in Control System Theory, vol. 311 of Lecture Notes in Control and Information Sciences, chapter 2, pp. 23–64. Springer, Berlin, Germany (2005)

6.

Schlanbush, R., Loria, A., Nicklasson, P.J.: Cascade-based controlled attitude synchronization and tracking of spacecraft in leader-follower formation. Int. J. Aerosp. Eng. 2011, article ID 151262, 12 pages (2011). https://​doi.​org/​10.​1155/​2011/​151262. Hindawi Publishing Corporation

7.

Panteley, E.D., Loria, A.: Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems. Automatica 37, 453–460 (2001)MathSciNet

8.

Sontag, E.D., Teel, A.: Changing supply functions in input/state stable systems. IEEE Trans. Autom. Control 40(8), 1476–1478 (1995)MathSciNet

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Woolf, P.J.: Chemical Process: Dynamics and Controls, Publisher: Openmichigan (2009). https://​open.​umn.​edu/​opentextbooks/​textbooks/​614. Accessed 21 July 2024

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Loria, A., Nijmeijer, H., Egeland, O.: Cascaded synchronization of two pendula. In: Proceeding of the American Control Conference, pp. 3751–3752, Philadelphia, Pennsylvania, June 1998

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_36

Time Filtered Finite Difference Schemes for Linear Hyperbolic Problems

K. Boatman1  , L. Davis2  , C. Drapaca3  , F. Pahlevani1   and T. Rajan1  

(1)

Division of Science & Engineering, Penn State Abington, Abington, USA

(2)

Department of Mathematical Sciences, Montana State University, Bozeman, USA

(3)

Department of Engineering Sciences and Mechanics, Penn State University, State College, USA

K. Boatman

Email: ajb7584@psu.edu

L. Davis (Corresponding author)

Email: lisa.davis@montana.edu

C. Drapaca

Email: csd12@psu.edu

F. Pahlevani

Email: fxp10@psu.edu

T. Rajan

Email: tms6516@psu.edu

Abstract

The focus of this paper is to investigate the effects of combining a time filter with three finite difference methods for numerically solving a linear hyperbolic equation. The examples in this paper show that, in some cases, the filter technique can be successfully combined with an explicit scheme. The three examples demonstrate different possible outcomes when one attempts to apply filtering. Von Neumann analysis proves to be a useful tool for examining the stability properties of filtered versions of the upwind and leapfrog methods as it allows one to deal with the filter parameter systematically. In each case, the analysis of the filtered scheme is done with insight into that of the original un-filtered method. With a careful choice of the filter parameter, the filtered upwind scheme is shown to be more accurate than its upwind counterpart, but the analysis also shows that a new CFL condition must be used in order to obtain accurate, stable results. This paper also demonstrates that the type of filter considered here cannot remedy certain kinds of stability properties. The filtered leapfrog scheme is shown to inherit the same type of spurious mode that its original un-filtered counterpart is known to possess. We also show that combining the filter with the Crank-Nicolson method leads to an implicit method where no value of the filter parameter provides for a consistent filtered version of the method. We conclude with numerical computations that support the theoretical results for the improved accuracy of the filtered upwind scheme.

1 Introduction

Time filters have been used for more than fifty years, with the first version introduced by Robert in 1966 [10], and subsequently analyzed and expanded by many scholars [1, 2, 6, 8, 11, 14]. The work presented here is motivated by a time filter proposed by Guzel and Layton in [5] which was developed for a first order system

y′=f(t,y),t>0,y(0)=y0

using a fully implicit (backward Euler) method. The filter introduced in [5] is constructed around tn+1 and takes the form

yn+1=yn+1∗-γ2(yn+1∗-2yn+yn-1),

(1)

where γ is the time filter parameter, yn+1∗ and yn+1 denote unfiltered values and filtered values for an approximation to y(tn+1). The filter is used as a post-processing step for the implicit method. The filter significantly increases accuracy from first- to second-order, with minimal computational cost, and it is A-stable, a stronger stability property.

Here we explore the application of the time filter to classic finite difference schemes for numerically solving the linear advection equation. We use a von Neumann analysis to investigate the stability and consistency properties of the filtered versions of the upwind, leapfrog and Crank-Nicolson schemes. The filtered leapfrog scheme is not guaranteed to be stable, and the filtered Crank-Nicolson method fails to be consistent. Lastly, one benchmark example numerically illustrates the improved accuracy of the filtered upwind scheme.

2 Time Filter Schemes for the Linear Advection Problem

Consider the linear advection problem

ρt+v¯ρx=0,0<x≤1,t>0

(2)

ρ(x,0)=ρ0

(3)

f(ρ(0,t))=v¯ρ(0,t)=ρI

(4)

where v¯ is a constant, ρ0 and ρI are constant initial and boundary conditions respectively. Equations (2)–(4) can be used to describe many physical phenomena, one of which is an idealized traffic flow behavior, where ρ(x,t) represents the car density in a one-way road as a function of space and time with a constant traffic velocity v¯, the traffic flux f(ρ)=v¯ρ, initial traffic density ρ0 and traffic inflow ρI.

The analysis for the classical upwind, leapfrog and Crank-Nicolson schemes applied to (2)–(4) can be found in many texts [7, 9, 12, 13]. Here we combine the time filter introduced above with each of these schemes, and we derive the new consistency and stability properties that arise for each method. The core idea of using the time filter introduced in [5] is to increase the accuracy of the scheme and to reduce the discrete curvature of the solution. For an implicit scheme, this result is accomplished by simply adding a single line of code that uses a linear combination of approximated solutions at time levels tn-1,tn and tn+1 with a scaling factor parameter γ known as the time filter parameter. The algorithm can be viewed as a two-step method where the unfiltered approximation at each time step is computed using a finite difference method as step 1, and the approximation is then filtered in step 2. The filtered approximation becomes the next iterate in the scheme.

In order to derive the filtered explicit schemes considered here, one defines the approximation of density as Pjn≈ρ(xj,tn) at the mesh grid points described by (xj,tn)=(jΔx,nΔt), for j=1,2,…M and n=1,2,…N. The discretizations of the Upwind, Leapfrog and Crank-Nicolson are viewed as the first step and then combined with a time filter as the second step shown here

Step 1:

Upwind

P~jn+1=(1-ν)Pjn+νPj-1n

(5)

Leapfrog

P~jn+1=Pjn-1-ν[Pj+1n-Pj-1n]

(6)

Crank-Nicolson

-14νP~j-1n+1+P~jn+1+14νP~j+1n+1=14νPj-1n+Pjn-14νPj+1n

(7)

Step 2:

Pjn+1=P~jn+1-γ2(P~jn+1-2Pjn+Pjn-1)

(8)

where P~jn+1 and Pjn+1 denotes the unfiltered and filtered density, respectively, for each method. The parameter ν is the well-known Courant number,

ν=v¯ΔtΔx,

(9)

and for a fixed choice of the parameters, the value of ν is constant. The parameter γ in the second step is the time filter parameter whose value is considered to be non-zero. Note that γ=0 reduces each filtered scheme to its original finite difference method, i.e. upwind, Leapfrog or Crank-Nicolson respectively. The linearity of the filter allows each explicit scheme to be simplified into a single step method.

Algorithm 2.1

Finite difference methods with the time filter assuming a constant time step with filter parameter γ can be written equivalently as one-step schemes.

Filtered Upwind: Given Pj-1n,Pjn and Pjn-1, find Pjn+1 satisfying the following

Pjn+1=[γ+(1-ν)(1-γ2)]Pjn-γ2Pjn-1+ν(1-γ2)Pj-1n

(10)

Filtered Leapfrog: Given Pj-1n,Pjn, Pj+1n and Pjn-1, find Pjn+1 satisfying the following

Pjn+1=(1-γ)Pjn-1+γPjn-ν(2-γ)2[Pj+1n-Pj-1n]

(11)

Filtered Crank-Nicolson: Given Pj-1n,Pjn,Pj+1n and Pjn-1, find Pj-1n+1,Pjn+1, and Pj+1n+1 satisfying the following

-14νPj-1n+1+Pjn+1+14νPj+1n+1=ν8[2-3γ]Pj-1n+12(2+γ)Pjn-ν8[2-3γ]Pj+1n-γ2-14νPj-1n-1+Pjn-1+14νPj+1n-1

(12)

Proof

Equations (10)–(12) are obtained by solving (8) for P~jn+1 in Step 2 that yields

P~jn+1=(22-γ)Pjn+1-γPjn+γ2Pjn-1

(13)

and replacing (13) into each of the schemes in Step 1 above.   □

3 Mathematical Preliminaries and Analysis of the Schemes

This section gives a von Neumann stability analysis for the filtered upwind and Leapfrog schemes developed above, and a consistency analysis for the filtered Crank-Nicolson scheme is presented. The stability analyses depend on the concepts of von Neumann and Schur polynomials, and we begin by briefly summarizing the mathematical preliminaries and two theorems from Chap. 4 of [12]. For a standard stability analysis, one examines the roots of the method’s amplification polynomial.

Definition 1

The polynomial ϕ(z) is a Schur polynomial if all roots, rk, satisfy |rk|<1. It is a von Neumann polynomial if all roots, rk, satisfy |rk|≤1. The polynomial ϕ is a simple von Neumann polynomial if it is a von Neumann polynomial and all of its roots on the unit circle are simple roots.

Given the polynomial ϕ2(z)=a0+a1z+a2z2, we define a new polynomial ϕ2⋆(z) by

ϕ2⋆(z)=a2¯+a1¯z+a0¯z2,

(14)

where a¯ denotes the complex conjugate of a. Using ϕ2 and ϕ2⋆, we construct a polynomial of degree 1, denoted ϕ1(z) by the expression below.

ϕ1(z)=ϕ2⋆(0)ϕ2(z)-ϕ2(0)ϕ2⋆(z)z

(15)

One can use ϕ2,ϕ2⋆ and ϕ1 to determine if a polynomial is von Neumann. The result that allows one to do this is in [12], and we state a simplified version in Theorem 1 for clarity. Once the amplification polynomial is shown to be von Neumann, the stability result in Theorem 2 can be applied.

Theorem 1

The amplification polynomial ϕ2 is a simple von Neumann polynomial if and only if either (a) or (b) holds.

(a.)

|ϕ2(0)|<|ϕ2∗(0)| and ϕ1 is a simple von Neumann polynomial.

(b.)

ϕ1 is identically zero and (ϕ2)′ is a Schur polynomial.

Theorem 2

If the amplification polynomial Φ(r,θ) is explicitly independent of Δt and Δx, then the necessary and sufficient condition for the numerical scheme to be stable is that all roots, r1,r2,…rm, of the polynomial satisfy the conditions:

(a.)

|rj|≤1, for all j=1,2,…m and

(b.)

If |rj|=1, then rj must be a simple root of Φ.

3.1 Stability Analysis for the Time Filtered Upwind Scheme

This section contains the von Neumann stability analysis for the time filtered upwind scheme in (10). We derive a CFL condition for stability which depends explicitly on the filter parameter.

Theorem 3

Assume that the filter parameter satisfies 0≤γ<2. If the CFL condition given by the following inequality holds

0<ν≤2-γ2+γ=CFL(γ),

(16)

then the filtered upwind scheme in (10) is stable.

Proof

We rewrite Eq. (10) as

2Pjn+1-(γν+γ-2ν+2)Pjn-ν(2-γ)Pj-1n+γPjn-1=0.

Assuming that Pjn=zneijθ, the amplification polynomial for this scheme is given by

ϕ2(z)=2z2-(γν+γ-2ν+2)z-e-iθν(2-γ)z+γ=2z2-(ν(γ-2)+γ+2+(2-γ)νcosθ+ν(γ-2)isinθ)z+γ

Define

ϕ2⋆(z)=2-[ν(γ-2)+γ+2+(2-γ)νcosθ-ν(γ-2)isinθ]z+γz2,

and observe that ϕ2(0)=γ and ϕ2⋆(0)=2; therefore, |ϕ2(0)|<|ϕ2⋆(0)| if and only if |γ|<2. Using the definition in (15), one can substitute, collect like terms and simplify to obtain the linear polynomial below.

ϕ1(z)=1z[2(γ-[ν(γ-2)+γ+2+(2-γ)νcosθ+ν(γ-2)isinθ]z+2z2)-γ(2-[ν(γ-2)+γ+2+(2-γ)νcosθ-ν(γ-2)isinθ]z+γz2)]=(2-γ)[ν(2-γ)-(γ+2)-(2-γ)νcosθ+(2+γ)νisinθ+(2+γ)z]

Since γ≠2, the root of ϕ1(z) can be shown to have the form

z=1+νγ-2γ+2(1-cosθ)-νisinθ

In order to verify that ϕ1 is a von Neumann polynomial as in Definition 1, we note that the square of the modulus of z is given by

|z|2=(1+νγ-2γ+2(1-cosθ))2+(-νsinθ)2=(1+2νγ-2γ+2(1-cosθ)+ν2(2-γ)2(2+γ)2(1-cosθ)2)+ν2(1-cos2θ)

The polynomial ϕ1 is a simple von Neumann polynomial if and only if |z|2≤1 for all θ∈[0,2π), and assuming the restriction 0≤γ<2 holds, then with some algebraic manipulations, this is equivalent to

2ν(1-cosθ)ν(γ2+4γcosθ+4)-(2-γ)(2+γ)≤0

Since ν>0 this implies

ν≤(2-γ)(2+γ)γ2+4γcosθ+4,

(17)

and it can be shown that

2-γ2+γ≤(2-γ)(2+γ)γ2+4γcosθ+4

for all θ∈[0,2π). If ν is chosen to satisfy (16), then ν necessarily satisfies (17) for all θ∈[0,2π). It follows that the single root of ϕ1 lies on or within the unit circle. Hence ϕ1 is a simple von Neumann polynomial under the given hypotheses, and Theorem 1 part (a) implies that ϕ2 is a simple von Neumann polynomial. Then ϕ2 satisfies the conditions of Theorem 2, and the filtered upwind scheme is stable.    □

A computational example demonstrating the accuracy and stability of the method on a benchmark example problem is given in Sect. 4. A more thorough numerical illustration of the scheme is planned for a subsequent work.

3.2 Stability Analysis for the Time Filtered Leapfrog Scheme

Using von Neumann analysis, the filtered leapfrog scheme is shown to possess a spurious or parasitic mode. This is consistent with the unfiltered leapfrog scheme, which is known to posess a spurious mode, see [9, 12] for example.

Theorem 4

The time filtered Leapfrog scheme given in (11) is unstable for any γ≠0.

Proof

By substituting Pjn=zneijθ in Eq. (11) and simple algebraic simplifications, the amplification polynomial ϕ2(z) and its conjugate ϕ2⋆(z) for this scheme are written as

ϕ2(z)=z2-[γ-ν(2-γ)isinθ]z-(1-γ)ϕ2⋆(z)=1-[γ+ν(2-γ)isinθ]z-(1-γ)z2

Therefore |ϕ2(0)|<|ϕ2⋆(0)| requires |1-γ|<1 enforcing 0<γ<2. According to the definition of the linear polynomial ϕ1 in (15), by substituting and simplifying, one obtains

ϕ1(z)=γ(2-γ)[z-1+νisinθ]

(18)

Setting (18) equal to zero and calculating the single root assuming 0≠γ≠2 yields z=1-νisinθ. By Theorem 2, stability requires |z|=|1-νisinθ|≤1, that is, νsinθ=0. This is only true if ν=0 or θ=0. In the case of the latter condition, the roots of ϕ1 are real for 0<γ<2.

Since ϕ1 cannot be a simple von Neumann polynomial for all θ, by Theorem 1, we inspect part (b). As a function of γ, ϕ1=0 if and only if γ=0 or γ=2. Furthermore, proving (ϕ2)′ is a Schur polynomial requires the norm of single roots for the linear equation (ϕ2)′=0 to be smaller than one. The calculated root is given by z=γ2-ν(2-γ)2isinθ. Requiring |z|=|γ2-ν(2-γ)2isinθ|<1, one obtains

γ2+ν2(2-γ)2sin2θ<4,orν2sin2θ<(2-γ)(2+γ)(2-γ)2.

Assuming γ≠2 yields (νsinθ)2<2+γ2-γ. The only γ that satisfies Theorem 1 conditions in part (b.) is γ=0 making this scheme functionally without a time filter for achieving stability.

3.3 Consistency for Time Filtered Crank-Nicolson Scheme

The analysis given here shows that there is no parameter choice for the type of filter in (8) that will yield a consistent filtered Crank-Nicolson algorithm for the advection equation. Expressing the PDE given in (2) using the operator form of the equation Lρ=0, where the linear operator L is defined as L(ϕ)=ϕt+ϕx, we follow Chap. 3 of [12] and compare the symbols of the difference operator to the symbol of the differential operator. The comparison allows one to calculate the order of the approximation using the symbols. If one assumes an argument of the form ϕ(s,ξ)=esteiξx, the symbol of the differential operator L is defined by the function ℓ(s,ξ) that satisfies L(ϕ)=ℓ(s,ξ)ϕ. Here the symbol of the differential operator is given by ℓ(s,ξ)=s+iξ. From Eq. (12) and some rearrangement of the time derivative, the expression for the filtered Crank-Nicolson algorithm is written in a form that exhibits the discretization of the original advection equation. To simplify the notation, we let k=Δt and h=Δx, and the expression for this is written as

12k2Pjn+1-(2+γ)Pjn+γPjn-1+14h(Pj+1n+1-Pj-1n+1)+12(2-3γ)(Pj+1n-Pj-1n)+γ2(Pj+1n-1-Pj-1n-1)=0,

(19)

and the definition for the difference operator is then given by

Lk,h(ϕjn)=12k[2ϕjn+1-(2+γ)ϕjn+γϕjn-1]+14h[12(2-3γ)(ϕj+1n-ϕj-1n)+γ2(ϕj+1n-1-ϕj-1n-1)+(ϕj+1n+1-ϕj-1n+1)].

The symbol of this discrete operator is defined as the function σk,h(s,ξ) satisfying Lk,h(ϕ)=σk,h(s,ξ)ϕ. If one defines ϕ(s,ξ)=eskneijhξ, then the filtered Crank-Nicolson scheme leads to a symbol satisfying

Lk,h(ϕ)=[1k(esk-1+γ2(e-sk-1))+14hisinhξ((2-3γ)+γe-sk+2esk)]ϕ

so that σk,h is determined to be

σk,h(s,ξ)=1k[esk-1+γ2(e-sk-1)]+14isinhξh[(2-3γ)+γe-sk+2esk].

Then the expression for the local truncation error for the filtered Crank-Nicolson is expressed by σk,h(s,ξ). Using a simple form of the theorem from page 66 of [12], a scheme Lk,hϕ=0 that is consistent with Lϕ=0 is accurate of order (p, q) if and only if for each value of s and ξ,

σk,h(s,ξ)=O(kp)+O(hq).

(20)

Following [12], one attempts to reduce the algebra by multiplying by e-αsk with α a constant, a term that is O(1) and doesn’t affect the order of the truncation error. Consistency for the unfiltered Crank-Nicolson algorithm is analyzed using this approach in Chap. 3 of [12]. There the analysis uses α=12, and one can regard this choice as being algebraically equivalent to analyzing consistency using the simple Taylor expansions at the half-step tn+1/2, see [9] for example. For the filtered Crank-Nicolson case considered here, we introduce the parameter α in order to determine the effect of the filter parameter γ on the truncation error calculation. That is, we multiply the expression in (20) by e-αsk and analyze the expression e-αskσk,h(s,ξ) to answer the question: Is there a pair α,γ where filtered Crank-Nicolson is consistent? With some tedious Taylor expansions, one can show that

e-αskσk,h(s,ξ)=1k[e(1-α)sk-e-αsk+γ2(e-(1+α)sk-e-αsk)]+isin(hξ)4h[(2-3γ)e-αsk+γe-(α+1)sk+2e(1-α)sk]=(1-γ2)(s+iξ)+ks2+14iξs(1-2α)(2-γ)+O(k2+h2)

Hence, we must choose γ=2 to obtain consistency for filtered Crank-Nicolson, but this choice of γ results in no filtering, see (8). Hence, no choice of γ in (8) yields a consistent filtered Crank-Nicolson algorithm for (2)–(4).

4 Convergence Results for a Benchmark Problem

Finally we present numerical illustrations of convergence and error calculations for the filtered upwind scheme with different values of filter parameter 0≤γ<2. All computations are implemented using Python. Results are obtained using a benchmark example problem referred to as the Sigmoid problem. For this example, we consider (2)–(3) with v¯=1 and ρ0=0. The inflow condition is given by ρI(0,t)=g(t)=0.941+e-20t-0.47,t>0, and the analytical solution is ρ(x,t)=g(x-t). With the exception of one corner, the solution is a smooth function with a steep derivative. We examine the computational error for density approximations using upwind and filtered upwind over the time interval [0, 1] for the Sigmoid example. Denote the error at the grid point (xj,tn) as Ejn=ρjn-Pjn, and our choice for the error norm is the L1-norm defined as ||En||=Δx∑j=0M|Ejn| for n=0,1,2,…,N. In the error calculations for (5) and (10), we assume a fixed ratio ν=ΔtΔx=0.8CFL(γ). The factor 0.8 is to guarantee a numerical solution free of oscillations [4]. Letting Δt→0 by refining the spatial grid, Δx, the error calculations at two fixed times, t=0.2 and t=1.0, are given in Table 1. The errors for both schemes become smaller converging to zero as Δx→0. Comparing the errors in Table 1 for Pjn obtained from upwind with those from filtered upwind with γ=1.75, the latter scheme shows a better accuracy at all computed times.

Calculations estimating the rate of convergence at t=1 for both schemes are in Table 2. For a grid size hi=2-(i+5) for i=1,2,3,…,5, the convergence rate of the error is determined by calculating the exponent a in the following expression

hihi+1a=||E(hi)||||E(hi+1)||.

(21)

The convergence rate values in Table 2 indicate that the L1-norm of the errors decay at the rate of about one as the grid size decreases in both cases. This means that error decreases like Δx validating that both methods are first order with the filtered scheme converging faster. Figure 1a, b present cross sectional plots of the solution for the Sigmoid example at two different times t=0.5 and t=1.0. As the solution propagates along the domain over time, the approximate solutions become less accurate. The filtered scheme provides better estimation around the corner and less dissipation overall. Both plots show increased accuracy and rate of convergence for the filtered upwind method as γ values increase. To balance the reduction in time step sizes, an optimal γ value for best results is 1.75 (Fig. 1).

Table 1

The density approximation error for upwind (γ=0) and filtered upwind with γ=1.75 using L1-norm and Δt=0.8CFL(γ)Δx

L1 Error at t=0.2

L1 error at t=1.0

Δx

γ=0

γ=1.75

γ=0

γ=1.75

2-7

3.857127 e-3

1.102405 e-3

1.1093878 e-2

3.806919 e-3

2-9

1.147664 e-3

2.74954 e-4

4.054965 e-3

1.117677 e-3

2-11

3.12214 e-4

6.89288 e-5

1.271363 e-3

3.03843 e-4

Table 2

The estimated convergence rate for upwind (γ=0) and filtered upwind with γ=1.75 at t=1 according to errors in Table 1 using values of i to estimate the value of a in Eq. (21)

Rate

i=1

2

3

4

5

γ=0

0.654319561

0.697141935

0.754860398

0.81096301

0.862350868

γ=1.75

0.846236316

0.868116094

0.900003931

0.928213372

0.950894343

Fig. 1

The graph in a contains the density plots for upwind (γ=0) and filtered upwind (γ>0) with 0<x<1 at t=0.5, and the graph in b contains the same information at t=1. All graphs use the choice of Δx=1128 and Δt=0.8CFL(γ)Δx

The numerical example illustrates the stability and improved accuracy of the filtered upwind scheme, which aligns with the analysis in Sect. 2 and the truncation error result in [3]. This paper demonstrates that a time filter can sometimes be successfully combined with a finite difference method, but stability and consistency properties of the filtered method depend explicitly on the analysis of the original method along with that of the choice of filter parameter.

References

1.

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Asselin, R.: Frequency filter for time integrations. Month. Weather Rev. 100(6), 487–490 (1972)

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Boatman, K., Davis, L., Pahlevani, F., Susai Rajan, T.: Numerical analysis of a time filtered scheme for a linear hyperbolic equation inspired by DNA transcription modeling. J. Comput. Appl. Math. 429, 115135 (2023)

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Davis, L., Pahlevani, F., Rajan, T.S.: An accurate and stable filtered explicit scheme for biopolymerization processes in the presence of perturbations. Appl. Comput. Math. 10(6), 121–137 (2021)

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Guzel, A., Layton, W.: Time filters increase accuracy of the fully implicit method. BIT Numer. Math. 58(2), 301–315 (2018)MathSciNet

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Guzel, A., Trenchea, C.: The Williams step increases the stability and accuracy of the hoRA time filter. Appl. Numer. Math. 131, 158–173 (2018)MathSciNet

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LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press (2002)

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Li, Y., Trenchea, C.: A higher-order Robert-Asselin type time filter. J. Comput. Phys. 259, 23–32 (2014)MathSciNet

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Robert, A.J.: The integration of a low order spectral form of the primitive meteorological equations. J. Meteorol. Soc. Jpn. Ser. II 44(5):237–245 (1966)

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Schlesinger, R.E., Uccellini, W., Louis, Johnson, D.R.: The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations. Mon. Weather. Rev. 111(3), 455–467 (1983)

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_37

Computational Considerations for Implementing the Collage Method for ODE Inverse Problems

Noah Boettger1   and Kimberly M. Levere1  

(1)

Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada

Noah Boettger

Email: nboettge@uoguelph.ca

Kimberly M. Levere (Corresponding author)

Email: klevere@uoguelph.ca

Abstract

The Collage Coding Method for solving inverse problems has been well established theoretically. However, in practice, a number of computational considerations arise during implementation that depend on the complexity of the problem at hand. In this paper, we review the theoretical underpinnings of the Collage Coding Method for solving inverse problems for first-order initial value problems. We then present a comprehensive investigation of computational challenges and choices that may be faced during numerical implementation. Examples showcasing the results of this investigation are presented and discussed.

Keywords

Inverse problemsParameter estimationCollage codingNumerical analysisDynamical systems

1 Introduction

The process of mathematical modelling often necessitates the use of parameter estimation techniques to inform aspects of models that are difficult to measure. The goal of many inverse problems is to minimize the approximation error; the distance between a given solution and the solution obtained using recovered parameters. A variety of approaches have emerged to tackle such problems: regularization techniques such as Tikhonov regularization [10], iterative schemes, and neural networks (see, for instance, [11]). While such methods have been shown to accurately solve a wide variety of parameter estimation problems, they are often computationally expensive to implement. In the realm of parameter estimation for ordinary differential equations (ODEs), the more recently proposed Collage Coding Method for parameter estimation has recognized a reduction in computational expense as well as gains in accuracy [8]. This approach to bounds the approximation error above by a new distance (the collage distance) that can be easier to minimize. Utilizing the Collage Theorem from fractal image compression [1], the Collage Coding Method has been extended to a wide variety of problems, including delay, fractional DEs, random DEs, variational problems, and partial differential equations [3, 5–7].

When it comes to implementation, there are a few avenues in which numerical challenges present themselves. This paper investigates methods for tackling these challenges and the impact that different choices have on the results. It also brings to light some complexities that can cause this method to struggle, much like other methods used in parameter estimation. The layout of this paper is as follows. In Sect. 2, we compile some mathematical preliminaries that will be used throughout this work. In Sect. 3, we review the theoretical construction of the Collage Coding Method as it applies to first-order IVPs. In Sect. 4, we will discuss a variety of computational considerations faced when implementing the Collage Coding Method. Finally, in Sect. 5 we present some concluding remarks and suggestions for researchers looking to implement this method.

2 Mathematical Preliminaries

In this Section we state some of the important theorems and definitions that will be utilized throughout this work.

Definition 1

(Contractivity) An operator T:X→Y is contractive if for any x1,x2∈X, ‖Tx1-Tx2‖Y≤c‖x1-x2‖X for some c∈[0,1).

Theorem 1

(Banach’s Fixed Point Theorem) Let (X,‖·‖X) be a complete metric space, and let T:X→X be a contractive operator with contraction factor c∈[0,1). Then there exists a unique fixed point x¯∈X such that Tx¯=x¯. Furthermore, for any x∈X, ‖T∘nx-x¯‖X→0 as n→∞.

Definition 2

(Lipschitz Continuity) Given two metric spaces (X,‖·‖X) and (Y,‖·‖Y), f:X→Y is Lipschitz continuous if there exists a real constant K≥0 such that for all x1,x2∈X, ‖f(x1)-f(x2)‖Y≤K‖x1-x2‖X.

Any such K is often referred to as a Lipschitz constant of f, the smallest possible constant is sometimes called the best Lipschitz constant of f.

3 The Collage Coding Method to Inverse Problems for ODEs

While the Collage Coding Method has been applied to a wide variety of problems, in this paper, we will focus on first-order IVPs of the form

dxdt=f(x,t)

(1)

x(t0)=x0

(2)

where x∈R, t∈R, and f:R×R→R is continuous. Notice that definite integration from t0 to t of both sides of the ODE in (1) yields

x(t)=x0+∫t0tf(x(s),s)ds.

(3)

This realization gives rise to the Picard operator, T associated with the IVP in (1) defined as

(Tx)(t)=x0+∫t0tf(x(s),s)ds

(4)

Defining I=[t0,τ], for τ>t0 and working on the complete space (C(I),‖·‖∞) one can place appropriate conditions on f and τ to establish that the Picard operator (4) is space preserving and contractive [3]. Thus, by Banach’s Fixed Point Theorem, T has a unique fixed point, x¯; that is, the IVP in (1) has a unique (forward) solution.

Instead of solving the forward problem (which seeks x(t) given f, x0 and t0), we now consider a related inverse problem: Given observational data representing the solution to (1) (interpolated via the function xtarget(t)), determine the form of the vector field f (which depends on some unknown parameters, λi) such that the resulting unique solution x¯λ agrees with the given data. Ultimately, this is really just an alternative statement of the approximation error, but in terms of the fixed point: minλi‖x¯λ-xtarget‖. Unfortunately, determining the fixed point, x¯λ in terms of the parameters of the problem λi can be a difficult or impossible task, so this restatement doesn’t seem to offer us much assistance in solving the inverse problem.

The Collage Coding Method first introduced by Kunze and Vrscay, [3] circumvents this challenge, by instead utilizing the Collage Theorem to bound the approximation error above by the so-called collage distance.

Theorem 2

(Collage Theorem) Let (X,‖·‖X) be a Banach space and T:X→X be a contractive operator with contraction factor c∈[0,1) and unique fixed point x¯∈X. Then ‖x-x¯‖X≤11-c‖x-Tx‖X.

Relating back to an ODE inverse problem, making use of the Picard operator, we notice that all of the parameters of the problem are present within the operator T, and thus one can control the approximation error by instead minimizing the collage distance. The choice of an operator T that satisfies the hypotheses of the Collage Theorem is not unique. The goal here is not necessarily to achieve the tightest upperbound on the approximation error, but rather to ensure that the upperbound can be made sufficiently small so that the approximation error is controlled within a desired tolerance. Indeed, the Collage Coding Method was shown to be sub-optimal in [4].

Having established the Collage Coding Method theoretically, we now turn our attention to numerical implementation of this method and the challenges that can arise.

4 Computational Considerations

In this Section we will investigate some computational considerations that arise when using the Collage Coding Method to solve an inverse problem for a first-order IVP of the form (1). While the space (C(I),‖·‖∞) was used in the theoretical development in Sect. 3, here we instead use the L2 norm (despite (C(I),‖·‖2) lacking completeness). This is not cause for concern as C(I)⊂L2(I).

4.1 Approximating the Target Function

Practically speaking, it is common for the solution to be given as a set of N data points and not as a function, x(t). Recall that the Picard operator T (see Eq. (4)) and the collage distance ‖x-Tx‖2 require integration involving the given solution, and thus, the given data must be interpolated (or the integration must be approximated discretely). In what follows, we will call this interpolated function xtarget(t). Here we investigate the impact of different interpolation choices on the resulting optimal parameters and collage distance.

First, we used a basic least squares polynomial approximation in which one assumes that the data can be fit with a polynomial of degree p, x(t)=∑i=0paiti, where ai, i=1,...,p are a set of unknown coefficients. The ai values are chosen by minimizing the sum of the squared error between this polynomial and the given data. The value of p isn’t known a priori, but certainly this can be chosen to obtain an error that is below a desired tolerance.

Secondly, we used Lagrange polynomials of different degrees to interpolate our data. A Lagrange polynomial L(x) of degree p passing through a given set of p+1 distinct data points (ti,x(ti)) for i=1,...,p+1 is the linear combination of the associated p+1 Lagrange basis functions, ℓj(t); L(x)=∑j=1p+1x(tj)ℓj(t). Each basis function ℓj(t), for j=1,...,p+1, is of degree p and has a value of 1 at tj and a value of 0 at all tk where k≠j, that is,

ℓj(t)=∏0≤k≤pk≠jt-tktj-tk.

This definition ensures that L(x) will pass through each of the p+1 data points used to construct it. Using this idea, we can build a piecewise continuous target function whose branches are the Lagrange polynomials of degree p. In this work, we considered a linear piecewise Lagrange polynomial fit, as well as a quadratic piecewise polynomial fit of our observational data. We compare these interpolation options in the following example.

Example 1

Suppose that we are given observational data for the solution x(t) of an IVP of the form x′(t)=f(x,t)=λ1-λ2x-λ3x2, x(0)=λ4 and are asked to recover the unknowns λi, for i=1,...,4. While in practice, the true value of f is not known, in this example we simulate N data points by evaluating the solution to x′(t)=9.5-8x-x2, x(0)=0 at equally spaced values of t∈[0,1]. Otherwise only use this information to check the quality of our results. Each of the aforementioned interpolation methods were used to interpolate the N simulated data points. The results of the Collage Coding process are presented in Table 1 

Table 1

Comparison of different interpolation methods for the target solution for Example 1; with exact form x′(t)=9.5-8x-x2, x(0)=0

Interpolation

N

Recovered f

Recovered IC

‖x-Tx‖2

Least squares

10

9.4259-7.7961x-1.1270x2

-2.1216×10-3

1.5040×10-3

20

9.4797-7.9252x-1.0532x2

-9.6910×10-4

1.4227×10-3

50

9.4413-7.8364x-1.1033x2

-6.3911×10-4

1.3043×10-3

Linear lagrange

10

8.5156-5.6498x-2.3520x2

-3.364×10-2

6.9617×10-3

20

9.3874-7.7461x-1.1402x2

-1.3541×10-2

1.8636×10-3

50

9.4916-7.9830x-1.0085x2

-2.4684×10-3

2.9688×10-4

Quadratic lagrange

10

9.2120-7.3441x-1.3650x2

-1.1862×10-2

2.1684×10-3

20

9.4795-7.9598x-1.0197x2

-2.1948×10-3

3.1858×10-4

50

9.4987-7.9976x-1.0011x2

-1.4644×10-4

2.1935×10-5

As expected, an increase in the number of data values, N produces stronger accuracy, regardless of interpolation method. This example suggests that with sparse data, least squares may produce very marginal gains over Lagrange interpolation. If data is plentiful, however, a quadratic Lagrange target function is significantly more accurate, although the results for all options are strong for large N. The shape of the target solution data may also play a role in which interpolation method is most accurate. These results may differ depending on the example.

The least squares results presented fit a polynomial of degree p=6. The results for this interpolation choice are quite sensitive to the choice of degree. For instance, for N=10, if instead we reduced the degree of the target function to 4, the resulting collage distance is 2.1921×10-2 which is indeed worse than either of the Lagrange results. Care must be taken to ensure that unnecessary error is not being introduced by this interpolation.

To simulate experimental error in data collection, we consider adding Gaussian noise to the data values prior to fitting the target function. For comparison, we use N=10 data points and set the relative noise level at 1%. With this set up, the collage distances for each interpolation method are as follows: least squares 1.5720×10-3, linear Lagrange 6.9593×10-3, and quadratic Lagrange 2.3366×10-3. Further investigations with higher relative noise levels performed similarly. This example shows reasonable tolerance to noisy data regardless of interpolation method. Least squares does perform marginally better than Lagrange interpolation, perhaps since Lagrange polynomials are forced to fit exactly at each data value and thus the noise has a heavier impact on the resulting target function. The impact of noise is even less pronounced as N increases; for instance, increasing to N=40 data values using linear Lagrange with a relative noise level 1% produces an improved collage distance of 7.7962×10-4.

4.2 Recovery of a Non-polynomial f

In Example 1, the unknowns to be recovered appeared as the coefficients within a polynomial. All integrals present in the collage distance could be easily computed and the result was a convex objective function that was quite easily minimized by solving a system of linear equations. Certainly if these unknowns appear in more complicated ways, this need not be the case. One solution to integration challenges is to use a Taylor series expansion of complicated integrands. Unlike fitting a target polynomial to observed data, the intention here is to simplify the complexity of the function, keeping its domain in mind. Optimizing a complicated (non-convex) objective function over the parameters of the problem is a well-studied problem. A variety of options have been proposed for searching parameter space for optimal values including Gradient Descent, the Nelder-Mead Algorithm, and Particle Swarm and/or Ant Colony Optimization (we will use the combination of these two methods in this work, PSACO, as discussed in [9]). These methods often suffer from locating local rather than global optima and are often sensitive to calibration and/or seed values. They can also be computationally expensive as it can take many iterations to settle to an optimum value of the objective. As a result, the quality of the results in this case are often due to the optimization method. We showcase these challenges in the following example.

Example 2

Suppose that we are given observational data for the solution x(t) of an IVP of the form x′(t)=λ1+λ2xλ3, x(0)=1, t∈[0,1], and are asked to recover the unknowns λi, for i=1,...,3. As we did in Example 1, we simulate data using the IVP x′(t)=0.2+0.3x0.4, x(0)=1, but otherwise only use this information to check the quality of our results. A 4th degree Taylor polynomial about t=0.5 of the integrand in the resulting Picard operator is utilized to aid in the integration process. A least squares polynomial fit of degree 6 is used to interpolate the observational data. The resulting collage distance is optimized using a combination of particle swarm and ant colony optimization, PSACO. The results of the Collage Coding process are presented in Table 2 for a variety of calibrations.

Table 2

Comparison of different calibrations of the PSACO algorithm for Example 2; with exact form x′(t)=0.2+0.3x0.4, x(0)=1

Search space

# ants

# iterations

N

Recovered f

‖x-Tx‖2

[0,1]3

100

100

10

0.16388+0.33605x0.3600

3.3946×10-6

20

0.19190+0.30808x0.3903

9.3175×10-7

50

0.30569+0.19465x0.5925

1.9664×10-5

[0,1]3

50

50

10

0.066786+0.44000x0.1853

1.1258×10-3

20

0.33378+0.16631x0.6935

6.8280×10-5

50

0.45096+0.061282x0.8075

1.7253×10-3

[0.1,0.5]3

50

50

10

0.23211+0.26796x0.4440

4.9211×10-6

20

0.23693+0.26314x0.4515

5.4013×10-6

50

0.21129+0.28874x0.4144

1.8724×10-6

[0.1,0.5]3

50

50

10

0.25057+0.24915x0.4777

3.5158×10-5

Looking at Table 2, while we might expect that decreasing iterations and/or ants would produce worse results, this seems to be sensitive to the restricted search space (which isn’t information that we often have a priori). As mentioned in Sect. 4.1, the degree of the target function must be chosen carefully so as not to propagate unnecessary error. The degree of the Taylor polynomial used to simplify integration is also an important factor in this process, for obvious reasons, as well as the value that the polynomial is centred around. For comparison we use the same setup as in the first row of Table 2, (a search space of [0,1]3, 100 ants, 100 iterations, N=10 data points) but instead of a Taylor polynomial of degree 4 we reduce to a degree of 2. With this reduced degree, we arrive at a recovered f of f(x)=0.17157+0.32780x0.3721 with a collage distance of 3.5145×10-5. This is still a reasonably accurate result, but we can see that the reduction in accuracy indeed negatively impacts our collage distance. In subsequent runs of this example, it is apparent that even small changes to the setup can produce dramatically different results which are not due to the quality of the Collage Coding Method, but rather the optimization scheme. Such search algorithms therefore must be used with care and widely tested.

4.3 Partitioning the Collage Method for ODEs

One of the requirements of the Collage Coding Method is that the Picard operator (4) is contractive with respect to the norm in play over the interval I. For the problem we are considering, contractivity is established through particular conditions on f, some of which depend on the length of interval I (see [3] for details). In the event that the operator T lacks contractivity (or c is near 1), one can often achieve contractivity (or reduce the value of c) by instead partitioning the interval I into a collection of disjoint subintervals Ik=[tk-1,tk], for k=1,...,m and solving the inverse problem over each partition. As a result, we now consider a set of m inverse problems each of the form x′(t)=fk(x(t),t), t∈Ik,k=1,...,m,x(t0)=x0, and each having their own Picard operator Tk and their own set of unknown parameter values. The Collage Coding process can be applied to each of these problems separately, producing a piecewise recovery of f that is defined collectively over all of I. Here we have to ensure that the hypotheses of the Collage Theorem are satisfied not only on each subinterval, but collectively over I to maintain the dependence of these problems. If we define T=(T1,T2,...,Tm), then to ensure that T:C(I)→C(I), we require that Tk[x(tk)]=Tk+1[x(tk)], k=1,...,m-1. For the Picard operator we are considering, this amounts to requiring that x(tk)=x(tk-1)+∫tk-1tkfk(s,x(s))ds, for k=1,...,m. For differentiability of x(t) at partition points, one can also impose that fk(tk,x(tk))=fk+1(tk,x(tk)) for k=1,...,m-1 (although this condition can often be relaxed to require just piecewise differentiability). This condition can be more challenging to impose, but can be done in some cases by solving this requirement for one of the unknowns, thus reducing the dimension of parameter space by 1 for each partition. Depending on the form of the fk functions though, this may not be feasible.

Example 3

Suppose that we are given observational data for the solution x(t) of an IVP of the form x′(t)=1λ1x+λ2, x(0)=0 and are asked to recover the unknowns λ1,λ2. As we did in Example 1, we simulate data using the IVP x′(t)=10.3x+0.2, x(0)=0, but otherwise only use this information to check the quality of our results. A 4th degree Taylor polynomial of the integrand in the resulting Picard operator is utilized to aid in the integration process. A least squares polynomial fit of degree 6 is used to interpolate the N=20 observational data points. The resulting collage distance is optimized using a combination of particle swarm and ant colony optimization, PSACO using 100 ants and 100 iterations with a search space of [0,1]2. This example seeks piecewise differentiability at partition points. The results of the Collage Coding process are presented in Table 3 with visual results in Fig. 1.

Table 3

Comparison of partitions of the interval I=[0,1] for Example 3; with exact form x′(t)=10.3x+0.2, x(0)=0

# Partitions

‖x-Tx‖2

1

2.4497×10-2

2

4.6607×10-3

3

8.6315×10-4

Fig. 1

Plots of the recovered f (blue) with various partitions of I=[0,1], together with the actual function f (red) used to build data

This example showcases how partitioning over shorter length intervals can produce positive improvements on the collage distance and resulting parameter values. Similar considerations from previous examples apply here: the degree of the target function and Taylor polynomial used to simplify integration should be chosen with care. The calibration of the PSACO algorithm is once again important. Appropriate restriction of the search space can produce stronger results even with fewer ants and iterations (for instance, restricting the search space from [0,1]2 as in Table 3 to [0.1,0.4]2 produces collage distances of 1.4497×10-2, 4.4572×10-3 and 2.1003×10-5 for partitions of 1, 2,  and 3, respectively). As partitioning requires us to perform the Collage Coding process (perhaps using a parameter search method depending on the problem) multiple times, it is no surprise that partitioning required much more computation time and power. This loss in computational speed means that the Collage Coding Method possesses less of a lead over other inverse problem methods when partitioning is necessary. This is also true to a degree when it comes to the use of parameter search algorithms like PSACO.

5 Concluding Remarks

In this paper we have reviewed the theoretical development of the Collage Coding Method as it applies to a first-order IVP. We have discussed a number of computational considerations for implementing this method numerically. While the few examples presented in this paper do not suggest results that will apply globally to any problem, they do serve as a reminder that there are several “moving parts” that should be considered when using this method. We investigated the use of different interpolation methods and degrees for fitting the given observational data. We concluded that the choice of interpolation method should consider the shape of the target solution data, as well the sparsity of the data. The noisiness of the data may also play a role in the interpolation choice. If a least squares interpolation is chosen, the degree of the polynomial should be carefully chosen to avoid unnecessary error. We discussed a few options for parameter search algorithms in the case that optimization of the collage distance is not convex, or is otherwise complicated enough that more simplistic optimization approaches are not appropriate. The overarching result here was that results tend to be very sensitive to the calibration of these schemes so care must be taken to thoroughly and systematically use these methods to find appropriate parameters. Finally, we investigated the use of partitions of the interval I in an effort to achieve contractivity or reduce the value of the contraction factor. While these results were strong and improved as the number of partitions increased, there was an obvious trade off with computational time and power.

As one last remark regarding the utility of the Collage Coding Method, we recognize that this method is data-driven. Obtaining data can be a challenge, particularly complete data that has been collected consistently and accurately. One can also envision a situation in which a model involves a system of differential equations such as a compartmental model perhaps used in infectious disease modelling. In this case, it may be feasible to collect data regarding one compartment of the system (perhaps number of infections) but it is unlikely that all other compartments will have associated data. In this case, data could be fabricated based on knowledge of the problem, but this would likely require assumptions and could also propagate additional error.

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Kunze, H.E., Hicken, J.E., Vrscay, E.R.: Inverse problems for ODEs using contraction maps and suboptimality of the ‘Collage Method’. Inverse Probl. 20(3) (2004)

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Kunze, H.E., La Torre, D., Levere, K.M., Vrscay, E.R.: Solving inverse problems for deterministic and random delay integral equations using the collage method. Int. J. Math. Stat. (IJMS) 11(1) (2012)

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Kunze, H., La Torre, D., Vrscay, E.R.: A generalized collage method based upon the Lax-Milgram functional for solving boundary value inverse problems. Nonlinear Anal.: Theory Methods Appl. 71(12), 1337–1343 (2009)MathSciNet

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_38

Penalty Ensembles for Navier-Stokes with Random Initial Conditions and Forcing

Rui Fang1  

(1)

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

Rui Fang

Email: ruf10@pitt.edu

Abstract

In many applications, uncertainty in problem data leads to the need for numerous computationally expensive simulations. This report addresses this challenge by developing a penalty-based ensemble algorithm. Building upon Jiang and Layton’s work on ensemble algorithms that use a shared coefficient matrix, this report introduces the combination of penalty methods to enhance its capabilities. Penalty methods uncouple velocity and pressure by relaxing the incompressibility condition. Eliminating the pressure results in a system that requires less memory. The reduction in memory allows for larger ensemble sizes, which give more information about the flow and can be used to extend the predictability horizon.

Keywords

Navier-Stokes equationsEnsemble calculationPenalty methodsNumerical analysisFinite element methods

1 Introduction

In fluid dynamics, uncertainties in problem data lead to the need for simulations with different initial and boundary conditions, body forces, and other data [7]. They often result in computationally expensive simulations. The trajectories of these simulations can diverge significantly under slightly different initial conditions [10–12]. The predictions are unreliable when they are O(1) apart.

We address these challenges using ensemble methods. Ensemble methods run multiple simulations with slightly different initial conditions. The best prediction is obtained by averaging the results from all simulations [9]. Let Ω be an open regular domain in Rd (d=2 or 3) and T be the final time. Consider the incompressible Navier–Stokes equations (NSE):

∂u∂t+u·∇u-νΔu+∇p=f,and∇·u=0inΩ×[0,T].

(1)

Here, u is the velocity, p is the pressure, f is the body force, and ν is the kinematic viscosity. ∇·u denotes the divergence of the velocity field.

The velocity and pressure in Eq. (1) are coupled by the incompressible constraint ∇·u=0. Penalty methods relax the incompressibility condition and uncouple velocity and pressure. One can eliminate the pressure through pϵ=-1ϵ∇·uϵ to speed up the calculation. The penalized NSE from [14] is described as the following:

∂uϵ∂t+uϵ·∇uϵ+12(∇·uϵ)uϵ-νΔuϵ+∇pϵ=f,∇·uϵ+ϵpϵ=0,where0<ϵ≪1.

(2)

Temam [14] proved that limϵ→0(uϵ,pϵ)=(u,p).

Instead of a single realization, we conduct J simulations. ujϵ and pjϵ represent the solutions to the NSE for the jth ensemble member and fj is the corresponding body force, where 1≤j≤J. We present the time discretization with a shared coefficient matrix for all ensemble members. We introduce the time step size Δt and the final step N=T/Δt. We denote the velocity and pressure at the nth step as ujϵ,n and pjϵ,n, respectively, where 0≤n≤N.

Definition 1

Denote the ensemble mean and fluctuation at the nth step:

<uϵ>n:=1J∑j=1Jujϵ,n,andUjϵ,n:=ujϵ,n-<uϵ>n.

For clarity, we suppress the spatial discretization. We use an implicit-explicit time discretization and keep the resulting coefficient matrix independent of ensemble members. The method is to find ujϵ,n+1∈X, pjϵ,n+1∈Q:

ujϵ,n+1-ujϵ,nΔt+<uϵ>n·∇ujϵ,n+1+12(∇·<uϵ>n)ujϵ,n+1+Ujϵ,n·∇ujϵ,n+12(∇·Ujϵ,n)ujϵ,n-νΔujϵ,n+1+∇pjϵ,n+1=fjn+1,∇·ujϵ,n+1+ϵpjϵ,n+1=0.

(3)

We can eliminate the pressure in Eq. (3). Here ϵ is the same across all ensemble members to have a shared coefficient matrix. Thus, we assemble the matrix once and reduce the time of assembling the matrix separately for each ensemble member.

2 Notation and Preliminaries

The L2(Ω) norm and the inner product are denoted by ‖·‖ and (·,·). Likewise, the Lp(Ω) norms and the Sobolev Wpk(Ω) norms are denoted by ‖·‖Lp and ‖·‖Wpk, respectively. Hk(Ω) is the Sobolev space W2k(Ω), with norm ‖·‖k. For functions v(t, x) defined for t∈[0,T], we define, for 1≤m<∞,

‖v‖∞,k:=EssSup[0,T]‖v(t,·)‖kand‖v‖m,k:=∫0T‖v(t,·)‖kmdt1/m.

(4)

Define the fluid velocity space and pressure space, respectively:

X:=(H01(Ω))d={v∈L2(Ω)d:∇v∈L2(Ω)d×dandv=0on∂Ω},andQ:=L02(Ω)={q∈L2(Ω):∫Ωqdx=0}.

Let Xh⊂X and Qh⊂Q be finite element spaces for fluid velocity and pressure respectively. We assume that Xh and Qh satisfy the LBBh condition:

Condition 1

Suppose (Xh,Qh) satisfies:

infqh∈Qhsupvh∈Xh(qh,∇·vh)‖vh‖‖qh‖≥βh>0,

(5)

where βh is bounded away from zero uniformly in h.

We summarize standard properties of finite element spaces from [6, 8]. Assume Xh and Qh satisfy the following approximation properties for 0≤s≤m:

infv∈Xh‖u-v‖≤Chs+1|u|s+1,∀u∈X∩Hs+1(Ω)d,infv∈Xh‖∇(u-v)‖≤Chs|u|s+1,∀u∈X∩Hs+1(Ω)d,infq∈Qh‖p-q‖≤Chs|p|s,∀p∈Q∩Hs(Ω).

(6)

Denote the skew-symmetric trilinear form: ∀u,v,w∈X,

b∗(u,v,w):=12(u·∇v,w)-12(u·∇w,v).

Lemma 1

(See [8], p.123 and p.155) ∀u,v,w∈X, the trilinear term b∗(u,v,w) is equivalent to

b∗(u,v,w)=(u·∇v,w)+12(∇·u)v,w.

Definition 2

PQh is the L2 projection of Q into Qh. That is, for any q∈Q, PQh(q) satisfies

(PQh(q)-q,qh)=0,∀qh∈Qh.

3 Algorithms

Let Δtn denote the time step size at the nth step. The fully discrete approximation is given (uj,hϵ,n,pj,hϵ,n)∈(Xh,Qh), find (uj,hϵ,n+1,pj,hϵ,n+1)∈(Xh,Qh) satisfying:

(uj,hϵ,n+1-uj,hϵ,n,vh)Δtn+1+b∗(<uhϵ>n,uj,hϵ,n+1,vh)+b∗(uj,hϵ,n-<uhϵ>n,uj,hϵ,n,vh)+ν(∇uj,hϵ,n+1,∇vh)-(pj,hϵ,n+1,∇·vh)=(fjn+1,vh),(∇·uj,hϵ,n+1,qh)+ϵ(pj,hϵ,n+1,qh)=0.

(7)

for all (vh,qh)∈(Xh,Qh). We reduce computation time by solving the ensemble members with a shared coefficient matrix and different right-hand side vectors. We can eliminate the pressure. The system will require less memory. The penalty-based ensemble method is presented in Algorithm 1.

Algorithm 1

Penalty-based ensembles

4 Error Analysis

We summarize stability and convergence theorems from our analysis, Fang [3]. Due to the explicit discretization of the stretching term b∗(uj,hϵ,n-<uhϵ>n,uj,hϵ,n,vh), a CFL condition [1] is necessary to ensure its stability and convergence of the method described by Eq. (7). The penalty method is a one-step method that allows us to assume a constant time step Δt for simplicity in the analysis.

Theorem 1

(Fang [3]) Consider the method in Eq. (7). Suppose for any 0≤n≤N-1,

CΔtνh‖∇(uj,hϵ,n+1-<uhϵ>n+1)‖2≤1.

Then for any N>1, the following holds

12‖uj,hϵ,N‖2+14∑n=0N-1‖uj,hϵ,n+1-uj,hϵ,n‖2+νΔt4‖∇uj,hϵ,N‖2+νΔt4∑n=0N-1‖∇uj,hϵ,n+1‖2+Δtϵ∑n=0N-1‖PQh(∇·uj,hϵ,n+1)‖2≤Δt2ν∑n=0N-1‖fjn+1‖-12+12‖uj,hϵ(0)‖2+νΔt4‖∇uj,hϵ(0)‖2.

The error of jth ensemble member at time tn is denoted by

ej,hϵ,n:=ujϵ(tn)-uj,hϵ,n.

Theorem 2

(Fang [3]) Let Ω be a convex polygonal/polyhedral domain. Consider the method in Eq. (7). Under the assumption of Theorem 1, there exists positive constants C and C0 independent of h and Δt such that

‖ej,hϵ,N‖2+12∑n=0N-1‖ej,hϵ,n+1-ej,hϵ,n‖2+Δtν‖∇ej,hϵ,N‖2+C0Δt∑n=0N-1ν‖∇ej,hϵ,n+1‖2≤exp(α){‖ej,hϵ,0‖2+Δtν‖∇ej,hϵ,0‖2+h2mC(ν)T‖|uj,tϵ|‖∞,02+1ν2‖|pj,tϵ|‖∞,02+(Δt)3C(ν)‖|uj,ttϵ|‖∞,02+h2mϵΔtC(ν,βh)‖|uj,tϵ|‖2,02+‖|pj,tϵ|‖2,02+h2mC(ν)T‖|ujϵ|‖2,02+1ν2‖|pjϵ|‖2,02+C(ν)(Δt)2‖|∇uj,tϵ|‖∞,02},

(8)

where

α=C(ν)Δt∑n=0N-1‖∇ujϵ,n+1‖4.

Combining Theorem 2 with the result of Shen [13], Theorem 4.1, p. 395, we have the following.

Corollary 1

(Fang [3]) Under the assumption of Theorem 1, for regular solutions, we have the following optimal estimates:

maxtn‖uj(tn)-uj,hϵ,n‖2+Δt∑n=1N‖∇(uj(tn)-uj,hϵ,n)‖2≤C(u,ν)(ϵ+Δt+hm)2.

5 Numerical Experiments

We conduct numerical experiments with simply J=2 ensemble members. In the first test, we applied perturbations on the initial conditions of the modified Green–Taylor vortex. We verified the accuracy of the algorithm and confirmed the predicted convergence rates. We studied a rotating flow on offset cylinders and adapted the time step in the second test. The kinetic energy and enstrophy suggested that our algorithm preserved the stability.

We employ the Taylor–Hood (P2-P1) finite element pair. Here, P2 represents a second-order polynomial for the velocity, while P1 represents a first-order polynomial for the pressure. The unstructured mesh was generated with GMSH, an open-source finite element mesh generator [4].

5.1 Convergence Experiment

The Green–Taylor vortex problem is commonly used for calculating convergence rates, since the analytical solution is known [6]. In Ω=(0,1)2, the exact solution of the modified Green–Taylor vortex is

u(x,y,t)=(-cos(x)sin(y)sin(t),sin(x)cos(y)sin(t))⊤,p(x,y,t)=14cos(2x)+cos(2y)sin2(t).

We take the exact initial condition u(x, y, 0) and perturb for the ensemble members with δ1=10-3 and δ2=-10-3:

u1(x,y,0)=(1+δ1)u(x,y,0),u2(x,y,0)=(1+δ2)u(x,y,0).

We set the kinematic viscosity ν=1, the characteristic velocity of the flow U=1, the characteristic length L=1, and the Reynolds number Re=ULν. To discretize the domain, we choose a sequence of mesh sizes h=1g, where g is successively defined as g=32i·27, where i=0,…,4. We set the time step size Δt=h10, penalty parameter ϵ=Δt, and the final time T=1. Denote the error e(h)=Chβ, solve β via

β=ln(e(h1)/e(h2))ln(h1/h2)

at two successive values of h, where β is the convergence rate. Set inhomogeneous Dirichlet boundary condition uh=utrueon∂Ω for all ensemble members.

Table 1

Errors and convergence rates for u1

g

maxtn‖u1(tn)-u1,hϵ,n‖

Rate

Δt∑n=1N‖∇(u1(tn)-u1,hϵ,n)‖2

Rate

(32)0·27

1.38·10-4

–

3.61·10-4

–

(32)1·27

9.37·10-5

0.99

2.38·10-4

1.05

(32)2·27

6.26·10-5

0.99

1.57·10-4

1.02

(32)3·27

4.14·10-5

0.99

1.03·10-4

1.01

(32)4·27

2.78·10-5

0.99

6.90·10-5

1.00

Table 2

Errors and convergence rates for u2

g

maxtn‖u2(tn)-u2,hϵ,n‖

Rate

Δt∑n=1N‖∇(u2(tn)-u2,hϵ,n)‖2

Rate

(32)0·27

1.38·10-4

–

3.59·10-4

–

(32)1·27

9.34·10-5

0.99

2.38·10-4

1.06

(32)2·27

6.24·10-5

0.99

1.56·10-4

1.02

(32)3·27

4.13·10-5

0.99

1.02·10-4

1.01

(32)4·27

2.76·10-5

0.99

6.87·10-5

1.00

Tables 1 and 2 show that the convergence rates of u1 and u2 are first order as predicted.

5.2 Stability Verification

The domain is a disk with a smaller off-center obstacle inside. Let outer circle radius r1=1 and inner circle radius r2=0.1, c=(c1,c2)=(12,0). The domain is

Ω={(x,y):x2+y2≤r12and(x-c1)2+(y-c2)2≥r22}.

A counterclockwise rotational force drives the flow,

f(x,y,t)=(-4y(1-x2-y2),4x(1-x2-y2))⊤,

with no-slip boundary conditions. Note that the outer circle remains stationary. We set mesh size h=0.05. We chose the final time T=100, time step size Δt=h/10, penalty parameter ϵ=Δt, ν=1/150,L=1,U=1, and Re=ULν.

Initial condition u(x,y,0)=0 and the Dirichlet boundary condition u=0 on ∂Ω. We denote two perturbation values δ1=0.1 and δ2=-0.1 to introduce perturbations. The perturbed initial conditions for u1 and u2:

uj(x,y,0):=δj(1-x2-y2)(0.12-(x-0.5)2-y2),wherej=1,2.

This formulation preserves the no-slip boundary condition. For the evaluation, we calculated flow statistics, enstrophy, and kinetic energy:

enstrophy=12ν‖∇×u‖2andkineticenergy=12‖u‖2.

To ensure the algorithm is stable, we adopt C=1200 from [5] and adapt Δt to satisfy

Δth‖∇(uj,hϵ,n+1-<uhϵ>n+1)‖2≤1200Re,wherej=1,2.

(9)

If Eq. (9) is violated, we reduce Δt by half and repeat the process until Eq. (9) is satisfied.

Fig. 1

The enstrophy of the ensemble average uave shows that the flow is smoothed out on average

Fig. 2

The kinetic energy is similar across all members and has reached a statistically steady state

In Fig. 1, the enstrophy of ensemble average uave is lower than the ensemble members and no perturbed one, which shows that the flow is smoothed out on average. In Fig. 2, the kinetic energy is similar across all members and has reached a statistically steady state.

Fig. 3

The plot of the relative error ‖uj-u‖/‖u‖steady

Figure 3 plots the relative error ‖uj-u‖/‖u‖steady, the horizontal axis is the time, and the vertical axis is the relative error. We normalized by ‖u‖ after the norm of the flow velocity reaches a statistically steady state. The predictability horizon is defined as the duration until ‖uj-u‖/‖u‖steady reaches a chosen threshold 0.1. Computing the maximum error at each time step for ensemble members 1 and 2, we found that the 0.1-predictability suggests that the predictability horizon for a single realization is T=32.6, while T=47.5 for the ensemble average. When we set the threshold to approximately 0.2, it suggests that the ensemble average remains accurate until the final time T=100, which is significantly longer than a single realization. Our algorithm uses a shared coefficient matrix, reducing assembling time. Eliminating pressure reduces memory. It allows for a larger size of ensemble members. Large ensemble sizes improve statistical robustness, reduce sampling error, better represent uncertainty and thus yield more reliable results.

6 Conclusions

In this report, we test the stability and convergence of the penalty-based ensemble method in Eq. (7). Our algorithm uses a shared coefficient matrix and reduces the memory, which allows for larger ensemble sizes. It gives a more accurate and reliable simulation. The ensemble spread is related to the forecast uncertainty, but the nature of this relationship remains unclear and is ongoing research [15]. Open problems include extending this to a higher Reynolds number and adapting penalty parameters [2].

Acknowledgements

I thank Professor William Layton for his instruction and support throughout this research. This research herein of Rui Fang was supported in part by the NSF under grant DMS 2110379 and DMS 2410893.

References

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Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen differenzengleichungen der mathematischen physik. Mathematische Annalen 100(1), 32–74 (1928)MathSciNet

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Fang, R.: Numerical analysis of locally adaptive penalty methods for the Navier–Stokes equations. SeMA (2025). https://​doi.​org/​10.​1007/​s40324-025-00394-8

3.

Fang, R.: Numerical analysis of penalty–based ensemble methods. Numer. Algor. (2025). https://​doi.​org/​10.​1007/​s11075-025-02014-y

4.

Geuzaine, C., Remacle, J.F.: Gmsh: a 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)MathSciNet

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Jiang, N., Layton, W.: An algorithm for fast calculation of flow ensembles. Int. J. Uncertain. Quantif. 4(4) (2014)

6.

John, V.: Finite Element Methods for Incompressible Flow Problems, vol. 51. Springer, Berlin (2016)

7.

Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press (2003)

8.

Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows. SIAM, Philadelphia (2008)

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Leith, C.E.: Theoretical skill of monte Carlo forecasts. Mon. Weather Rev. 102(6), 409–418 (1974)

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Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)MathSciNet

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Lorenz, E.N.: The predictability of hydrodynamic flow. Trans. N. Y. Acad. Sci. 25(4), 409–432 (1963)

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Lorenz, E.N., Haman, K.: The essence of chaos. Pure Appl. Geophys. 147(3), 598–599 (1996)

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Shen, J.: On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32(2), 386–403 (1995)MathSciNet

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Temam, R.: Une méthode d’approximation de la solution des équations de Navier-Stokes. Bull. de la Société Mathématique de France 96, 115–152 (1968)MathSciNet

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_39

Explicit Periodic Solutions in a Delay Differential Equation

Anatoli Ivanov1   and Sergiy Shelyag2  

(1)

Pennsylvania State University at Wilkes-Barre, Dallas, PA, USA

(2)

Flinders University, Tonsley, SA, Australia

Anatoli Ivanov (Corresponding author)

Email: afi1@psu.edu

Sergiy Shelyag

Email: sergiy.shelyag@flinders.edu.au

Abstract

We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known idea of reduction to interval maps is used in the case under consideration, when both the defining nonlinearity and the periodic coefficient are piecewise constant functions. The stable periodic dynamics persist under a smoothing procedure in a small neighborhood of the discontinuity set. This work continues the research in recent paper [7] on stable periodic solutions of differential delay equations with periodic coefficients.

Keywords

Delay differential equationsNegative feedbackSlow oscillationExplicit periodic solutionInterval mapsStabilityPeriodicity

1 Introduction

Scalar delay differential equations of the form

x′(t)=-μx(t)+f(x(t-τ))

(1)

are used as mathematical models in a broad variety of applications. Some of the most well-known ones are the Mackey-Glass physiological models [10], Nicholson’s blowflies models [4, 11], and many others [3, 9]. In spite of its formal simplicity equation (1) can exhibit quite complex dynamical behaviors. Unlike scalar ordinary differential equations the DDE (1) can have complicated dynamics of solutions including oscillations about equilibrium, stable and unstable periodicity, and complex/chaotic behaviors [1, 5].

More accurate mathematical models of real world phenomena take into account some periodicity factors such as seasonal changes or circadian rhythm parameter fluctuations and several others. Such improved models with respect to Eq. (1) lead to similar type DDEs with periodic coefficients:

x′(t)=-μ(t)x(t)+a(t)f(x(t-τ)),

(2)

where μ(t) and a(t) are periodic functions with the same period. One of the natural questions that appears in this context is whether DDE (2) has periodic solutions with the same period as the periodic coefficients. This work is an attempt to answer such question in a particular case.

Note that the periodicity problem we consider is different from similar ones in many other publications, such as in e.g. [2] (also see additional references therein). We assume the negative feedback conditions on f,x·f(x)<0,∀x≠0, implying that DDE (2) admits the trivial solutions x≡0 (for any functions μ and a). This situation is similar to the case of autonomous Eq. (1), when it admits a constant solution (with the negative feedback in many cases). In [2] and elsewhere, when such constant positive equilibrium is perturbed by a multiplicative periodic coefficient, the new problem does not admit constant solutions (positive equilibrium) any longer.

A particular case of Eq. (2) is considered when μ(t)≡0 (see Eq. (3) below). The initial choice of the nonlinearity f and the periodic coefficient a(t) is as piecewise constant functions. Such simple form allows for an explicit calculation of solutions for all forward times as piecewise affine functions (made up continuously of line segments). The problem of existence of slowly oscillating solutions and their stability is reduced to the existence of fixed points of a simple interval map and their attractivity. The dynamics of both is shown to persist when the two discontinuous functions are replaced by close to them continuous functions (or even of C∞ class).

The results of this paper are a continuation and an expansion of recently obtained similar results by the same authors [7]. The new more challenging case of DDE (2) with μ(t)=μ0>0 is considered in our other recent work [8].

2 Preliminaries

As in [7] we consider the special case of DDE (2) when μ(t)≡0:

x′(t)=a(t)f(x(t-1))

(3)

Here initially the nonlinear function f is given by f(x)=f0(x)=-sign(x) and the periodic coefficient a=a(t,a1,a2,p1,p2) is defined as

a(t)=a0(t)=a1,if0≤t<p1a2,ifp1≤t<p1+p2periodicextensionoutside[0,p1+p2)forallt∈R,

(4)

with all the values a1,a2,p1,p2 being positive and p1+p2:=T>1.

For arbitrary initial function φ(s)∈C=C([-1,0],R) such that φ>0∀s∈[-1,0] the corresponding unique solution x=x(t,φ) exists for all t>0 (it is calculated by the consecutive integration, the "step method"). The solution is a continuous piecewise affine function made up of straight line segments. It is differentiable everywhere except a countable set of points where two line segments with different slopes match.

3 Main Results

3.1 Explicit Periodic Solutions

Given f=f0 and a=a0 as above for arbitrary initial function ϕ∈C the corresponding solution x=x(t,ϕ) is constructed for all forward times t≥0. It is a continuous piecewise differentiable function composed of consecutive segments of affine (linear) functions.

We assume that periodic solutions have the form as depicted in Fig. 1 (this will be shown to be the case for particular values of the parameters a1,a2,p1,p2). For arbitrary initial function ϕ∈C such that ϕ(s)>0∀s∈[-1,0], the corresponding forward solution x(t),t≥0, depends only on the value ϕ(0):=h>0 and does not depend on the other values ϕ(s),s∈[-1,0) of the initial interval. There exist consecutive zeros t1>0 and t2=t1+2 of the solution such that x(t)>0,t∈[0,t1), x(t)<0,t∈(t1,t1+2), and x(t)>0,t∈[t1+2,p1]. The necessary condition for such first two zeros to exist is that p1>2. We also assume that p1<t1+3<p1+p2 and that the solution remains positive on the interval [p1,p1+p2]. For the latter to be true one only has to require that x3:=x(p1+p2)>0.

Fig. 1

Slowly oscillating piecewise affine solution

We next calculate the explicit values of t1,x1=x(p1),x2=x(t1+3),x3=x(p1+p2) in terms of the parameters a1,a2,p1,p2 and given initial value h>0. It is straightforward to find that:

t1=ha1,x1=x(p1)=-h+a1p1-2a1,

x2=x(t1+3)=a2a1-1h+a1p1-2a1+3a2-a2p1,

x3=x(p1+p2)=2a2a1-1h+a1(p1-2)+a2[6-(2p1+p2)]:=mh+b

In addition to the previous assumptions the condition b>0, i.e.

b=a1(p1-2)+a2[6-(2p1+p2)]>0,

(5)

guarantees that x3>0, and thus that the solution x(t),t∈[0,p1+p2], is of desired shape (as shown in Fig. 1).

A fixed point h∗>0 of the affine map F(x)=mx+b gives rise to a slowly oscillating periodic solution x=x(t,h∗) of Eq. (1). Moreover, by the linearity of F and the construction, such periodic solution is asymptotically stable if |m|<1. The latter is equivalent to 0<a2<a1. The unique fixed point x=h∗ is easily found as

h∗=b1-m=a1[a1(p1-2)+a2[6-(2p1+p2)]]2(a1-a2).

(6)

Therefore, we arrive at the following statement

Theorem 1

Suppose that the parameters a1,a2,p1,p2 are such that the inequality (5) is satisfied and a1>a2. Then DDE (3) has an asymptotically stable slowly oscillating periodic solution. The periodic solution is generated by the initial function ϕ(s)≡h∗,s∈[-1,0], where h∗ is given by (6).

Note that due to the symmetry property of the nonlinearity f (oddness) and the procedure of construction of the periodic solution xh∗, under the assumption of Theorem 1 there also exists the symmetric to xh∗ periodic solution generated by the initial function ϕ(s)≡-h∗. The two periodic solutions are related by x-h∗(t)≡-xh∗(t).

3.2 Smoothing of Nonlinearities

In this subsection we demonstrate that the stable periodic solution derived in the previous subsection persists under the standard smoothing procedure for functions f and a. The procedure is a well known one which has been used in many publications (see e.g. [6, 12] and further references therein). It consists in replacing each jump discontinuity by a continuous affine function in a small neighborhood of a discontinuity point. We repeat here the exact constructions as in [7].

Consider the functions f and a as defined in Sect. 3.1: f(x)=f0(x) and a(t)=a0(t). Let δ0>0 be small, and for every δ∈(0,δ0] introduce the continuous functions fδ(x) and aδ(t) by:

f(x)=fδ(x)=+1ifx≤-δ-1ifx≥δ-(1/δ)xifx∈[-δ,δ],

(7)

and

a(t)=aδ(t)=a2+a1-a22δ(t+δ)ift∈[-δ,δ]a1ift∈[δ,p1-δ)a1+a2-a12δ[t-(p1-δ)]ift∈[p1-δ,p1+δ]a2ift∈[p1+δ,p1+p2-δ)a2+a1-a22δ[t-(p2-δ)]ift∈[p1+p2-δ,p1+p2+δ]periodicextensiononRoutsideinterval[0,p1+p2).

(8)

The following statement is an extension of the main Theorem 1 to the case of continuous functions fδ and aδ.

Theorem 2

Suppose the assumptions of Theorem 1 are satisfied. There exists δ0>0 such that for every δ∈[0,δ0] DDE (3) with f=fδ and a=aδ has an asymptotically stable slowly oscillating periodic solution xδ(t). The solution xδ(t) is close in the uniform metric to the periodic solution xh∗ of Theorem 1 for small δ≥0, moreover limδ→0{supt∈R|xδ(t)-xh∗(t)|}=0.

The proof is based on the explicit calculation of the similar value x3 of the solution x(t, h) of the corresponding DDE (3). Its shape is similar to that shown in Fig. 1, except that the corner points (where the derivative x′ is not defined) are replaced by a smooth C1 curve, based on the affine representation of both f and a around their discontinuity points. The exact calculations repeat those in [7]. The final result is a representation of x3(h) in the form x3(h)=F~(h), where F~(h) is C1-close to the F defined in Sect. 3.1 (see pages 74–75 of [7] for more details). Therefore, the existence, stability, and closeness follow.

3.3 Numerical Demonstration

Periodic solutions described in Sects. 3.1 and 3.2 can be easily verified numerically. It is clear that sets of parameters leading to the existence of a stable periodic solution, as described by Theorem 1, are not exceptional. Any sufficiently small perturbation of a particular quadruple of the parameters will produce an asymptotically stable periodic solution. They are open in the set of all parameters. The exemplary parametric values for which numerical periodic solutions have been obtained and confirmed to be of the described form are given in Table 1.

Table 1

Parametric values for which stable periodic solutions of Eq. (3) with the coefficient defined in Eq. (4) have been demonstrated numerically

a1

a2

p1

p2

h∗

T

1

0.25

2.5

1.5

0.25

4

2

0.5

2.5

2

1/3

4.5

2

0.25

2.5

1

4/7

3.5

1

0.5

3

1

0.5

4

2

1

3

1.5

0.5

4.5

2.5

0.5

3

4

0.31

7

3

0.5

3

4.5

0.45

7.5

5

0.5

3

3

1.94

6

5

1

3

2

1.88

5

4 Conclusions and Discussion

In this paper, we have explicitly constructed stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic nonlinear negative feedback of a multiplicative form. The initial construction is performed for discontinuous piecewise constant functions used for both the feedback nonlinearity and the multiplicative periodic coefficient. Such choices of functions allow for an explicit form of the solutions of the equation in forward times, thus reducing the problem of dynamics in the original DDEs to that of a simple interval map. The existence and the stability of the fixed points in the latter translates back to the analogous properties for periodic solutions of the DDE. We then demonstrated that the stable periodic dynamics persist if the jump discontinuities in the equation are smoothed in a sufficiently small neighborhood around them, thus making the dynamics of DDE of the C1-class. The new existence and stability results on the periodic solutions are complimentary to those recently derived in our paper [7].

There is a possibility of existence of additional types of stable periodic solutions in (3) with piecewise constant defining functions f and a, the question we intend to tackle in our next research considerations. It is of interest to show the existence of stable periodic solutions in a general DDE of the form (3), where f and a are smooth functions of arbitrary type. An interesting and seemingly very nontrivial problem is the existence of (stable) periodic solutions in DDE (3) with smooth functions, when f is arbitrary and fixed nonlinearity with the negative feedback property and the periodic coefficient a is parameter dependent, e.g., a=a(t,ε), such that a(t,0)=a0>0 and a(t+T,ε)=a(t,ε), for some T>0. As the value ε=0 gives periodic solutions in the well-known autonomous case of Eq. (1), and some larger positive values of ε0 result in T-periodic solutions with the same period as a, the intermediate values of ε∈(0,ε0) may yield varied dynamics and complex transition patterns. This is an open problem worthy of further investigation.

Acknowledgements

The authors thank the mathematical research institute MATRIX in Australia where part of this research was performed. Its final version resulted from collaborative activities of the authors during the workshop “Delay Differential Equations and Their Applications” (https://​www.​matrix-inst.​org.​au/​events/​delay-differential-equations-and-their-applications/​) held in December 2023. The authors are also grateful for the financial support provided for these research activities by Simons Foundation (USA), Flinders University (Australia), and the Pennsylvania State University (USA). A.I.’s research was also supported in part by the Alexander von Humboldt Stiftung (Germany) during his visit to Justus-Liebig-Universität, Giessen, in June–August 2023.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_40

Proportional Consistency of Apportionment Methods

Michael A. Jones1  , David McCune2   and Jennifer M. Wilson3  

(1)

Mathematical Reviews, Ann Arbor, MI, USA

(2)

William Jewell College, Liberty, MO, USA

(3)

The New School, New York City, NY, USA

Michael A. Jones

Email: maj@ams.org

David McCune

Email: mccuned@william.jewell.edu

Jennifer M. Wilson (Corresponding author)

Email: wilsonj@newschool.edu

Abstract

We analyze a little-known property of apportionment methods that captures how allocations scale with the size of the house: specifically, if, for a fixed population distribution, the house size and allocation can be scaled down within the set of integers, then the apportionment should be correspondingly scaled down. Balinski and Young [1] include this property among the minimal requirements for a “reasonable” apportionment method. We argue that this property is better understood as a consistency requirement since quota-based apportionments that are “less proportional” meet this requirement while others that are “more proportional” do not. We also show that the family of so-called quotatone methods based on stationary divisors (including the quota method [2]) do not satisfy this property.

Keywords

ApportionmentProportional consistencyDivisor methodsQuota methodsQuotatone method

1 Introduction

Apportionment methods are typically used to determine the number of representatives allocated to each state or district in a national parliament in proportion to their population. They are also used to determine both the number of seats awarded to political parties based on their share of the popular vote and the number of delegates awarded to presidential candidates in the US presidential primaries based on their performance in the state primaries. The difficulty is that the apportioned values must be nonnegative integers. A thorough introduction into the methods and their properties in these contexts can be found in [1, 3, 5], respectively.

Proportional consistency addresses in an indirect way that as the total number of seats increases, the apportionment should get closer to proportionality. This mirrors our intuitive sense that if there are more seats available then it should be easier to allocate the seats to states in a way that is closer to their fair share. Although Balinski and Young [1, p. 97] include proportional consistency as one of three properties required for an apportionment method to be proportional, the property has not been systematically studied. They state, without proof, that all reasonable methods satisfy this requirement. Although true for all standard apportionment methods, it is not true of all apportionment methods currently in use or well-known in the literature. In particular, proportional consistency is not satisfied by several methods used in the US presidential primaries [3], nor does it apply to the quotatone methods that were introduced by Balinski and Young [2] to satisfy quota and monotonicity conditions.

In Sect. 2, we recall several well-known apportionment methods and give a formal definition of proportional consistency and proportionality. In Sect. 3, we prove that stationary and other well-known divisor methods satisfy proportional consistency, but that not all divisor methods do. We do the same for quota-based methods. In Sect. 4 we show that the primary family of quotatone methods do not satisfy proportional consistency. We offer some final thoughts as a conclusion.

2 Preliminaries

Let H>0 denote the total number of representatives (i.e., the house size). Assume state Si has population vi, i=1,…,n, and that v1≥v2≥⋯≥vn>0. Let V=∑ivi be the total population, and let v=(v1,…,vn). An apportionment method F is a mapping that, for each (v,H), assigns one or more n-tuples of nonnegative integers h=(h1,…,hn) satisfying ∑i≤nhi=H where hi is the number of seats allocated to state Si. The range of F is set-valued because of the possibility of ties at some point in the apportionment process. Because ties occur rarely, we will write F(v,H)=h.

Historically, the two-most important classes of apportionment methods are divisor and quota-based methods. A divisor method corresponds to a rounding rule f which assigns to each nonnegative integer k, a value f(k)∈[k,k+1] such that there do not exist positive integers a and b for which f(a)=a-1 and f(b)=b. The value f(k) acts as a bifurcation point in which positive numbers between k and f(k) are rounded down to k and numbers between f(k) and k+1 are rounded up to k+1.

Definition 1

The divisor method F with rounding rule f satisfies F(v,H)=h if there exists a divisor x>0 such that minhi>0vif(hi-1)≥x≥maxhj≥0vjf(hj), where the hi are nonnegative integers that satisfy ∑i≤nhi=H.

Divisor methods with rounding rules f(k)=k+s for s∈[0,1] are called stationary. These include Jefferson’s (or D’Hondt’s), Webster’s (or Sainte-Laguë’s), and Adams’ methods for s=1, s=1/2, and s=0, respectively. Among stationary methods, Jefferson’s and Webster’s methods were used to apportion the US House of Representatives from 1791 to 1842 and from 1842 to 1852 and 1901–1941, respectively. Non-stationary methods include Hill-Huntington’s method—f(k)=k(k+1), which was used to apportion the US House from 1941-present—and Dean’s method—f(k)=k(k+1)k+0.5. Among legislative houses using proportional representation, D’Hondt’s and Sainte Laguë’s methods are the most common. Balinski and Young [1] and Pukelsheim [5], respectively, provide the definitive account for both the mathematics and the history of the use of apportionment methods for the US House and for countries in the European Union.

Quota methods are the other well-known class of apportionment methods in which each state i’s quota or fair share of H, qi=HviV, is rounded to integers that sum to H. The most well-known apportionment method is Hamilton’s method, also called the greatest remainder method; it is one of a larger family of apportionment methods: shift-quota methods. Shift-quota methods, which are based on a parameter s∈[0,1), replace the usual notion of quota with an adjusted quota qis=(H+s)viV. Let ⌊qis⌋ denote the quota rounded down to the nearest integer. Note that qis-1<⌊qis⌋≤qis, which implies H+s-n<∑i⌊qis⌋≤H+s, and so H-n<∑i⌊qis⌋<H+1. Integer constraints tighten these inequalities to H-(n-1)≤∑i⌊qis⌋≤H, ensuring that an initial allocation of ⌊qis⌋ leaves at most n-1 remaining seats to distribute. When s=0, the shift-quota method reduces to Hamilton’s method. Other apportionments are introduced in subsequent sections.

Definition 2

Under the shift-quota method with parameter s∈[0,1), each state receives an initial number of seats equal to ⌊qis⌋. The remaining H-∑i⌊qis⌋ seats are given one each to the states with the highest fractional remainders qis-⌊qis⌋.

Proportional Consistency and Other Properties. We introduce the properties of apportionment methods that are most relevant to our discussion. For additional properties, see [1, 3, 5]. The first property is homogeneity, which requires apportionments to be invariant under a uniform scaling of the state populations.

Definition 3

A method is homogeneous if F(v,H)=F(λv,H) for all λ>0.

Thus if an apportionment method is homogeneous, we can refer interchangeably to F(v,H), F(q,H) or F(p,H) where q and p are the vector of quotas qi and population proportions pi=vi/V, respectively. The next two properties are weak proportionality and proportional consistency. Weak proportionality requires an apportionment method to return the quotas when all quotas are integers.

Definition 4

F is weakly proportional if whenever qi is an integer for all i, then F(v,H)=q.

Definition 5

A method is proportionally consistent if whenever F(v,H)=h and for all rational λ<1 such that λhi are integers for every i, then F(v,λH)=λh.

Proportional consistency means, for instance, that if an apportionment method allocates 24 seats to 4 states by h=(9,6,6,3), then it should allocate 23(24)=16 seats by 23h=(6,4,4,2) and 13(24)=8 seats by 13h=(3,2,2,1). The scaling should work in only one direction: if 48 seats were apportioned instead, we do not want to require that the apportionment be (18, 12, 12, 6). Intuitively, more seats should allow for the possibility that another apportionment could be closer proportionally.

Balinski and Young [1, p. 97] define an apportionment method to be proportional if it satisfies homogeneity, weak proportionality and proportional consistency. In Sects. 3 and 4, we examine which methods satisfy proportional consistency. The methods defined above are proportional, but other methods—including ones used in delegate apportionment—are not proportionally consistent.

3 Divisor and Quota Methods and Proportional Consistency

Let’s begin this section by proving that well-known divisor methods are proportionally consistent. Because the methods of Adams, Webster and Jefferson are stationary methods, they are included in the following proposition.

Proposition 1

(a)

Stationary divisor, Hill-Huntington, and Dean methods are proportionally consistent.

(b)

Not all divisor methods are proportionally consistent.

Proof

(i) If F(p,H)=h, then there is an x>0 such that minhi>0viλf(hi-1)≥xλ≥maxhj≥0vjλf(hj). It suffices to show that λf(h-1)≥f(λh-1) for all h>0 and λf(h)≤f(λh) for all h≥0, since this implies

minhi>0vif(λhi-1)≥minhi>0viλf(hi-1)≥x′≥maxhj≥0vjλf(hj)≥maxhj≥0vjf(λhj);

hence F(v,λH)=λh. These inequalities are easily verified when λ<1 for stationary divisor methods as well as for Dean’s and Hill-Huntington’s methods.

(ii) For a counterexample, let f(k)=k+0.5 for k=15,9,3 and f(k)=k+1 for all other values of k. Then if H=27, λ=2/3, and q=(463,253,103) then F(q,27)=(15,9,3) but F(q,18)=(11,5,2)≠(10,6,2).

There is no formal mathematical distinction between quota-based and non-quota apportionment methods since any homogeneous apportionment method can be described in terms of quotas by a change of variables. But for convenience, we loosely define a quota-based apportionment method as one that is easiest to define in terms of quotas, or which requires calculating the quotas as a first step and then rounding the quotas. Let ⌈·⌉ and [·] denote the ceiling and nearest integer rounding functions, respectively. The proof of the following lemma is straightforward, appearing in [4].

Lemma 1

For all λ<1, if ki, λki, and t are positive integers, then

(a)

If ⌊qi⌋=ki then ⌊λqi⌋=λki and if ⌈qi⌉=ki then ⌈λqi⌉=λki.

(b)

If [qi]=ki+t for some t≥0, then λki≤[λqi]≤λki+t. If [qi]=ki-t for some t≥0, then λki-t≤[λqi]≤λki. In both cases, if λ≤2t+1 then [λqi]=λki.

(c)

If [qi]=ki then [λqi]=λki.

An apportionment method is said to satisfy quota if, for each i, hi∈{⌊qi⌋,⌈qi⌉}.

Lemma 2

If F satisfies quota, F(v,H)=h, and for some λ<1, the λhi are integers, then qi is an integer if and only if λqi is an integer.

Proof

If qi is an integer then ⌊qi⌋=⌈qi⌉ and, since F satisfies quota, ⌊qi⌋=⌈qi⌉=hi. From Lemma 1(i), this implies λhi=⌊λqi⌋=⌈λqi⌉, so that λqi is an integer.

Alternatively, if qi is not an integer, then either ⌊qi⌋=hi or ⌈qi⌉=hi. Without lost of generality, assume ⌊qi⌋=hi. Hence, hi<qi<hi+1 and so λhi<λqi<λhi+λ<hi+1, since λ<1. Thus λqi is not an integer.

To investigate which quota methods satisfy proportional consistency, suppose F satisfies quota and let L(v,H)={i∣hi=⌊qi⌋} and U(v,H)={i∉L(v,H)∣hi=⌈qi⌉} be the set of states who receive their lower quota or upper quotas, respectively. Note that L(v,H) contains those states whose quotas are integers so that L(v,H) and U(v,H) partition the set of states. Define L(v,λH) and U(v,λH) analogously.

Suppose F(v,H)=h and the λhi are integers. By Lemma 1, i∈L(v,H) implies ⌊qi⌋=hi and ⌊λqi⌋=λhi. Similarly, i∈U(v,H) implies ⌈qi⌉=hi and ⌈λqi⌉=λhi. Since λqi is not an integer, this implies ⌊λqi⌋=λhi-1. So

∑i⌊λqi⌋=∑i∈L(v,H)⌊λqi⌋+∑i∈U(v,H)⌊λqi⌋=∑i∈L(v,H)λhi+∑i∈U(v,H)(λhi-1)=λH-|U(v,H)|.

But, if F(v,λH)=h′, then

∑i⌊λqi⌋=∑i∈L(v,λH)⌊λqi⌋+∑i∈U(v,λH)⌊λqi⌋=∑i∈L(v,λH)hi′+∑i∈U(v,λH)(hi′-1)=λH-|U(v,λH)|,

so that |U(v,λH)|=|U(v,H)| and |L(v,λH)|=|L(v,H)|. Thus if a method satisfies quota, the same number of states receive their quota rounded up (down) after scaling by λ.

Proposition 2

If F satisfies quota then

(i)

if F is proportionally consistent, F(v,H)=h, and for some λ<1 the λhi are integers, then L(v,H)=L(v,λH) and U(v,λH)=U(v,H); and

(ii)

if, for every v and H, where F(v,λH)=λh and the λhi are integers for some λ<1, L(v,λH)=L(v,H) and U(v,λH)=U(v,H), then F is proportionally consistent.

Proof

(i) Let F satisfy the hypotheses and F(v,λH)=λh. By Lemma 1, if i∈L(v,H), then ⌊qi⌋=hi and ⌊λqi⌋=λhi so that i∈L(v,λH). Likewise, if i∈U(v,H) then ⌈qi⌉=hi and ⌈λqi⌉=λhi. As qi is not an integer, neither is λqi. Hence i∈U(v,λH).

(ii) Let F(v,H)=h, the λhi be integers and L(v,λH)=L(v,H) and U(v,λH)=U(v,H). Let F(v,λH)=h′. If i∈L(v,H), then λhi=⌊λqi⌋=hi′, since i∈L(v,λH). Likewise, if i∈U(v,H), then λhi=⌈λqi⌉=hi′, since i∈U(v,λH). Hence h′=λh.

Proposition 2 implies that, for any method satisfying quota, proportional consistency is equivalent to the condition that the same states receive their upper (or lower) quota after scaling by λ<1. This property is true for a number of quota-based methods, including shift-quota methods. Propositions 1 and 2 remain true if all references to quota and qi are replaced with the shifted quota and qis. To show these methods are proportional, it suffices to show that the same states receive their shifted quota rounded up or down after scaling. For s∈[0,1), let L(v,λH)s and U(v,λH)s denote the set of states who receive their lower (resp. upper) shifted quota.

Proposition 3

The shift-quota methods are proportionally consistent.

Proof

If i∈L(v,H)s and j∈U(v,H)s, then qis-⌊qis⌋≤qjs-⌊qjs⌋ or qis-hi=qis-⌊qis⌋≤qjs-⌊qjs⌋=qjs-(hj-1). Multiplying by λ yields λqis-λhi≤λqjs-λ(hj-1)≤λqjs-(λhj-1) or λqis-⌊λqis⌋≤λqjs-⌊λqjs⌋. This implies that after rounding down, the remaining seats go the same states, and hence the shift-quota methods are proportionally consistent by Proposition 2.

While Hamilton’s method is the best known apportionment method satisfying quota, Proposition 2 suggests that any quota method which first awards ⌊qi⌋ seats and then allocates the remaining seats in a consistent way will be proportional. This leads to consideration of methods such as the following.

Definition 6

Let each state receive an initial number of seats equal to ⌊qi⌋. Under the Large (LAR) method, the remaining seats are given one each to the H-∑i⌊qi⌋ states with the largest populations whose quotas are not integers. Under the Small (SML) method, the remaining seats are given one each to the H-∑i⌊qi⌋ states with the smallest populations whose quotas are not integers.

Proposition 4

The LAR and SML methods are proportional.

The proof of Proposition 4 follows directly from Lemma 1 and Proposition 2. The LAR and SML methods are clearly biased in the sense that they reward large or small states disproportionately. While such methods would not be appropriate for awarding seats to states in the US House, they may be relevant in other settings. The LAR method, for instance, is used by the US Republican Party in several states to award delegates to presidential candidates based on the candidates’ share of the primary vote. In this context, it is considered advantageous to use an apportionment method that is slightly biased towards stronger candidates; see [3, §5.3]. Rather than using population to select which states receive an additional seat, an alternate method that is also proportional (but not anonymous) is to designate a priority order of which states with non-integer quotas should receive one of the extra H-∑i⌊qi⌋ seats. We turn to other types of quota methods.

Many quota-based methods can be defined by first rounding quotas up, down or to the nearest integer and then adjusting in ways that may not satisfy quota. We consider several of these methods, determining which are proportionally consistent.

The Lower Quota Extremes (LQE) method takes the bias of the LAR method to the extreme.

Definition 7

Let each state receive an initial number of seats equal to ⌊qi⌋. Under the Lower Quota Extremes method, the remaining H-∑i⌊qi⌋ seats are given to the largest state.

While the LQE method is even more biased than the LAR and SML methods, it is straightforward to prove the following. The proof appears in [4].

Proposition 5

The LQE method is proportionally consistent.

Similar arguments apply to methods based on upper quota rounding in which state i initially receives ⌈qi⌉ seats and then ∑i⌈qi⌉-H seats are removed in a systematic way. One such example is the Sequential Upper Quota method, which, like LAR, is used by some Republican state parties to allocate delegates to candidates during the US presidential primaries; see [3, §4.3.5] for which states use this method.

Definition 8

The Sequential Upper Quota (SUQ) method initially assigns each state ⌈qi⌉ seats. The ∑i⌈qi⌉-H over-allocated seats are removed from the smallest state; if the smallest state does not have a sufficient number of seats, the rest are removed from the second-smallest state, and so on, until the house size is met.

The SUQ method can alternatively be described as giving ⌈qi⌉ seats to states in descending order of size until no more seats are available.

Proposition 6

The SUQ method is proportionally consistent.

Proof

Let F(v,H)=h and λ<1 be such that the λhi are integers. Let F(v,λH)=h′. Assume h=(⌈q1⌉,⌈q2⌉,…,⌈qk-1⌉,H-∑i<k⌈qi⌉,0,…,0). If i>k then hi=0. If i<k, then ⌈qi⌉=hi which, by Lemma 1, implies ⌈λqi⌉=λhi. If ⌈qk⌉=hk, then ⌈λqk⌉=λhk; otherwise hk<⌈qk⌉ which implies, since hk is an integer, hk≤⌊qk⌋≤qk, or λhk≤λqi≤⌈λqk⌉. In either case, ⌈λqk⌉≥λhk, so

∑i<k⌈λqi⌉=∑i<kλhi≤λH=∑i<kλhi+λhk≤∑i≤k⌈λqi⌉.

Hence when apportioning λH seats, the largest k-1 states receive ⌈λqi⌉ seats, with the kth-largest state receiving the remaining seats. So

h′=(⌈λq1⌉,…,⌈λqk-1⌉,λH-∑i<k⌈λq1⌉,0,…,0)=(λh1,…λhk-1,hk′,0,…,0)

where hk′=λH-∑i≠kλhi=λhk. Thus, SUQ is proportionally consistent.

Other apportionment methods are most easily described when quotas are rounded to the nearest integer and then allocations are adjusted. Note that since qi-1/2<[qi]≤qi+1/2, then H-n/2<∑i[qi]≤H+n/2. The upper bound is strict unless n is even and [qi]=qi+1/2 (so the fractional part of the quota is equal to 0.5) for all i. Thus, apart for this rare exception, |∑i[qi]-H|≤⌊(n-1)/2⌋, and at most ⌊(n-1)/2⌋ seats can be under or over-allocated.

For comparison, we introduce two methods: Nearest Integer Extremes (NIE) and Nearest Integer Sequential (NIS), which are both used in the US Republican Party presidential primary [3]. These are similar to the LQE and LAR methods respectively: the NIE method corrects for under- or over-allocation by adjusting the apportionment of the largest or smallest state; the NIS method corrects for under- or over-allocation by adjusting the allocations of multiple large or small states.

Definition 9

Under the Nearest-Integer Extremes (NIE) method, each state initially receives [qi] seats. If H≥∑i[qi], the remaining c=H-∑i[qi] seats are given to state 1. If H<∑i[qi], the extra c=∑i[qi]-H seats are removed from state n; if state n does not have enough seats, additional seats are taken from state n-1, etc.

Definition 10

Under the Nearest-Integer Sequential (NIS) method, each state initially receives [qi] seats. If H≥∑i[qi], the remaining c=H-∑i[qi] seats are given one each to states i≤c. If H<∑i[qi], the extra c=∑i[qi]-H seats are removed one each from the smallest states.

To investigate proportional consistency, we require the following lemma, whose proof follows from Lemma 1.

Lemma 3

Suppose F(v,H)=h and for some λ<1, the λhi are all integers.

(a)

If [qi]=hi for all i, then ∑i[λqi]=λH.

(b)

If ∑i[qi]=H+c for some c>0, then λH≤∑i[λqi]≤λH+c. If ∑i[qi]=H-c, then λH-c≤∑i[λqi]≤λH. In both cases, if λ≤1/(2k+1) where k satisfies k≥maxi|[qi]-hi|, then ∑i[λqi]=λH.

Lemma 3 shows that any failures of proportional consistency for the NIE or NIS methods occur when ∑i[qi]≠H or when λ≥2k+1. Under the NIS method, [qi]=hi±1, so, Lemma 3 implies F(v,λH)=λh for all λ≤1/(2(1)+1)=1/3. Under the NIE method, as discussed previously, unless n is even and the fractional parts of each quota is 1/2, |∑i[qi]-H|≤⌊(n-1)/2⌋ so F(v,λH)=λh for all λ≤1/(2(⌊(n-1)/2⌋+1)=1/n if n is odd and λ≤1/(n-1) if n is even.

The NIE method strays further from proportionality than the NIS method. However, the NIE method is proportionally consistent while the NIS method is not. (The proofs of Propositions 7 and 9 appear in [4].)

Proposition 7

The NIE method is proportionally consistent.

Proposition 8

The NIS method is not proportionally consistent.

Proof

Let n=8, H=70, and v=(1000,965,965,965,965,965,625,550). Then, F(v,70)=(10,10,10,10,10,10,5,5). If λ=25, then λH=28 and F(v,28)=(4,4,4,4,4,4,3,1), which is not equal to 25F(v,70)=(4,4,4,4,4,4,2,2).

While the NIS method fails proportional consistency generally, it does satisfy proportional consistency for n≤4 and also in part for n=5 (when λ≤1/2).

Proposition 9

Under NIS, if F(vH)=h and there exists a λ<1 such that the λhi are integers, then F(v,λH)=λh if: (i) n≤4; or (ii) n=5 and λ≤1/2.

When 12<λ<1 and n=5, it is not necessarily true that F(v,λH)=λh. For a counterexample, let H=40, λ=45, p=175(14.375,9.35,5.425,5.425,5.425). Then F(v,40)=(15,10,5,5,5) however, F(v,λ40)=(13,7,4,4,4)≠(12,8,4,4,4).

4 Stationary Quotatone Methods

One of the central problems in apportionment theory is the incompatibility of satisfying quota (as does Hamilton’s method) and satisfying monotonicity properties (as do divisor methods). In [2], Balinski and Young define the Quota method which satisfies both house monotonicity and quota. Subsequently, they identified a more general family of apportionment methods—the quotatone methods—which they show uniquely satisfy both properties.

Quotatone methods are defined inductively, in a manner analogous to divisor methods which can be described equivalently by assuming that if H seats are allocated by F(v,H)=h, then the (H+1)st seat is given to the state satisfying maxivif(hi). Quotatone methods require also that the (H+1)st seat be given to an eligible state, where a state is eligible if awarding it the next seat would not: (i) break upper quota; or (ii) break lower quota in the next several induction steps.

To define eligibility more precisely, suppose that H seats have been apportioned with F(v,H)=h. Let U(v,h)={i∣hi<(H+1)viV}. If i∉U(v,h), it follows that hi≥⌈(H+1)viV⌉ or hi+1>(H+1)viV, which implies giving Si seat H+1 would break upper quota. Hence, eligible states must lie in U(v,h).

To define conditions for not breaking lower quota, for each integer α≥1, let Lα(v,h)={i∣⌊(H+α)viV⌋-hi≥1} and let g(α)=∑i∈Lα(v,h)(⌊(H+α)viV⌋-hi).

We consider what it means for g(α)≥α. L1(v,h)={i∣⌊(H+1)viV⌋-hi≥1} consists of all states that must receive a seat in the next step to not break lower quota. If g(1)≥1, there is at least one state in L1(v,h) who must be given the next seat. L2(v,h)={i∣⌊(H+2)viV⌋-hi≥1} consists of all states that must receive another seat in the next two steps to not break lower quota. If g(2)≥2, at least one state in L2(v,h) must be given the next seat. This is because L2(v,h) must contain either: two states for whom ⌊(H+2)viV⌋-hi≥1, who must get 1 more seat in the next two steps; or one state for whom ⌊(H+2)viV⌋-hi≥2, who must get 2 more seats in the next two steps. Similarly, if g(α)≥α, the next seat must be given to a state in Lα(v,h) to not break lower quota in the next α steps.

Let α~=min{α≥1∣g(α)≥α} and let L(v,h)=Lα~(v,h) or, if no such α~ exists, let L(v,h)={1,2,…,n}. Combining the criteria, the set of eligible states is L(v,h)∩U(v,h). In [1, pp. 137–8], Balinski and Young prove: (i) L(v,h)∩U(v,h)≠∅; and (ii) if α~ exists, it satisfies α~≤maxi⌈hi-H(vi/V)vi/V⌉. They show the following.

Proposition 10

([1, Theorem 7.2]) A house monotonic method F satisfies quota if and only it can be defined inductively as follows. For each v, let F(v,0)=0 and, assuming F(v,H)=h, give seat H+1 to some state i∈L(v,h)∩U(v,h).

Chief among quotatone methods are those based on divisor methods.

Definition 11

Given a rounding rule f, its associated quotatone apportionment method assigns seat H+1 to the state in L(v,h)∩U(v,h) that maximizes vif(hi).

The next two propositions show that quotatone methods based on stationary divisor methods and Hill-Huntington’s method are not proportionally consistent. The proof of the first proposition requires the lemma below, the proof of which is in [4].

Proposition 11

Suppose f(k)=k+s is a rounding rule for some s∈[0.5,1] and let F be the corresponding quotatone method. If v=(48569,41012,8200,1115,1095), then F(v,40)=(20,16,4,0,0) but F(v,30)=(15,13,2,0,0)≠34F(v,40).

Lemma 4

Let F and v be as in Proposition 11 with F(v,H)=h. Then h4=h5=0 for all H≤36.

Proof

(Proposition 11) If H=24, then ⌊q⌋=(11,9,1,0,0). Since ∑i⌊qi⌋=21 and h4=h5=0 by Lemma 4, we have F(v,24)=(12,10,2,0,0). Next, since p1h1+s>p32+s if h1≤14 and p2h2+s>p32+s if h2≤12 for all s∈[0.5,1], S3 cannot receive additional seats until h1≥15 and h2≥13. It easily follows that F(v,28)=(14,12,2,0,0), F(v,29)=(15,12,2,0,0) and F(v,30)=(15,13,2,0,0).

Similarly, if H=36, ⌊q⌋=(17,14,2,0,0), implying F(v,36)=(18,15,3,0,0). For H=37,38,39,40, the results are, respectively, the following: (18, 16, 3, 0, 0);  (19, 16, 3, 0, 0); (19, 16, 4, 0, 0); (20, 16, 4, 0, 0).

The next proposition, proved in [4], is a counterexample when λ=4/5 for quotatone methods where f(k)=k+s for s∈[0,0.5] and Hill-Huntington’s method.

Proposition 12

Let f(k)=k+s be a rounding rule for s∈[0,0.5] or f(k)=k(k+1) and F be the associated quotatone method. If v=(57535,56825,4027, 3318, 3295), then F(v,175)=(80,80,5,5,5) but F(v,140)=(64,63,5,4,4)≠45F(v,175).

Concluding remarks. The fact that quotatone methods fail proportional consistency indicates that, despite their positive attributes, these methods have serious flaws. More generally, the behavior of quota methods with respect to proportional consistency suggests that proportional consistency does not capture the “proportionality” of an apportionment method but its “consistency” as H is scaled, since NIE, SUQ and LAR satisfy proportional consistency but NIS, which stays closer to quota, does not. This raises the question whether there is a better-formulated property that captures the idea of an apportionment method getting more proportional as the house size increases.

References

1.

Balinski, M.L., Young, H.P.: Fair Representation: Meeting the Ideal of One Man, One Vote, Brookings Institute, Washington D.C (2001)

2.

Balinski, M.L., Young, H.P.: Quotatone apportionment methods. Math. Oper. Res. 4(1), 31–38 (1979)MathSciNet

3.

Jones, M.A., McCune, D., Wilson, J.M.: Delegate Apportionment in the US Presidential Primaries: A Mathematical Analysis, Studies in Choice and Welfare. Springer, Cham. (2023). https://​doi.​org/​10.​1007/​978-3-031-24954-9

4.

Jones, M.A., McCune, D., Wilson, J.M.: Proportional consistency of apportionment methods (2023). arXiv:​2311.​06969 [math.GM]

5.

Pukelsheim, F.: Proportional Representation: Apportionment Methods and Their Applications, 2nd edn. Springer, Cham (2017)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_41

Observer-Based Adaptive Robust Fractional Order Control of a Class of Nonlinear Delayed Constrained Systems with Singularity

Ladan Khoshnevisan1   and Xinzhi Liu1  

(1)

Applied Math Department, Faculty of Mathematics, University of Waterloo, Waterloo, Canada

Ladan Khoshnevisan

Email: lkhoshnevisan@uwaterloo.ca

Xinzhi Liu (Corresponding author)

Email: xzliu@uwaterloo.ca

Abstract

Control input signals are usually transferred to real applications with an inevitable transmission delay. On the other hand, most of the nonlinear models, which are compatible with industrial processes, include constrained inputs and singular control gains. Therefore, designing a robust control approach, which is compatible with limited inputs and can overcome singular gain, input delay and external disturbances is crucial. In this study, an observer-based adaptive fractional order sliding mode control (OBAFOSMC) is proposed for a class of nonlinear delayed systems with singular control gain and external disturbances. To this end, firstly an adaptive fractional order sliding mode observer is designed to estimate the virtual disturbance induced by the singularity in the system, through which the estimation error converges to zero exponentially fast. Then, an adaptive fractional order sliding mode control is designed for the predicted nonlinear system to control the states to desired values exponentially. To the authors’ knowledge, this is the first time that an adaptive fractional order sliding mode control is proposed for a nonlinear delayed system with constrained input signals and singular control gain. Analytical results through Lyapunov theorem demonstrate that the designed method effectively stabilizes the system and avoids chattering phenomenon via adaptive smooth functions. Finally, the simulation results confirm the effectiveness of the analytical attainments.

Keywords

Adaptive robust controlFractional order sliding mode methodNonlinear delayed systemsSingularity

1 Introduction

Fractional order control as an interdisciplinary method, which has started since 1988 by Oustaloup in [1] has attracted the attention of many researchers in the field of engineering and practical applications during recent years. For instance, fractional order controller designing for DC motor speed control [2], economic systems [3], AC microgrid [4], and microgrid power systems [5], are some examples of these applications.

Sliding mode control (SMC), as a robust method to tackle with the external disturbances and uncertainties, which are inevitable issues in real applications, is mostly designed through integer order procedures [6, 7]. However, different applications necessitate a various of system conditions, for which fractional order SMC (FOSMC) provides more designing parameters, and enhances the control design in terms of applicability and less controller constraints [8–10].

Real applications usually suffer from different aspects, such as singular control gain, constraint input signals, and input delay. However, most of the proposed procedures in the field of FOSMC are designed for non-singular control gain [8, 10]. On the other hand, constraint input is usually provided by limiting designing parameters [7, 11], which results in being limited to be applicable to many systems; or by just considering the input signal as a saturation function [9], which is not analytically complete. Many authors have designed procedures suitable for input-delayed systems with linear models [12–15]. Some others have considered input delay systems with nonlinear models, but they have supposed to propose a delayed signal directly, which is inappropriate as the delay is an inherent characteristic of the system [16].

The main contribution of our study is to design an observer-based adaptive fractional order sliding mode control (OBAFOSMC) procedure for nonlinear delayed systems with singular control gain and constrained input signals. To this end, a model-based predictor is designed to obtain a predicted system with no input delay. Then, using such predicted states, a compensation system is proposed through which the system with constrained input vector is transformed to another one with unconstrained input. Furthermore, using the fractional order SMC method alongside designing a virtual disturbance observer, the control objectives are guaranteed to be achieved as well as the singular control gain problem is ensured to be overcome. Stability analysis results and the simulation achievements confirm the effectiveness of the proposed observer-based adaptive fractional order procedure on nonlinear systems with external disturbances, singular control gains, and input constraints. Furthermore, adaptive methods prevent chattering phenomenon, which is a common issue in sliding mode approaches.

The rest of the paper is arranged as follows: The nonlinear delayed system description with constrained input vector, external disturbances, and singular control gain, as well as the predicted and compensated systems are formulated in Sect. 2. Then, the observer-based adaptive fractional order sliding mode control procedure suitable for delayed nonlinear systems with singular control gain is proposed in Sect. 3. Afterwards, simulation results in Sect. 4 show the effectiveness of the proposed procedure and validate the analytical achievements. Finally, Sect. 5 presents some concluding remarks.

2 Nonlinear Delayed System Description

A nonlinear system description is assumed as follows:

X˙1t=N1-gX1t,X1t-τ,tut-τ+d1t,X˙2t=AX1t+N2+d2t,

(1)

where X1t∈Rn×1 and X2t∈Rm×1 are system states, u∈Rl×1 is the input signal vector in which l is the number of input signals, τ is a known constant time delay, N1∈Rn×1 and N2∈Rm×1 are constant vectors, A∈Rm×n is a constant matrix, d1t and d2t are disturbances with the appropriate dimensions. It is also considered that X1 and X2 are available, and gX1t,X1t-τ,t∈Rn×l represents a nonlinear matrix function that satisfies gX1t,X1t-τ,t=gX1t-τ,X1t,t-τ for a known fixed τ. As it is shown through system description (1), time-delayed state is included in the time-varying control gain vector g., which may cause singular control gain. On the other hand, the system is also suffering from input delay, which usually happens in the communication links between the actuators and the systems. In addition, external disturbances, exposed by contextual effects wind, fractions, fading effects and communication errors, are proposed by d1t and d2t, as common challenging factors that deteriorate the system performance. Furthermore, the following relationships are assumed for the disturbance and its derivative:

Ad1t≤ε1,

(2)

d˙2t≤ε2,

(3)

Equation (1) demonstrates a delay differential equation (DDE) with input and state delay. Moreover, the delayed state is included in the input signal gain which may cause singular control gain situation. On the other hand, each element of the input vector is assumed to be limited to the following constraint:

umin≤uit≤umax,fori=1,…,l.

(4)

So, firstly the following predictors are designed to dominate the input delay obstacle to obtain an input-delay-free system [17].

PX1θ=X1t+∫t-τθN1-gX1s,X1s-τ,susds,t-τ≤θ≤t,PX2θ=X2t+∫t-τθAX1s+N2ds,t-τ≤θ≤t.

(5)

where PX1 and PX2 are τ-time units ahead predictors of the states X1 and X2. Replacing t=θ+τ in (1), in the disturbance-free situation, results in

X˙1θ+τ=N1-gX1θ+τ,X1θ,θ+τuθ,X˙2θ+τ=AX1θ+τ+N2.

(6)

Based on the assumption gX1t,X1t-τ,t=gX1t-τ,X1t,t-τ and using the fact that τ is supposed to be a known constant time delay we have

X˙1θ+τ=N1-gX1θ,X1θ+τ,θuθ,X˙2θ+τ=AX1θ+τ+N2.

(7)

Integrating of (7) from θ=t-τ to θ=t and comparing with (5) result in

PX1θ=X1θ+τ,t-τ≤θ≤t,PX2θ=X2θ+τ,t-τ≤θ≤t.

(8)

Therefore, signals PX1 and PX2 represented in (5) are τ-time units ahead predictions of X1 and X2, respectively. Consequently, the stability of PX1 and PX2 implies the stability of X1 and X2 and vice versa [18]. Substituting θ=t in (5) and taking the time derivative results in

P˙X1t=N1-gPX1t,PX1t-τ,tut+d1t,P˙X2t=APX1s+N2+d2t.

(9)

Therefore, the predicted system (9) is the input-delay-free version of (1). Because the input signal is limited, represented by (4), it can be rewritten as follows

uit=umaxuCit>umax,uCitumin≤uCit≤umax,uminuCit<umin.

(10)

where uCi is the ith element of uCt=uC1,⋯,uClT, as the unlimited input vector. For the sake of simplicity, P1 and P2 are represented instead of PX1 and PX2, respectively in the rest of the paper. A compensation system will be designed as follows to deal with the input constraint.

χ˙P1t=-K^χtanhρ^χχP1-gP1t,P1t-τ,tut-uCt-ATχP2t-ηCχP1t,χ˙P2t=AχP1t-ηCχP2t,

(11)

where uCt is assumed as the unsaturated input signal, K^χ and ρ^χ are the adaptive positive parameters, which will be proposed later. Now, we define

ZP1t=P1t-χP1t,ZP2t=P2t-Xref-χP2t.

(12)

in which Xref is the reference constant vector, P1t and P2t are respectively τ-time units ahead predictors of X1 and X2, which are represented by (5). Compensated system is achieved by taking the time derivative of (12) as follows:

Z˙P1t=P˙1t-χ˙P1t=N1+K^χtanhρ^χχP1+ATχP2t+ηCχP1t-guCt+d1t,Z˙P2t=P˙2t-χ˙P2t=AZP1t+N2+ηCχP2t+d2t

(13)

where g stands for gZP1t,χP1t,ZP1t-τ,χP1t-τ,t. Therefore, an input-delay-free system with unsaturated input signal is achieved from the nonlinear system (1) by applying predictor (5). Moreover, it is supposed that

ρ^χt>0.

(14)

In the next section, a stabilizing controller ut will be designed that satisfies constraint relationship (4).

3 Observer-Based Adaptive Fractional Order Sliding Mode Control Procedure

Most of the industrial systems deal with the links between different parts of the system, tackling with communication delays. In addition, singularity in the system description is a common aspect of practical applications due to lack of communication or inherent specifications of the system. Therefore, many practical and industrial applications can be considered, such as vehicular systems, communication networks, and manufacturing applications, to be controlled through the proposed procedure. In this section, an adaptive robust controller designing method is designed based on fractional order predictive method empowered by adaptive sliding mode control for a nonlinear system with singular control gain and saturated input signal to ensure that the output vector (X2t) pursues the reference constant vector (Xref) exponentially fast. Basically, in sliding mode approaches, the procedure tries to control the states of the system to reach the sliding surface. To this end, the following sliding mode surface is assumed for the controller:

SCt=Z^P2t+λCDα-1sigZP2γ,

(15)

where λcϵRm×m is the controller designing matrix, 0<α,γ<1, and sig.γ is a function [19] defined by sigeγ=eγsigne. From the sliding mode control point of view, the unsaturated input vector of system (uCt) consists of an equivalent control vector signal (uCeqt) and a switching signal (uCswt) as uCt=uCeqt+uCswt, through which the system achieves the sliding surface by uCswt and tends to the sliding surface (SCt=0) by uCeqt, which will be designed later. As the delayed state is included in the input signal gain g in (13), singular control gain occurs. As a result, the following modification is considered for the unsaturated input signal in (13) to deal with this problem [17].

uCt=gTggT+ε-1u1t,

(16)

where ε∈Rn×n is a positive definite diagonal designing parameter matrix and u1t is the auxiliary input signal, which will be designed later. Moreover, we have the following equation:

ggTggT+ε-1=ggT+ε-εggT+ε-1=I-εggT+ε-1.

(17)

where I denotes the identity matrix with suitable dimensions. Therefore, substituting (16) into (13) and using (17) result in

Z˙P1t=N1+K^χtanhρ^χχP1+ATχP2t+ηCχP1t-u1t+udt+d1t,Z˙P2t=AZP1t+N2+ηCχP2t+d2t.

(18)

where udt=εggT+ε-1u1t is the virtual disturbance. Furthermore, a positive constant ε3 is supposed to satisfy the following assumption:

Audt≤ε3.

(19)

An adaptive fractional order sliding mode observer is designed as follows, to estimate the virtual disturbance udt.

Z^P1t=N1+K^χtanhρ^χχP1+ATχP2+ηOχP1-u1t+u^dtZ^P2t=AZ^P1t+N2+ηOχP2

(20)

where Z^P1t, Z^P2t and u^dt are the estimation of the real variables. The observer sliding surface is considered as follows

SOt=E^˙ZP2t+λODα-1sigE^ZP2γ

(21)

in which λOϵRm×m is the observer designing matrix, 0<α,γ<1, E^ZP2=ZP2-Z^P2 and sig.γ has been defined previously. The following theorem obtains the virtual disturbance estimation.

Theorem 1

The estimation error E^ZP2 and its derivative tend to zero exponentially fast if the adaptive fractional order predictive sliding mode observer (AFOPSMO) (20) and (21), the virtual disturbance estimation (22) and the adaptive laws (23) and (24) are proposed.

u^dt=A†ηOSOt+K^Otanhρ^OTtSOt+λODαsigE^ZP2tγ

(22)

where λO,α, γ and E^ZP2 are introduced previously, K^O∈Rm×1 and ρ^O∈Rm×1 are the adaptive observer parameters. Furthermore, † stands for the pseudo-inverse of the related non-squared matrices [20], which is A† =ATAAT-1. Moreover, the adaptive laws are considered as follows for the observer parameters K^O and ρ^O.

K^O˙t=SOttanhρ^OTtSOt-ηOK^Ot-KOref,

(23)

ρ^O˙t=-ηOρ^Ot-ρOref

(24)

It is also assumed that

KOref>ε1+ε2+ε3

(25)

Proof

The following Lyapunov function is considered.

VOΨt=12ΨTtΨt=12SOTtSOt+K^Ot-KOrefTK^Ot-KOref+ρ^Ot-ρOrefTρ^Ot-ρOref

(26)

in which Ψt=SOK^Ot-KOrefρ^Ot-ρOrefT. Taking the time derivative of (26), using the virtual disturbance estimation (22), observer adaptive laws (23) and (24), and assumption (25) gives

V˙OΨt≤-ηOSOTtSOt+K^Ot-KOrefTK^Ot-KOref+ρ^Ot-ρOrefTρ^Ot-ρOref=-2ηOVOΨt

(27)

which is negative definite for VOΨt≠0. As a result, the adaptive fractional order sliding mode observer (20) with the observer sliding surface (21), the virtual disturbance estimation (22) and the observer adaptive laws (23) and (24) is asymptotically stable. Moreover, due to the following equation

lim‖Ψt‖→∞VOΨt=∞

(28)

the Lyapunov function VOΨt is radially unbounded. So, the virtual disturbance observer is globally asymptotically stable. Therefore, VOΨt, Ψt, 12ΨTtΨt and V˙OΨt are bounded, which imply that Ψ(t)∈L∞∩L2 and Ψ˙t∈L∞. So, based on the Barbalat’s lemma we have limt→∞Ψt=0 which means that Ψt converges to zero. Furthermore, integrating (27) results in

∫0t1VOtdVO≤∫0t-2ηOdt⇒VOt≤VO0e-2ηOt

(29)

Consequently, the Lyapunov function VOΨt converges to zero exponentially fast and the virtual disturbance observer (20) and (21) with the virtual disturbance estimation (22) is globally exponentially stable. Moreover, it is assured that the adaptive parameters converge to the predefined reference values.

The following theorem gains the stabilizing controller, through which the desired objectives would be obtained.

Theorem 2

If the nonlinear control signal (16) with the auxiliary input signal (30), the controller sliding surface (15), the adaptive laws (31), (32), (33) and (34) that satisfy assumptions (12), (25) and (35), the adaptive fractional order sliding mode observer (20) with the observer sliding surface (21) and the virtual disturbance estimation (22) are applied, exponential stability of the compensated system (18) is guaranteed and the output vector signal is controlled exponentially fast to the desired vector value (Xref).

u1t=A†AN1+AK^χttanhρ^χTtχP1t+1+ηCAχP1t+Au^dt+AATχP2t-ηC2χP2t+λCDαsigEP2tγ+ηCSCt+K^Cttanhρ^CTtSCt

(30)

where χP1 and χP2 are represented through the compensation system (11), 0<α,γ<1, K^C∈Rm×1 and ρ^C∈Rm×1, K^χ∈Rn×1 and ρ^χ∈Rn×1 are the adaptive parameters, EP2=ZP2-Xref, and ZP2 is introduced in (12). Moreover, the adaptive control laws for the controller parameters are as follows:

K^˙Ct=SCttanhρ^CTtSCt-ηCK^Ct-KCref

(31)

ρ^˙Ct=-ηCρ^Ct-ρCref

(32)

K^˙χt=χP1ttanhρ^χTtχP1t-ηCK^χt

(33)

ρ^˙χt=-sgnρ^χt-ρχrefχP1TtgZP1t,χP1t,ZP1t-τ,χP1t-τ,tut-uCt-ηCρ^χt-ρχref

(34)

Moreover, the following assumption is considered:

KCref>ε1+ε2

(35)

The equivalent and the switching control laws are as follows:

uCeqt=gTggT+ε-1uCeq1t

(36)

uCswt=gTggT+ε-1uCsw1t

(37)

Where

uCeq1t=A†AN1+AK^χttanhρ^χTtχP1t+2AχP1t+Au^dt+AATχP2t-χP2t+λCDαsigEP2tγ

(38)

uCsw1t=A†ηCSCt+K^Cttanhρ^CtSCt

(39)

Following analysis steps of the proof of theorem 2 in [19] completes the proof of the above theorem.

4 Numerical Results

In this section, the numerical results obtained by applying the proposed procedure on a system with constrained input and time delay, through Matlab Toolbox (FOMCON) [21], are provided to scrutinize the designed method. In addition, some comparison results with the Predictive Sliding Mode Control procedure proposed in [7], where the input constraint is provided by limiting the designing parameters, are illustrated to show the efficacy of our method. To this end, a network with transmission control protocol and active queue management (TCP/AQM), with the following additive increase multiplicative decrease (AIMD) description is assumed [17, 22]:

X˙1t=1τ-12τX1tX1t-τut-τ+d1t;X˙2t=NτX1t-C+d2t

(40)

where u is the packet drop probability, N and C are constant parameters considered as 50 and 3750, respectively, and the time delay τ is assumed to be equal to 40 ms. Moreover, d1t and d2t stochastic disturbance signals with the following distributions [7]:

d1t=-12τX1tX1t-τ1-1-PBER(t-τ)S

(41)

d2t=-B×log1+SNRh2

(42)

fy=12σ2e-yσ2;y=h2

(43)

where PBER is the bit error rate probability caused by bit errors in the acknowledgement packets, which is considered to be distributed within 1.25×10-7 and5×10-5, each of which is kept for 5 s before jumping to another, and S is the acknowledgement packet, assumed to be equal to 40. Furthermore, B stands for the channel bandwidth, SNR represents the signal to noise ratio and h is the time varying channel gain, which are considered to be 10 dB, 0.5 and Gaussian (0,0.01), respectively. Moreover, h2 is described by the normal distribution (43). The following parameters are considered in the procedure:λO=2, α=0.5,γ=0.5, λC=20 and Xref=100. Applying the proposed procedure on system (40), we have the following results:

It can be inferred from Figs. 1 and 2, the control signals converge to a value through which the desired reference value is achieved by the second state of the system, shown in Fig. 4. Furthermore, it is shown through Fig. 2 that the control signal satisfies that constraints required for the system. On the other hand, as the system is stable, the first state (illustrated in Fig. 3) tends to the operating point, C∗τN=3. Therefore, simulation results illustrate the effectiveness of the proposed procedure. Figure 5 presents the results of the second state, showing the queue length of the TCP/AQM system (40), as simulated through Network Simulator 2 (NS2), which emulates a real communication network. The figure demonstrates that the PSMC procedure proposed in [7] fails to control the system with the specified parameters, highlighting the protocol’s limitations in practical applications. In contrast, our proposed method effectively regulates the network’s queue length to the desired value.

Fig. 1

Unsaturated input signal

Fig. 2

Saturated input signal

Fig. 3

The first state

Fig. 4

The second state

Fig. 5

Comparison results of the second state obtained through NS2

5 Conclusion

In this study, an adaptive robust control procedure is proposed for a nonlinear delayed system with constrained inputs, singular control gain, and external disturbances, which are common issues in practical applications. To this end, firstly, an adaptive fractional order sliding mode observer is designed for the virtual disturbance imposed by the singular control gain. Then, based on this estimation, an adaptive fractional order sliding mode control procedure is designed to control the system states to the desired values. To demonstrate the effectiveness of the proposed procedure, Lyapunov stability analysis is considered through which it is proved that the estimation error tends to zero exponentially and the system states exponentially converge to the desired values. Finally, simulation achievements and comparison results of a TCP/AQM network, through Matlab/Simulink using FOMCON toolbox and via NS2, validate the analytical achievements.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_42

Stability Analysis of Impulsive Neutral Switched Systems

Yingqi Li1   and Xinzhi Liu1  

(1)

Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

Yingqi Li (Corresponding author)

Email: yingqi.li@uwaterloo.ca

Xinzhi Liu

Email: xinzhi.liu@uwaterloo.ca

Abstract

Impulsive switched neutral systems have a broad and profound research significance owing to their rich dynamics and applications in various applied areas such as science, engineering, physics, and biology. This paper studies the stability problem of nonlinear impulsive switched neutral systems. Based on the common Lyapunov function method and a state-dependent switching rule, sufficient conditions for asymptotic stability are established for nonlinear impulsive switched neutral systems, where impulses and switchings are assumed to occur at the same time. It should be noted that a suitable sequence of impulses and switchings may be used to stabilize unstable neutral systems. Examples are given to illustrate the theoretical results.

Keywords

StabilityImpulsive neutral switched systemsCommon Lyapunov functionState-dependent switching ruleLinear convex combination

Research supported by NSERC Canada.

1 Introduction

In ordinary differential equations, the future states of the systems are assumed to be independent of the past states. However, we know time-delay commonly appears in various real-world phenomena, for example, drivers need reaction time to brake after seeing the vehicles in front slowing down on highways; it takes some time for the drug concentration in the blood to reach a certain level after the patient takes the pills. For the sake of more accurate analysis, it is crucial to consider time-delay in the problems dependent to the past when we are building mathematical models. Neutral differential equations are a class of time-delay systems which depend on not only the past and present values of the state variable but also their past and present derivatives. Some fundamental theorems and properties of neutral systems can be found in [1]. Neutral systems are important in many application fields, namely distributed networks containing lossless transmission lines, control of epidemics, continues induction heating of a thin moving body and population dynamic models. On the other hand, impulsive effects and parameter switching occur frequently in many physical systems. Consequently, there has been an increasing interest in the study of impulsive switched systems.

Recently, there are many results related to stability of neutral systems. Sanchez et al. [10] addressed the practical stabilization of operating points for switched affine systems by using a hybrid systems approach. Some hybrid control logic in switched neutral systems can be seen in [9, 17]. By applying Lyapunov-Krasovskii functionals, Wang [13] and Li [4] investigated some stability results of switched neutral systems. Conditions of existence and stability of some specific switched neutral systems have been presented (see [6, 7, 14–16]). Criteria for asymptotic stability of a type of homogeneous linear impulsive delay control neutral-type systems were established by Stamova et al. [12]. Some exponential stability results of a nonlinear impulsive neutral differential equation with fixed impulsive moments were attained by Xu et al.; see [18]. Yeniçerioğlu [20] obtained some stability results of linear impulsive neutral delay differential equations. More results about impulsive switched neutral systems can be seen in [2, 8].

In this paper, we investigate the stability problem for a class of impulsive switched neutral systems. The remaining of this paper is organized as follows. In Sect. 2, we introduce some definitions and lemmas. The nonlinear switching impulsive neutral systems and a state-dependent switching rule will be defined. In Sect. 3, we establish some sufficient stability conditions of the nonlinear impulsive switched neutral systems. In Sect. 4, we provide several numerical examples to illustrate the role of impulses and switching in stabilizing unstable neutral systems.

2 Preliminaries

Let R+ denote the set of nonnegative real numbers, Rn the n-dimensional real space equipped with the Euclidean norm ||·||, N+ the set of positive integers, i.e., N+={1,2,…}. Define ψt+=lims→t+ψ(s) and ψt-=lims→t-ψ(s). For a,b∈R with a<b and for S⊂Rn, we define a class of functions PC([a,b],S)=ψ:[a,b]→S∣ψ(t)=ψt+;ψt-existsinS,andψt-=ψ(t) for all but at most a finite number of points, ∀t∈[a,b). Define ‖ϕ‖τ=sup-τ≤θ≤0‖ϕ(θ)‖. Given t0∈R+, ϕ∈PC([-τ,0],Rn), and a sequence i0,t0,i1,t1,…,ij,tj,…, we consider the following impulsive switched neutral system,

x˙(t)=Aikx(t)+Bikx(t-τ)+Cikx˙(t-τ)+fikt,x,xt,x˙t,t∈(tk-1,tk],xtk+=Dikxtk,t=tk,k=1,2,…,

(1)

where x˙(t) represents the upper right-hand derivative of x(t), x∈PC(R+,Rn), for τ>0, xt∈PC[-τ,0],Rn is defined by xt(s)=x(t+s) for -τ≤s≤0, Aik,Bik,Cik,Dik∈ Rn×n, ik∈{1,2,…,N}, k,N∈N+, and for any t∈Jt0, fik∈PCJt0×Rn×PC([-τ,0],Rn) ×PC([-τ,0],Rn),Rn. Let x(t)=x(t,t0,φ) be a solution of system (1) subject to the initial condition xt0=φ defined for all t≥t0. We assume f(t,0,0,0)=0 so that system (1) admits the trivial solution, i.e. x≡0 is a solution.

Definition 1

The trivial solution of system (1) is said to be

(a)

stable if for every ε>0 there exists some δ=δε>0 such that if φ∈PC([-τ,0],Rn) with ‖φ‖τ<δ, and xt,φ<ε, for all t≥t0;

(b)

asymptotically stable if (a) holds and there exists some λ>0 such that for every ε>0, there exists some γ=γ(λ,ε)>0 such that if φ∈PC([-τ,0],Rn) with ‖φ‖τ<λ, then xt,φ<ε for t≥t0+γ.

It should be noted that system (1) is a multi-mode system, where the ith (i=1,2,…,N) mode is the subsystem

x˙(t)=Aix(t)+Bixt-τ+Cix˙(t-τ)+fit,x,xt,x˙t,t∈tk-1,tk,xtk+=Dixtk,k∈N+.

(2)

We make the following assumption.

Assumption 1

There exists a linear convex combination U=∑i=1Nαi(I-Ci)-1Ai, where αi∈(0,1) with ∑i=1Nαi=1, such that all the eigenvalues of matrix U have negative real parts (i.e., U is a Hurwitz matrix).

Then for a positive definite matrix Q∈Rn×n, there exists a unique symmetric positive definite matrix P∈Rn×n such that

UTP+PU=∑i=1Nαi(AiT((I-Ci)-1)TP+P(I-Ci)-1Ai)=-Q.

(3)

Construct the region Ωi as

Ωi=x∈Rn:xT(AiT((I-Ci)-1)TP+P(I-Ci)-1Ai)x≤-1ξ∗xTQx,

(4)

where i=1,2,…,N, ξ∗≥1 is some constant and P,Q are defined in Eq. (3).

Lemma 1

Rn=⋃i=1NΩi, where Ωi is defined in (4).

Proof

Apply the method in [13]. Suppose the result is not true, then there must exist some nonempty region S⊂Rn such that S=Rn\⋃i=1NΩi. Thus for any x∈S, we have xT(AiT((I-Ci)-1)TP+P(I-Ci)-1Ai)x>-1ξ∗xTQx,∀i=1,2,…,N.

By Eq. (3), we obtain

∑i=1NαixT(AiT((I-Ci)-1)TP+P(I-Ci)-1Ai)x=-xTQx>-∑i=1Nαi1ξ∗xTQx=-1ξ∗xTQx,∀x∈S,

which is a contradiction since ξ∗≥1, and hence the proof is complete.

Minimum rule. At each switching we make the choice of the next mode according to

j(x)=mini∈{1,2,…,N}xT(AiT((I-Ci)-1)TP+P(I-Ci)-1Ai)x.

(5)

Switching rule S.

(1)

Switch only when x hits the boundary of the current switching region.

(2)

Choose each mode (including the first mode) through Minimum Rule.

Remark. We can adjust the value of ξ∗ to allow those switching regions overlapping each other to avoid a sliding motion or chattering behavior (see [5, 11]).

We shall need the following lemmas.

Lemma 2

([3], Theorem 1.5.1) Assume that

A0 the sequence tk satisfies 0≤t0<t1<t2<…, with limk→∞tk=∞;

A1 m∈PC1R+,R and m(t) is left-continuous at tk,k=1,2,….

Assume further that p∈CR+,R+, m(t)≤C+∫t0tp(s)m(s)ds+∑t0<tk<tβkmtk,t≥t0, where k=1,2,…, and βk≥0, C are constants. Then,

m(t)≤C∏t0<tk<t1+βkexp∫t0tp(σ)dσ,t≥t0.

Lemma 3

([19]) Given a positive definite matrix P∈Rn×n, any symmetric matrix Q∈Rn×n and x∈Rn, then λminP-1QxTPx≤xTQx≤λmaxP-1QxTPx.

3 Main Results

In this section, we present our main results. Let us begin by proving two lemmas.

Lemma 4

Let x(t)=x(t,t0,φ) be a solution of system (1) for t≥t0. Assume that there exists a constant M>0 such that

‖φ‖τ<M,andsup0≤t≤τfit,x(t),xt,x˙t<M,i=1,2,…,N.

(6)

Then, for 0<t≤τ, the inequality ‖x(t)‖≤M(1+2c+τb+τM)ek∗ln(1+d)+aτ is valid, where a=maxi∈{1,2,…,N}Ai,  b=maxi∈{1,2,…,N}Bi, c=maxi∈{1,2,…,N}Ci, d=maxi∈{1,2,…,N}Di.

Proof

For t∈tk-1,tk, without loss of generality, we assume that the ith mode is switched on. Let fi(s) denote fis,x(s),xs,x˙s. Then system (2) is equivalent to the following integral equation

x(t)=φ(0)+Ci[x(t-τ)-φ(-τ)]+∫0t[Aix(s)+Bix(s-τ)+fis]ds+∑0≤tk<tDixtk.

(7)

By (7) we have, for 0<t≤τ,

‖x(t)‖≤‖φ‖τ+2c‖φ‖τ+∫0ta‖x(s)‖ds+τb‖φ‖τ+τsup0≤t≤τ‖fi(t)‖+∑0≤tk<td‖xtk‖.

By (6), we gain ‖x(t)‖≤M(1+2c+τb+τM)+∫0ta‖x(s)‖ds+∑0≤tk<td‖xtk‖. By Lemma 2, we have when 0<t≤τ,

‖x(t)‖≤M(1+2c+τb+τM)∏0≤tk<t(1+d)eaτ=M(1+2c+τb+τM)ek∗ln(1+d)+aτ,

here k∗ is the number of impulsive moments when 0≤tk<t.

Lemma 5

Assume conditions of Lemma 4 hold, c=maxi∈{1,2,…,N}Ci<1, then for t>τ,t≠tk,k=1,2,…, the following estimate is valid

‖x˙(t)-x˙(t-τ)‖≤sups∈0,τ‖x˙(s)-x˙(s-τ)‖+2q(a+b)+2M1-csups∈-τ,t‖x(s)‖.

(8)

Proof

Denote fit,x(t),xt,x˙t as fit. For t>0,t≠tk, we have the solution of system (1) x˙(t)=Cix˙(t-τ)+Aix(t)+Bix(t-τ)+fi(t), and t>τ,t≠tk, we have x˙(t-τ)=Cix˙(t-2τ)+Aix(t-τ)+Bix(t-2τ)+fi(t-τ). Therefore,

‖x˙(t)-x˙(t-τ)‖≤‖Ci‖‖x˙(t-τ)-x˙(t-2τ)‖+‖Ai‖‖x(t)-x(t-τ)‖+‖Bi‖‖x(t-τ)-x(t-2τ)‖+‖fi(t)-fi(t-τ)‖,t>τ,t≠tk.

Let In:=[nτ,(n+1)τ], b0:=sups∈0,τ‖x˙(s)-x˙(s-τ)‖, bn:=sups∈In‖x˙(s)-x˙(s-τ)‖,n∈N+, q:=sup-τ≤s≤t‖x(s)‖,

wehave‖x˙(t)-x˙(t-τ)‖≤bn≤cbn-1+[2q(a+b)+2M]≤cnb0+[2q(a+b)+2M]1-cn1-c<b0+2q(a+b)+2M1-c,byc=maxi∈{1,2,…,N}Ci<1.

Theorem 1

Assume that

(i)

Assumption 1 and conditions of Lemmas 4 and 5 hold;

(ii)

Equation (3) holds with positive definite matrices P and Q;

(iii)

the following inequalities are satisfied:

maxi∈{1,2,…,N}λmax(P-1DiTPDi)≤1,

(9)

maxi∈{1,2,…,N}‖PBi~‖+maxi∈{1,2,…,N}‖PCi~‖a+2q(a+b)+2M1-c+Mαλmin(P)≤141ξ∗λmin(Q)λmin(P)λmax(P).

(10)

Then the trivial solution of system (1) is asymptotically stable.

Proof

The common Lyapunov function method was applied to analyze the stability of system (1). Define the common Lyapunov function

V(x)=xTPx,forallx∈Rn,

(11)

where P is defined in Eq. (3). To prove that the trivial solution of system (1) is stable, i.e. for ∀ε>0, we shall show that there exists a constant δ=δ(ε)>0 such that if φ∈PC[[-τ,0],Rn],‖φ‖r<δ then ‖x(t)‖r<ε, for all t>0.

Step 1. To prove x(t)=x(t,φ)∈vα,for0≤t≤τ. By Lemma 4, for 0<t≤τ we have

‖x(t)‖≤δ(1+2c+τb+τM)ek∗ln(1+d)+aτ.

(12)

By Lemma 3, we have λmin(P)‖x‖2≤V(x)≤λmax(P)‖x‖2, for all x∈Rn. Let α>0,α∈Rn, define vα=x∈Rn:V(x)<α,∂vα=x∈Rn:V(x)=α. Choose δ>0 such that

δ(1+2c+τb+τM)ek∗ln(1+d)+aτλmax(P)≤α.

(13)

Since ‖φ‖τ<δ, and by (12) and (13), we can obtain x(t)=x(t,φ)∈vα,fort0≤t≤τ.

Step 2. We shall prove x(t)=x(t,φ)∈vα,fort>τ. By (9) and Lemma 3, we have

V(x(tk+))=x(tk)TDiTPDix(tk)≤maxi∈{1,2,…,N}λmax(P-1DiTPDi)x(tk)TPx(tk)≤V(x(tk)).

(14)

That means x(t) cannot leave vα by jumps.

Now, we shall prove x(t)=x(t,φ)∈vα,fort>τ, by contradiction. Assume there exists a T>τ, T≠tk, k=1,2,… such that x(T)∈∂vα and x(t)∈vα, for τ≤t<T. Transform system (2) into the following form

(I-Ci)x˙(t)=Ci[x˙(t-τ)-x˙(t)]+Aix(t)+Bixt-τ+fi(t),t∈tk-1,tk,xtk+=Dixtk,t=tk,

and let Ai~=(I-Ci)-1Ai, Bi~=(I-Ci)-1Bi, Ci~=(I-Ci)-1Ci, and fi~=(I-Ci)-1fi(t), then we have x˙(t)=Ci~[x˙(t-τ)-x˙(t)]+Ai~x(t)+Bi~xt-τ+fi~, t∈tk-1,tk. By the fact of (4), we have

V˙(x(t))≤-1ξ∗λmin(Q)‖x(t)‖2+2maxi∈{1,2,…,N}‖PBi~‖‖x(t-τ)‖‖x(t)‖+2maxi∈{1,2,…,N}‖PCi~‖‖x(t)‖‖x˙(t-τ)-x˙(t)‖+2maxi∈{1,2,…,N}‖xTPfi~‖.

By Lemma 3 and V(x)=α at t=T, we have λmin(P-1)xTPx≤xTx≤λmax(P-1)xTPx, thus

-‖x‖≤-λmin(P-1)xTPx≤-αλmax(P);‖x‖2≤V(x)λmax(P)-1≤αλmin(P).

(15)

Let t=T, by Lemma 5 we have

V˙(x(T))≤‖x(T)‖-1ξ∗λmin(Q)αλmin(P)+2maxi∈{1,2,…,N}‖PBi~‖αλmin(P)+‖PCi~‖sups∈0,τ‖x˙(s)-x˙(s-τ)‖+2q(a+b)+2M1-cαλmin(P)+‖Pfi~‖.

(16)

We have

sups∈0,τ‖x˙(s)-x˙(s-τ)‖≤sups∈0,τ(‖Ai‖‖x(s)‖+‖Bi‖‖x(s-τ)‖+‖Ci‖‖x˙(s-τ)‖+‖fi‖+‖x˙(s-τ)‖)≤(1+c+b)δ+aαλmin(P)+M.

(17)

Combine (16) and (17),  we have

V˙(x(T))≤‖x(T)‖-1ξ∗λmin(Q)αλmax(P)+2maxi∈{1,2,…,N}‖PCi~‖(1+c+b)δ+‖PBi~‖αλmin(P)+‖PCi~‖a+2q(a+b)+2M1-cαλmin(P)+M.

(18)

We apply (10) and obtain

V˙(x(T))≤‖x(T)‖-121ξ∗λmin(Q)αλmax(P)+2maxi∈{1,2,…,N}‖PCi~‖(1+c+b)δ.

(19)

Choose δ>0 such that δ<λmin(Q)α4ξ∗λmax(P)maxi∈{1,2,…,N}‖PCi~‖(1+c+b), then we have V˙(x(T))<0. Thus there exists a δ∗ such that V(x(T-δ∗))>V(x(T))=α,0<T-δ∗<T. On the other hand, we have V(x(T-δ∗))∈Vα, then V(x(T-δ∗))<α, which is a contradiction. Therefore, x(t,φ)∈vα for all t>0.

Let α≤λmin(P)ε2, then we have ‖x(t,φ)‖<ε for all t>0, which means the trivial solution of system (1) is stable.

Step 3. We shall prove there exists some λ>0 such that for every ε>0, there exists some γ=γ(λ,ε)>0 such that if φ∈PC([-τ,0],Rn) with ‖φ‖r≤λ, then xt,φ≤ε for t≥t0+γ.

Choose δ=δ(ε)>0 such that δ<λmin(Q)α4ξ∗λmax(P)maxi∈{1,2,…,N}‖PCi~‖(1+c+b)

-λmin(Q)α-αλmax(P)4ξ∗λmax(P)maxi∈{1,2,…,N}‖PCi~‖(1+c+b), then let t=T in (19) and by (15), we have

V˙(x(t))<-λmin(P)‖x(t)‖2,t>t0,t≠tk.

(20)

Choose η=η(ε) such that λmax(P)η2<λmin(P)ε2, and γ=γ(λ,ε) such that γ>λmax(P)λ2λmin(P)η2. Assume ‖x(t,φ)‖≥η for t∈[0,η], then by (20) we have V(x(t))≤V(φ(0))-∫0tλmin(P)‖x(s)‖2ds≤λmax(P)λ2-λmin(P)η2γ<0, which contradict with the definition of V(x(t)). Hence, there exists a t∗∈[0,η] such that ‖x(t∗,φ)‖<η. Thus, λmin(P)‖x(t)‖2≤V(x(t))≤Vxt∗≤λmax(P)xt∗2 <λmax(P)η2<λmin(P)ε2, which implies ‖x(t)‖<ε for all t>γ.

4 Examples and Simulations

Example 1

(Switching between stable and unstable system matrices)

Consider the impulsive neutral switching system given by Eq. (1) with n=2, t0=0, φ∈PC[-1,0],R, τ=1, tk=k for k∈N+, φ=e-t,cos(t) and

A1,B1,C1,D1=0.01-0.20.1-0.4,0.0010.0010.003-0.002,0.0040.00020.00030.00020.10.2-0.1-0.2,A2,B2,C2,D2=0.02-0.30.40.2,0.0020.0010.001-0.003,0.00020.00010.00010.004-0.30.20.03-0.2.

We denote this system as (∗). It is not difficult to check that system matrix (I-C1)-1A1 has negative real eigenvalues λ1=-0.0468,λ2=-0.3433, which is stable and (I-C2)-1A2 is unstable, since its eigenvalues are λ1,2=0.1104±0.3352i which have positive real parts. Let α1,α2=0.5, then U=α1(I-C1)-1A1+α2(I-C2)-1A2=0.0151-0.25100.25015.9554, and has eigenvalues λ1,2=-0.0425±0.1853i. Equation (3) holds since all eigenvalues of U have negative real parts.

Choose Q=0.5I in Eq. (3), here I is a 2×2 matrix, then the symmetric positive definite matrix P is given by P=6.44891.38411.38414.7955, with λmin(P)=4.7962 and λmax(P)=7.6080.

Choose ξ∗≥1, this gives

Ω1=x=(x1,x2)∈Rn:2x12+37x1x2+21x22≤52(x12+x22),

Ω2=x=(x1,x2)∈Rn:5x12-3x1x2-7x22≤-52(x12+x22). These switching regions are illustrated in Fig. 1a.

We have maxi∈{1,2}λmax(P-1DiTPDi)=0.16736≤1,which satisfies condition (9) in Theorem 1. c=maxi∈{1,2}Ci=0.004017<1 satisfies the condition of Lemma 5. maxi∈{1,2}‖PBi~‖+maxi∈{1,2}‖PCi~‖a+2q(a+b)+2M1-c+Mαλmin(P)-141ξ∗λmin(Q)λmin(P)λmax(P)=-0.036946<0 satisfies the condition (10) of Theorem 1. The simulation of system (∗) under the switching rule S with initial function φ=e-t,cos(t) for t∈[-1,0] is given in Fig. 1b, which shows the asymptotically stability of the trivial solution of switching system (∗) with switching rule S.

Fig. 1

Simulation results of system (∗). a Switching regions, the shaded areas are overlapping regions; b t-x graph

5 Conclusion

We have proposed a type of nonlinear neutral system with impulses occurring at fixed moments and a state-dependent switching rule. Using the common Lyapunov function method, we have established sufficient stability conditions. Notably, we discovered that impulses and switching could have a stabilizing effect on unstable neutral systems. We can see from the proof of Lemma 5 that the key point to estimate the neutral term x˙(t-τ) is the assumption c=maxi∈{1,2,…,N}Ci<1. A future research direction is to consider loosening or even removing this constraint. In this article, we examined fixed time delays in system states and linear autonomous impulses. Another future research topic is to consider time-varying delays in both system states and general non-autonomous impulses.

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Wang, Q., Liu, X.: Stability criteria of a class of nonlinear impulsive switching systems with time-varying delays. J. Frankl. Inst. 349(3), 1030–1047 (2012)MathSciNet

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Wang, Y.E., Zhao, J., Jiang, B.: Stabilization of a class of switched linear neutral systems under asynchronous switching. IEEE Trans. Autom. Control 58(8), 2114–2119 (2013). https://​doi.​org/​10.​1109/​TAC.​2013.​2250076MathSciNetCrossref

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_43

pythOS: An Operator-Splitting Library in Python

Mohammad Mahdi Moayeri1  , Victoria Guenter1  , Siqi Wei2   and Raymond J. Spiteri1  

(1)

Department of Computer Science, University of Saskatchewan, Saskatoon, Canada

(2)

Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada

Mohammad Mahdi Moayeri

Email: m.moayeri@usask.ca

Victoria Guenter

Email: v.guenter@usask.ca

Siqi Wei

Email: siqi.wei@usask.ca

Raymond J. Spiteri (Corresponding author)

Email: spiteri@cs.usask.ca

Abstract

Operator-splitting (OS) methods are powerful and popular divide-and-conquer methods used to solve differential equations. Implementations of OS methods, however, are rarely performed in a general-purpose way. This paper introduces pythOS, a Python library designed to facilitate the systematic implementation, exploration, and evaluation of OS methods. Two examples are presented to demonstrate the library’s functionality and versatility: a model of neutron transport in a high-scattering medium and the monodomain model of cardiac electrophysiology. Both examples employ different OS methods and numerical sub-integration techniques to demonstrate some of pythOS’s capabilities. The results show the utility of pythOS as a general-purpose tool for researchers and practitioners for the time integration of differential equations using OS methods.

Keywords

Operator splittingFractional-step methodsAdditive Runge-Kutta methodsMulti-rate methodsMulti-rate infinitesimal methods

1 Introduction

Large complex systems modeled by differential equations typically require computationally efficient time integration methods in order to obtain solutions in an acceptable amount of time. A common approach to achieving such solutions is via operator splitting (OS); i.e., the differential equation is decomposed into various operators, which are then treated separately with their own numerical methods. This approach is especially appealing for models composed of various sub-components from different physical processes, such as advection-diffusion-reaction systems, because specialized solvers can be used for the individual operators, thus offering the potential for improved computational efficiency. Despite their popularity in practice, OS methods are typically implemented in an ad hoc manner. There is no software library of which we are aware that allows the user to experiment with OS methods in a systematic manner.

The concept of OS methods dates back to Lie in 1875 [12], with significant developments by Trotter and Godunov in 1959 [9, 19], and later by Strang [18] and Marchuk [14] in 1968, both of whom proposed a second-order method. OS methods of order greater than two, however, remain less common, perhaps partially due to their increased complexity and lack of intuitive appeal. But perhaps more importantly, such methods require backward time steps in each operator, potentially leading to instability for certain processes like diffusion. Accordingly, higher-order OS methods have mainly been considered only for time-reversible flows (e.g., [15]) or with complex time steps (e.g., [4]). Recent advances have addressed some of these challenges, demonstrating the viability of higher-order methods in specific contexts [5–7].

In this paper, we introduce pythOS, a Python library that facilitates the systematic implementation, exploration, and evaluation of OS methods. It offers a range of high-order and complex-coefficient OS methods. Moreover, pythOS supports spatial discretization with finite element (FEM) methods via compatibility with tools such as Firedrake [11] for high-performance simulations. pythOS is publicly available at: https://​github.​com/​uofs-simlab/​pythOS.

The remainder of this paper is structured as follows. Section 2 provides some background on OS methods. Section 3 describes the pythOS implementation. Section 4 presents examples including a model of neutron transport in a high-scattering medium and the monodomain model of cardiac electrophysiology. Section 5 concludes the paper.

2 Background

We consider an initial-value problem (IVP) for a system of ordinary differential equations (ODEs),

U˙=F(t,U),U(t0)=U0,t∈[t0,tf].

(1)

We assume the right-hand side of Eq. (1) is split into Nop additive components,

U˙(t)=∑k=1NopF[k]t,U[k],U(t0)=U0,t∈[t0,tf],

(2)

with each F[k] acting on a relevant subset U[k] of U(t).

An s-stage operator-splitting method is characterized by a set of coefficients {αj[k]}j=1,2,⋯,sk=1,2,⋯,Nop. At each stage j, we solve the sub-system

U˙=F[k]t,U[k],k∈{1,2,⋯,Nop},

(3)

over a time interval of length αj[k](tn+1-tn) successively using the solution from the previous sub-system integration as an initial condition. Each sub-system Eq. (3) is solved using a sub-integration method Φj[k].

In summary, OS methods involve a set of operators, each applied in a specified order, with coefficients determining their sub-integration intervals, and methods associated with each sub-integration. To achieve implementational simplicity, we adopt a sparse representation of OS methods, characterizing them as sequences of operator-coefficient-integrator triplets for non-zero sub-integrations. Thus, we consolidate this information into a list defining the OS method’s execution,

Ψ=F[kℓ],Φℓ,αℓℓ=1nsub,

(4)

where kℓ is the active operator (with non-zero sub-integration coefficient) and nsub is the total number of non-zero sub-integrations. We note that the user only needs to input the OS coefficients {αj[k]}j=1,2,⋯,sk=1,2,⋯,Nop, and pythOS will convert it to the sparse representation Eq. (4). The algorithm for executing a single step as outlined above is given in algorithm 1.

Algorithm 1

Algorithm for taking a single step for a general operator-splitting method.

3 Implementation Details

3.1 The Principal Function

The pythOS library can be imported as import operator_splitting as os. The principal function, operator_splitting, takes inputs such as a list of operators, initial values, initial time t0, final time tf, the OS method, and sub-integration methods. Operators should be provided in a list and, when using Firedrake, t0 should be a Constant type. The OS method is specified as a string. Predefined methods are stored in a dictionary called alphas, e.g., alphas[‘Strang’] = [[0.5,1], [0.5,0]]. Users can easily add new OS methods to alphas.

In pythOS, sub-integration methods are defined in a dictionary named methods that allows for the assignment of a sub-integrator to an active operator. The forward Euler method is used as the default sub-integrator when no specific sub-integrator is specified in the methods dictionary. This default sub-integrator can be redefined using the dictionary key (0,). A sub-integrator can be assigned for operator k globally using the dictionary key (k,); a sub-integrator for sub-integration j of operator k can be assigned using (k,j). Section 3.3 describes the predefined sub-integration methods available in pythOS. When using Firedrake, operator_splitting can also accept boundary conditions and Firedrake solver parameters.

For example, using Forms F1 and F2 for the operators, Function y_0 and DirichletBC bc for initial and boundary conditions, respectively, the call for the Godunov OS method with forward and backward Euler sub-integrators using pythOS is as follows:

result = os.operator_splitting([F1,F2], dt, y_0 , t0, tf,

“Godunov”, methods={(1,): “FE”, (2,): “BE”} , bc = bc)

3.2 OS Methods

There are a number of different OS methods pre-defined in pythOS. All of the methods used in this paper can be found in the web-based resource [3], which is associated with Auzinger et al. [2].

First, we consider the following classical real-valued 2-splitting OS methods: the first-order Godunov (Lie–Trotter) method [9, 19] (Godunov), the second-order Strang (Marchuk) method [14, 18] (Strang), the third-order Ruth method [17] (R3), and the fourth-order Yoshida method [20] (Y4). The definitions for these methods are found in Fig. 1.

Fig. 1

Sub-integration coefficients for 2-splitting OS methods: a Godunov (Godunov), b Strang (Strang), c Ruth (R3), and d Yoshida (Y4) (θ=1/(2-23))

We also consider complex-valued 2-splitting OS methods: a second-order Milne¹ method [2] (Milne22_C_i) and a third-order method from a palindromic pair [2] (A33_C). The definitions for these methods are found in Fig. 2.

Fig. 2

Sub-integration coefficients for complex-valued 2-splitting methods from Auzinger et al. [1, 2]: a second order (Milne22_C_i) and b third order (A33_C)

3.3 Sub-integrators

There are also a variety of sub-integrators pre-defined in pythOS, including implicit and explicit Runge–Kutta (RK) methods. The RK methods used in this study are shown in Fig. 1. Further discussion of RK methods, in particular for stiff problems, can be found in [10]. There is also an option to use an EXACT solution in terms of a closed-form expression or a highly accurate numerical solution.

Table 1

Runge–Kutta sub-integrators. The first-order methods are backward Euler (BE) and forward Euler (FE). The second-order methods are a two-stage, L-stable SDIRK method (SD2O2) and Heun’s method (Heun). The third-order methods are a three-stage, L-stable SDIRK method (SD2O3) and a three-stage explicit RK method (RK3). The fourth-order methods are a five-stage, L-stable SDIRK method (SD5O4) and the classical explicit RK method (RK4)

Order

Implicit method

Explicit method

1

BE:

111

FE:

01

2

SD2O2:

γγ11-γγ1-γγ,γ=1-1/2

Heun:

0111/21/2

3

SD2O3:

λλ(1+λ)/2(1-λ)/2λ1b1(λ)b2(λ)λ1b1(λ)b2(λ)λ,λ=0.4358665215b1(λ)=-3λ2+4λ-1/4b2(λ)=-3λ2/2-5λ+5/4

RK3:

01/21/21-121/62/31/6

4

SD5O4:

1/41/43/41/21/411/2017/50-1/251/41/2371/1360-137/272015/5441/4125/24-49/48125/16-85/121/425/24-49/48125/16-85/121/4

RK4:

01/21/21/201/210011/61/31/31/6

4 Numerical Examples

In this section, we present two examples to demonstrate the functionality and versatility of pythOS in the context of standard convergence studies. The order of convergence is approximated using the errors for two different time-step sizes,

order(Δt1,Δt2)=logerror(Δt1)/error(Δt2)logΔt1/Δt2.

The methods tested are described as the OS method and the sub-integration methods. For example, Godunov-BE-FE is the Godunov OS method with BE applied to operator 1 and FE applied to operator 2.

4.1 Neutron Transport Equation

We consider a model for neutrons moving in a high-scattering medium. The medium is represented as a two-dimensional domain x∈Ω=[0,5]2 (with boundary ∂Ω) for time interval t∈[0,tf] with tf=10. Assuming a non-fissile medium, the neutron flux on Ω×[0,tf] satisfies the partial differential equation (PDE),

1v∂ϕ∂t=∇·(D∇ϕ)-Σaϕ+S(t,x),

(5)

where ϕ(t,x) is the neutron flux, v is the neutron velocity, D is the diffusion coefficient, Σa is the absorption cross section, and S(t,x) is a source term. Equation (5) is subject to mixed boundary conditions (BCs) on ∂Ω×[0,tf] as two disjoint parts of the boundary ∂Ω=∂ΩD∪∂ΩN. We use the following Dirichlet boundary condition on ∂ΩD×[0,tf],

ϕ=ϕD(t,x)|x∈∂ΩD,

and the following Neumann boundary condition on the boundary ∂ΩN×[0,tf],

(D∇ϕ)·n|x∈∂ΩN=ϕN(t,x)|x∈∂ΩN,

where ϕD(t,x) and ϕN(t,x) are specified functions on their respective sub-domains and n is the unit outward normal vector on ∂Ω. The initial condition is

ϕ(0,x)=ϕ0(x)|x∈Ω=0.

The functions ϕD(t,x) and ϕN(t,x) are zero on ∂ΩD=(0,y),(5,y)y∈[0,5] and ∂ΩN=(x,0),(x,5)x∈[0,5], respectively. The problem has a diffusion coefficient D=130, a scattering cross-section Σa=1.0, and neutron velocity v=1.0. The source term is

S(t,x)=10ωvbx-x2cosωt+Σabx-x2+2Dsinωt,

with frequency ω=π10, leading to the exact solution

ϕ(t,x)=Asin(ωt)(bx-x2).

(6)

We use the continuous Galerkin FEM with quadratic elements (polynomial degree 2) on a two-dimensional rectangular mesh as implemented in Firedrake for the spatial discretization, yielding the method-of-lines (MOL) ODEs,

ϕ˙=vM-1Dϕ⏟F[1]+v-Σaϕ+S(t)⏟F[2],ϕ(0)=ϕ0,

(7)

where ϕ(t) is the discrete vector of finite-element basis coefficients, M is the mass-matrix, D is the diffusion matrix, and S(t) a vector of source terms.

To test convergence of Eq. (7), we consider the methods Godunov-BE-FE, Strang-SD2O2-Heun, R3-SD2O3-RK3, and Y4-SD5O4-RK4. We also test the complex-valued OS methods Milne22_C_i-SD2O2-Heun and A33_C-SD2O3-RK3. Because the exact solution eq. (6) satisfies ϕ(π/ω,x)=0, we evaluate the l2-norm of the error only at t=π/ω. Tables 2 and 3 represent the errors and convergence rates for real-valued and complex-valued methods, respectively. The observed convergence rates are as expected.

Table 2

Errors and convergence rates for neutron scattering problem with real OS methods

OS method sub-integrators

Godunov

BE-FE

Strang

SD2O2-Heun

R3

SD2O3-RK3

Y4

SD5O4-RK4

Δt

l2–error

Order

l2–error

Order

l2–error

Order

l2–error

Order

2.000e–1

0.77

–

0.06

–

4.17e–3

–

1.59e–4

–

1.000e–1

0.39

0.98

0.01

2.59

4.88e–4

3.10

9.02e–6

4.15

5.000e–2

0.20

0.96

3.26e–3

1.62

5.89e–5

3.06

1.17e–6

2.96

2.500e–2

0.10

1.00

7.99e–4

2.03

7.25e–6

3.03

9.28e–8

3.67

1.250e–2

0.05

1.00

1.97e–4

2.02

8.98e–7

3.02

6.41e–9

3.87

Table 3

Errors and convergence rates for neutron scattering problem with complex OS methods

OS method sub-integrators

Milne22_C_i

SD2O2-Heun

A33_C

SD2O3-RK3

Δt

l2–error

Order

l2–error

Order

2.000e–1

0.02

–

1.72e–04

–

1.000e–1

0.05

2.09

2.08e–05

3.04

5.000e–2

1.37e–3

2.04

2.56e–06

3.02

2.500e–2

3.38e–4

2.02

3.18e–07

3.01

1.250e–2

8.40e–5

2.01

3.96e–08

3.00

4.2 Monodomain Model

Our second example is the monodomain model of cardiac electrophysiology, e.g., [5]. The problem is considered on the one-dimensional domain x∈Ω (with boundary ∂Ω) and time interval t∈[0,tf]. The dynamics on Ω×[0,tf] satisfy

χCm∂v∂t=λ1+λ∇·(σi∇v)-χIion(s,v),∂s∂t=g(s,v),

where v(t, x) is the transmembrane potential at time t and position x, χ is the cell membrane surface-to-volume ratio, Cm is the capacitance of cell membranes, σi(x) is the intracellular conductivity tensor, and λ(x) is a parameter representing the ratio of extracellular to intracellular conductivities at position x. The vector s(t,x) represents the state variables involved in a cell model, and the functions Iion(s,v) and g(s,v) define the dynamics of the cell model.

Neumann boundary conditions are used on ∂Ω×[0,tf],

(σi∇v)·n|x∈∂Ω=vN(t,x)|x∈∂Ω,

where n is the unit outward normal vector. The initial conditions are

v(0,x)=v0(x)|x∈Ω,s(0,x)=s0(x)|x∈Ω.

In this example, the spatial domain Ω is [0, 1], and the time interval is [0, 15]. The parameters used are χ=1400, Cm=1.0, σi(x)=1.75, and λ(x)=1. The boundary conditions for the problem are homogeneous Neumann boundary conditions. The FitzHugh–Nagumo (FHN) cell model [8] is used, with dynamics

Iion(s,v)=-kv-vrs+v-vthv-vp,g(s,v)=Lv-vr-bs,

where the constants are set k=4.0 e–4, vr=-85, vth=-70, vp=40, L=0.63, and b=0.013. These constants define specific characteristics of the cell dynamics, such as rest potential, threshold potential, peak potential, and parameters governing the ion channel kinetics. The initial condition for the transmembrane potential v is

v0(x)=-85+100cos2(πx/0.2),x∈[0,0.1],-85,otherwise,

and the initial state for the gating variable is s0(x)=0.

In this example, we consider a uniformly distributed spatial grid of Nx=131 grid points; i.e., Δx=1/130. Central differences are employed to approximate the spatial derivative. The resulting system of MOL ODEs can be written as

where V and S are the spatially discretized representations of transmembrane and cell states, respectively, Iion and G constitute the discrete cell membrane model, and M is the Laplacian matrix constructed using central differences. A reference solution is obtained by solving the MOL ODEs with Δt=0.000234375.

Table 4

Errors and convergence rates for the monodomain problem

OS method sub-integrators

Godunov

BE-FE

Strang

SD2O2-Heun

R3

SD2O3-RK3

Y4

EXACT-EXACT

Δt

MRMS

Order

MRMS

Order

MRMS

Order

MRMS

Order

2.5e–2

6.664

–

0.1574

–

0.00082

–

6.3055e-06

–

2.0e–2

5.2852

1.0388

0.10077

1.9986

0.00045

2.69

2.6524e-06

3.8807

1.5e–2

3.9266

1.0328

0.056707

1.9985

0.00020

2.7662

8.6025e-07

3.914

7.5e–3

1.9336

1.022

0.014187

1.999

2.7605e-05

2.8819

5.5253e-08

3.9606

3.75e–3

0.9592

1.0114

0.0035479

1.9995

1.744e-06

3.9845

3.2634e-09

4.0816

We consider the time-integration methods Godunov-BE-FE, Strang-SD2O2-Heun, R3-SD2O3-RK3, and Y4-EXACT-EXACT. In this example, we use the mixed root-mean-square (MRMS) error [13] defined by MRMS=[MRMS]v2+[MRMS]s2, where [MRMS]y=1N∑i=1N(yiref-yi1+|yiref|)2, with yiref and yi, respectively, denoting the reference and numerical solution for quantity y at space point i and time t and N=NxNt is the number of space-time points used, where Nt=10. The error values and convergence rates for these methods are shown in Table 4. The observed convergence rates are as expected.

5 Conclusion

This paper provides a brief overview of a Python library called pythOS for the systematic implementation, exploration, and evaluation of OS methods for differential equations. The general viewpoint presented allows OS methods to be expressed in a simple algorithm. Support for converting PDEs to ODEs is offered in pythOS via spatial discretization using the Firedrake FEM software library. Two examples are employed to highlight the functionality and versatility of pythOS, including high-order splitting and OS methods with complex coefficients. As a final note, the current release of pythOS does not support the important practical feature of adaptive step-size control. This feature will be added and explored in a future release.

Acknowledgements

The authors thank the reviewers for their constructive feedback. The authors also gratefully acknowledge support received from the Natural Sciences and Engineering Research Council of Canada under its Discovery Grant program (RGPIN 2020-04467) and the Co-operative Institute for Research to Operations in Hydrology (RJS).

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Footnotes

1

One of a pair of integrators from a Milne predictor-corrector method [16].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_44

Sensitivity and Optimal Control Theory for Linear Complementarity Systems

Peter Stechlinski1  , Hesham Abdelfattah2   and Sameh A. Eisa2  

(1)

University of Maine, Orono, ME, USA

(2)

University of Cincinnati, Cincinnati, OH, USA

Peter Stechlinski (Corresponding author)

Email: peter.stechlinski@maine.edu

Hesham Abdelfattah

Email: abdelfhm@mail.uc.edu

Sameh A. Eisa

Email: eisash@ucmail.uc.edu

Abstract

This article focuses on sensitivity and control theory for linear complementarity systems (LCSs), a type of dynamical system that experiences hybrid continuous/discrete behavior and is therefore nonsmooth. In particular, a sensitivity theory is given that characterizes generalized derivative information of solutions of LCSs with respect to parametric perturbations. With this theory in hand, a computationally-relevant open-loop optimal control theory is provided using a direct method (i.e., the control is parametrically discretized and generalized gradients of the objective function are described). The approach here is based on lexicographic directional differentiation theory, a relatively new tool in nonsmooth analysis, being applied to nonlinear complementarity systems (NCSs). The optimal control theory is illustrated with an example. As a byproduct of the sensitivity theory, well-posedness results for a new class of hybrid dynamical system, called the lexicographic linear complementarity system (LexLCS), are also established.

Keywords

Complementarity systemsHybrid systemsGeneralized derivativesSensitivity analysisOptimal control

1 Introduction

Nonlinear complementarity systems (NCSs) are systems of ODEs that are coupled with complementarity constraints that cause the system to experience “switches” between modes whenever the active constraint changes. Because of this, NCSs display hybrid-like behavior (i.e., a mixture of continuous and discrete events). The NCS modeling framework finds use in a variety of problems, such as electrical systems, mechanical systems, and economics [18]. NCSs were formalized and their basic properties were studied in [16, 17] (see also [18]).

Linear complementarity systems (LCSs) [9] are an important subclass of NCSs with linear RHS functions. Authors have investigated their well-posedness [9], non-Zenoness [19, 20], passivity [3], stability [4], observer-based stabilization control [8], and controllability and stabilizability [2]. In the paper by Vieira et al. [27], the authors developed quadratic optimal control theory and methods for LCSs by deriving necessary optimality conditions (and provided sufficient conditions in some special cases). Vieira et al. [26] established theory and methods for minimal time control problems for NCSs and then specialized their results to LCSs, because of difficult-to-verify conditions, challenges in practical implementations, etc. in the general nonlinear setting (see their discussion in [26, Sect. 3.1]).

In the papers [21, 22], the author uses nonsmooth differential-algebraic equation (DAE) theory to analyze NCSs, including their sensitivity to parameters and optimal control, under an appropriate regularity condition (strong regularity of [14]). In this paper, we are concerned with establishing optimal control theory and methods for LCSs, using the above-mentioned theory of NCSs. In particular, we use a direct method (sequential approach), wherein the control is parametrically discretized by a finite truncation of basis functions so as to use sensitivity theory to determine how to update parameters to decrease the objective function value. This is in contrast to the indirect approaches outlined above (whose methodology falls under the umbrella of calculus of variations/Pontryagin’s maximum principle). See [10] for more details. In doing so, we establish some existence/uniqueness results for a new class of hybrid dynamical system that arises from the sensitivity system associated with an LCS. In summary, we make the following contributions: (i) a sensitivity theory for parametric LCSs is given (Sect.  2); (ii) a computationally-relevant optimal control method for LCSs is provided (Sect. 3); (iii) simulations are shown to illustrate the optimal control results (Sect. 4); and (iv) well-posedness theory for a new class of hybrid dynamical system, called the lexicographic linear complementarity system, is established (Sect. 5). Future directions are given in Sect. 6.

2 Parametric LCS Sensitivity Theory

In this part, we give sensitivity theory for the following parametric LCS:

dxdt=A(p)x+B(p)v,x(0)=f0(p),

(1a)

0≤v⊥C(p)x+D(p)v≥0,

(1b)

with independent variable t, differential and algebraic state variables x(t)∈Rnx and v(t)∈Rnv, respectively, matrix-valued C1 functions A:Dp→Rnx×nx,B:Dp→Rnx×nv,C:Dp→Rnv×nx,D:Dp→Rnv×nv, initial value C1 function f0:Dp→Rnx, and problem parameters p∈Dp⊆Rnp. The complementarity conditions are understood in the following sense: given two vectors v and y, 0≤v⊥y≥0 is equivalent to v≥0 and y≥0 (component-wise) and vTy=0. The complementarity conditions implicitly define an “active set” of the constraints, with the dynamical system experiencing “switches” between different modes whenever the active set changes: Given a solution (x~,v~) on [0,tf] through initial data (p0,x0,v0), we define the inactive, weakly active, and strongly active index sets associated with (1b), respectively, as follows:

α(t)={i:v~i>0=y~i},β(t)={i:v~i=0=y~i},γ(t)={i:v~i=0<y~i},

(2)

where y~(t)=C(p0)x~(t)+D(p0)v~(t). The LCS system satisfies the P-property if D is a P-matrix (all its principal minors are positive). In this case, the embedded linear complementarity system in (1b) is uniquely solvable given any Cx [7], and it follows that the solution (x~,v~) of the LCS is strongly regular in the sense of [14] (i.e., the matrix-theoretic condition on [14, Page 243]) and unique.

However, because of the complementarity conditions in (1b), the solutions x~ and v~ of the LCS system may behave nonsmoothly with respect to parametric perturbations. Thus, the classical (forward) parametric sensitivity functions Sx(t)=∂x~∂p(t,p0) and Sv(t)=∂v~∂p(t,p0) may not exist. Consequently, we characterize the solution parametric sensitivities using generalized first-order derivative information instead, using the concept of Clarke [5]; given a locally Lipschitz function F:Rn→Rm, its Clarke generalized derivative [5] at z=z0 is

∂F(z0)=conv{H∈Rm×n:∃zn→z0s.t.zj∈domF\ΩF∀jandJF(zn)→H},

where JF(zj)∈Rm×n is the Jacobian matrix of F evaluated at zj and ΩF is the zero measure subset on which F is not differentiable. For example, for F(z)=|z|, ∂F(0)=conv{-1,1}=[-1,1]. Such first-order generalized derivative information is computationally relevant in dedicated nonsmooth numerical methods (such as nonsmooth equation solving and nonsmooth optimization—see [1]). However, it is difficult to calculate generalized derivative elements in an accurate and easy-to-implement way for complex problems/models, such as dynamic LCS systems (see [1] for a detailed explanation of the difficulties).

To overcome these challenges, a new tool in nonsmooth analysis, called the lexicographic directional derivative [11] can be used (again, see [1] for details) to furnish lexicographic derivatives [12] which are just as computationally relevant as Clarke generalized derivatives in the above-mentioned nonsmooth numerical methods. Using this approach, we provide theory below that describes how to obtain “lexicographic sensitivity functions”—in particular, an auxiliary sensitivity system is obtained whose unique solution yields (lexicographic) sensitivity functions SxL(t) and SvL(t), that are analogous to Sx(t)=∂x~∂p(t,p0) and Sv=∂v~∂p(t,p0), and are indistinguishable from Clarke generalized derivative information of the solutions with respect to parameters, i.e., ∂[x~(t,·)](p0) and ∂[v~(t,·)](p0). Said auxiliary sensitivity system is a lexicographic linear complementarity system with (matrix-valued) lexicographic complementarity conditions [21, Sect. III.B.]: given V,H∈Rm×n,

0⪯V⊥H⪰0⟺(V⪰0)∧(H⪰0)∧(rowi(V)=0∨rowi(H)=0∀i)

(3)

where V⪰0 holds if, for each i, fsign(rowi(V))=1 or rowi(V)=0, i.e., the first nonzero element in each row is positive or the row is the zero vector. For example,

Before presenting the main result, we introduce the folloting notation: Let In denote the n×n identity matrix. Given two matrices M∈Rm×n and N∈Rp×q, let M⊗N∈Rmp×nq denote the Kronecker product of the matrices. Let MJ∈R|J|×n denote the submatrix of M∈Rm×n constructed by its rows as indexed by the properly ordered subset J⊆{1,…,m}. Given a matrix-valued C1 function W:Rq→Rn×m, let ∂W∂z denote the mn×q partial derivative matrix obtained by first vectorizing W and then taking its partial derivative (see [13, Eq. (2.5)]). We now give the main result of this section.

Theorem 1

Let p0∈Dp. Let D(p0) be a P-matrix. Let (x~,v~) be a solution of () on [0,tf] through initial data (p0,x0,v0). Then the (lexicographic) sensitivity functions (SxL,SvL) uniquely solve the following (lexicographic) sensitivity system:

dXdt=(Inx⊗x~T)∂A∂p+AX+(Inv⊗v~T)∂B∂p+BV,X(0)=Jf0(p0),

(4a)

0=(I|α|⊗x~T)∂Cα∂p+CαX+(I|α|⊗v~T)∂Dα∂p+DαV,

(4b)

0⪯Vβ⊥(I|β|⊗x~T)∂Cβ∂p+CβX+(I|β|⊗v~T)∂Dβ∂p+DβV⪰0,

(4c)

0=Vγ,

(4d)

on [0,tf] through initial data (X0,V0), where X0=Jf0(p0)∈Rnx×np and V0∈Rnv×np uniquely solves (4b)–(4d) at t=0.

Proof

□Letting f(p,x,v)↦A(p)x+B(p)v and h(p,x,v)↦C(p)x+D(p)v, Eq. () can be cast in the form

dxdt=f(p,x,v),x(0)=f0(p),

(5a)

0≤v⊥h(p,x,v)≥0.

(5b)

Moreover, (5b) is equivalent to the nonsmooth algebraic equation 0=g(p,x,v) where g:(p,x,v)↦min(v,h(p,x,v)). Following the arguments in [21, Sect. 5] (based on [15, Example 17]), D(p0) being a P-matrix implies that the mapping g is completely coherently oriented (in the sense of [15]) with respect to v at (p0,x,v)∈Dp×Rnx×Rnv. It follows that g is completely coherently oriented with respect to v at (p0,x~(t,p0),v~(t,p0)) for each t∈[0,tf]. Hence, from [21], (x~,v~) is a strongly regular solution (in the sense of [14]) on [0,tf]. Thus, the sensitivity theory in [22, Sect. III.B] is applicable: the sensitivity system of () is given by [22, Eq. (18)], which, using the definitions of f and h above, setting P=Inp, and using the product rule ∂[MN]∂z=(M⊗Ir)∂N∂z+(In⊗NT)∂M∂z for matrix-valued C1 functions M and N [13], simplifies to ().    □

Remark 1

If β(t)=∅ for all t∈[0,tf], then SxL(t)=Sx(t)=∂x~∂p(t,p0) and SvL(t)=Sv(t)=∂v~∂p(t,p0) in Theorem 1.

3 A Direct Method for Solving LCS Optimal Control Problems

In this part, we apply the sensitivity theory from the previous section to solve the following quadratic optimal control LCS problem:

minpϕ(p)=∫0tf(xTQx+vTRv+uTNu)dt,

(6a)

s.t.dxdt=Ax+Bv+Eu,x(0)=x0,

(6b)

0≤v⊥Cx+Dv+Fu≥0,

(6c)

with state variables (x,v) and (parametrically discretized) controls u, constant initial conditions x0, positive semidefinite Q,R and positive definite N, and objective function ϕ. The controls have been parametrically discretized as follows:

u(t,p)=u(t,p(1),…,p(ns))=ψ(1)(t)p(1)+⋯+ψ(ns)(t)p(ns),

(7)

where {ψ(1),…,ψ(ns)} is a finite truncation of basis functions (e.g., Lagrange polynomials, Legendre polynomials, etc.), and p=(p(1),…,p(ns))∈Dpns is a vector of control parameters. The approach here is a sequential direct method: the optimal control problem has been transformed into a finite-dimensional nonlinear optimization problem thanks to the parametric discretization of the controls (hence the “direct” part of this approach). Said optimization can be solved by obtaining gradient information associated with ϕ(p) in order to update the control parameters, in the spirit of gradient descent, for example (hence the “sequential” part of this approach). The objective function ϕ may change nonsmoothly under parametric perturbations, because of the dependence of x and v on p, and hence we require the (lexicographic) sensitivity functions from the previous section. The main result is given as follows.

Theorem 2

Let p0∈Dpns. Let D be a P-matrix. Let u be parameterized according to (7) with {ψ(i)} a finite truncation of C1 basis functions. Let (x~,v~) be a solution of (6b) and (6c) on [0,tf] through initial data (p0,x0,v0). Then a generalized Clarke derivative element of the objective function ϕ in (6a) is given by

μ=2x~T(tf)QSxL(tf)+2v~T(tf)RSvL(tf)+2uT(tf,p0)NU(tf)∈∂ϕ(p0),

(8)

where U(t)=[ψ(1)(t)⋯ψ(ns)(t)]⊗Inu and where (SxL,SvL) uniquely solve the following sensitivity system on [0,tf] through initial data (X0,V0):

dXdt=AX+BV+EU,X(0)=0,

(9a)

0=CαX+DαV+FαU,0⪯Vβ⊥CβX+DβV+FβU⪰0,0=Vγ,

(9b)

where X0=0 and V0∈Rnv×np uniquely solves (9b) at t=0.

Proof

□The proof follows from [22, Theorem II.1] with the following considerations: (i) the sensitivity system in Equation (6) in [22, Theorem II.1] is replaced by () above; (ii) we add an auxiliary variable xaux via dxauxdt=xTQx+vTRv+uTNu, xaux(0)=0; and (iii) we set Q=Inpns in [22, Theorem II.1] (which is not to be confused with the Q appearing in (6a) above), with p and P in [22, Theorem II.1] absent here. For Item (i), we define the mappings f(t,p,x,v)↦Ax+Bv+Eu(t,p) and h(p,x,v)↦Cx+Dv+Fu(t,p) (in this case, g is absent) and proceed in a similar fashion to the proof of Theorem 1 to derive the sensitivity system () . Item (ii) is to transform the Lagrange form of ϕ here into the Mayer form of ϕ as used in [22]. Item (iii) is to furnish lexicographic derivatives, as is done in Theorem 1 above. With this in place, a generalized gradient of ϕ can be obtained via [22, Eq. (5)] as η=SxauxL(tf), which simplifies to (8).    □

4 Simulations

In this part, we illustrate the theory with an example. First, we provide a special case of the optimal control method in Sect. 3 for piecewise constant controls in the differential equations (i.e., F=0 in (6c)): suppose that (7) is replaced by:

u(t,p)=p(i)ift∈(τi-1,τi],i=1,2,…,ns,

(10)

where the time horizon [0,tf]=⋃[τi-1,τi] is split up into ns subintervals, i.e., τ0=0 and τns=tf, and where p(i)∈Dp as above. Then, following the same arguments as in the proof of Theorem 2, combined with the piecewise constant control theory from [23, Example 3.4], we get the following if D is a P-matrix: a generalized Clarke derivative element of the objective function ϕ in (6a) is given by

μ=2x~T(tf)QSxL(tf)+2v~T(tf)RSvL(tf)+2p(ns)TNP(tf)∈∂ϕ(p0),

where (SxL,SvL) uniquely solve the following sensitivity system on [0,tf] through initial data (X0,V0):

dXdt=AX+BV+EP,X(0)=0,

(11a)

0=CαX+DαV,0⪯Vβ⊥CβX+DβV⪰0,0=Vγ,

(11b)

where P(t)=e(i)T⊗Inp for t∈(τi-1,τi], e(i) denotes the ith unit coordinate vector in Rnp and X0=0 and V0∈Rnv×np uniquely solves (11b) at t=0.

Example 1

Consider the quadratic optimal control problem in () with nx=2, nv=1, np=nu=1, LCS matrices A=01-10,B=-11,C=[-11],D=1,E=01,F=0, and objective function matrices Q=I, R=0, N=I. Suppose that u(t) is parametrically discretized according to (10) where the time horizon [0,tf]=[0,1] is split into ns=50 subintervals for the piecewise constant controls. The optimal control can then be constructed numerically by a sequential approach, using the theory outlined above; the optimal control and corresponding optimal trajectories are given in Fig. 1, using initial control parameters p0=(p(1),…,p(ns))=(-0.5,…,-0.5), i.e., u(t,p0)=-0.5 for each t∈(τi-1,τi], and initial conditions x(0)=x0=(1,3), v(0)=v0=0, X(0)=[00], V(0)=0. Observe that v(t)=0 for t∈[0,t∗], and then v(t)>0 for t>t∗≈0.35, i.e., γ([0,t∗))={1}, β(t∗)={1}, α((t∗,tf])={1}, as, at t=t∗, the constraint sets switch from inactive through weakly active to active, and the system shifts accordingly. 

Fig. 1

Simulations of optimal control and state trajectories in Example 1, using the direct method

5 Theory of Lexicographic Linear Complementarity Systems

The NCSs and LCSs studied above are dynamical systems coupled with complementarity problems. Given q∈Rnq and F:Rnq×Rnv→Rnv, consider the nonlinear complementarity problem (NCP) [7] of finding v∈Rnv such that 0≤v⊥F(q,v)≥0. Linear complementarity problems (LCPs) [6] are NCPs with affine participating functions, often arising from linearizing NCPs, and can be formulated as follows: given q∈Rnv and D∈Rnv×nv, find v∈Rnv such that

0≤v⊥q+Dv≥0.

(12)

This LCP is uniquely solved for any q∈Rnv if and only if D is a P-matrix [6]. As mentioned earlier, a matrix D is a P-matrix if all its principal minors are positive, a condition which implies nonsingularity of D and which is implied if D is positive definite [6].

Mirroring the connection between LCSs and LCPs (i.e., LCSs being ODEs with LCPs embedded), we note that the lexicographic sensitivity systems associated with LCSs contain lexicographic complementarity conditions, and hence are dynamical systems with “lexicographic complementarity problems” embedded. Motivated by this, we formally introduce lexicographic complementarity problems and systems. The lexicographic linear complementarity problem (LexLCP) is stated as follows: given k∈N, Q∈Rnv×k and D∈Rnv×nv, find V∈Rnv×k such that

0⪯V⊥Q+DV⪰0.

(13)

Solution properties of the LexLCP are captured in the next result.

Proposition 1

If D is a P-matrix, then the LexLCP in (13) admits a unique solution for any Q∈Rnv×k.

Proof

□Let D be a P-matrix. Choose q0=0∈Rnv. Then the LCP (12) has the unique solution v0=0∈Rnv. Thus, the piecewise affine (and thus PC1) function g:(q,v)↦min(v,q+Dv) satisfies g(q0,v0)=0 and is completely coherently oriented (in the sense of [15]) with respect to v by the arguments made in the proof of Theorem 1 above. Hence, [25, Theorem 3.4] is applicable; there exists a PC1 implicit function r:N→Rnv, for some neighborhood N of q0, such that for any q∈N, (q,r(q)) is the unique solution of g(q,r(q))=min(q,r(q))=0. Moreover, the LD-derivative r′(q0;Q)=V0 is the unique solution of g′(q0,v0;(Q,V0))=0. By [25, Lemma 4.2], g′(q0,v0;(Q,V0))=slmmin([v0V0],[q0+Dv0Q+DV0]) where slmmin is the shifted-lexicographic-matrix minimum function [25]. Then, g′(q0,v0;(Q,V0))=0 is equivalent to lmmin(V0,Q+DV0)=0, where lmmin is the lexicographic-matrix minimum [25], because v0=0=q0+Dv0. The result then holds by [22, Proposition A.1].    □

We can establish existence and uniqueness of solutions of lexicographic linear complementarity systems (LexLCSs), introduced as follows:

dXdt(t)=AX(t)+BV(t),X(0)=X0,

(14a)

0⪯V(t)⊥CX(t)+DV(t)⪰0,

(14b)

for some k∈N and X0∈Rnx×k, and matrices A, B, C, and D with dimensions outlined earlier. A mapping Z~=(X~,V~) is a solution of () on T⊆R if (i) X~ is absolutely continuous on T and V~ is Lebesgue integrable on T; and (ii) (X~,V~) satisfies (14a)–(14b) for almost every t∈T and satisfies the ICs in (14a) at t=0.

Theorem 3

If D is a P-matrix, then () admits a unique solution (X~,V~):R→R(nx+nv)×k on T=(-∞,+∞) through initial data (X0,V0), where V0 is the unique solution of 0⪯V0⊥CX0+DV0⪰0.

Proof

□Consider () with f0(p)=p and p0=0. Then, since D is a P-matrix, v0=0 is the unique solution of the embedded LCP 0≤v0⊥Cp0+Dv0≥0 and z~(t)=(x~(t),v~(t))=(0,0) is the corresponding unique solution of () on (-∞,+∞) through (p0,x0,v0)=(0,0,0). In this case, α(t)=γ(t)=∅ and β(t)={1,…,nv} for all t. Choose any [a,b]⊂R with a<0<b, then according to [21, Theorem 4.5], an additional conclusion in Theorem 1 above is that z~(t)=z~(t,p) is lexicographically smooth (in the sense of [12]) with respect to p near p0, for each fixed t∈T1=[a,b], and, given the directions matrix P=X0∈Rnx×k, Eq. () admits unique solution Z~=(X~,V~) on T1=[a,b], where the mappings X~ and V~ are absolutely continuous and Lebesgue integrable, respectively. Moreover, with ∂A∂p=0, ∂C∂p=0, etc., and α(t)=γ(t)=∅ for all t, Eq. () simplifies to Eq. (), with (14a) satisfied for almost every t∈T and (14b) satisfied for every t∈T.

It is straightforward to construct a continuation of Z~ to T2=[-2a,2b] by repeating the arguments above. Hence, the augmented graphs of continuations, Γext:={{(t,Z~(t)):t∈T^}:Z^isacontinuationofZ~onT^} is nonempty. Then solutions can be continued to Zmax on a maximal interval of existence Tmax=(tL,tU), where tL∈R∪{-∞} and tU∈R∪{+∞}, by arguments involving Zorn’s Lemma (see the proof of [24, Theorem 6.4]). In fact, it must hold that Tmax=(-∞,+∞) here, since if not, i.e., Tmax=(tL,tU) with tL or tU∈R, there exists a continuation to Tk=[-ka,kb] where k∈N is chosen so that -ka<tL or kb>tU, a contradiction to Zmax being the maximal continuation.    □

6 Future Work

One future direction for this work would be comparing the methods here with those put forth by Vieira et al. in [26, 27] (which can accommodate “high-index” LCS systems, since D need not be a P-matrix in their work—e.g. see [27, Example 2]).

Acknowledgements

This material is based upon work supported by the National Science Foundation under Award No. 2318772 and 2318773.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_45

On Some Directions in Researching the Oscillation of Solutions to First Order DDE

J. I. Stavroulakis1, 2  

(1)

Ariel University, Ariel, Israel

(2)

Georgia Institute of Technology, Atlanta, GA, USA

J. I. Stavroulakis

Email: jstavroulakis3@gatech.edu

Email: john.ioannis.stavroulakis@gmail.com

Abstract

We investigate the oscillation of the first-order delay differential equation

x′(t)+p(t)x(τ(t))=0,t≥t0∈R,

where p:R→[0,+∞), τ:R→R,τ(t)≤t, limt→∞τ(t)=∞. We trace the evolution of the relevant literature, from the seminal results of

lim inft→∞∫τ(t)tp(s)ds>1e,

and

lim supt→∞∫τ(t)tp(s)ds>1,

to the recent condition (for nondecreasing delayed argument τ(t))

lim supt→∞∫τ(t)tp(s)ds>κ(lim inft→∞∫τ(t)tp(s)ds),

where κ:[0,1e]→[1e,1] is strictly decreasing, continuous. This condition is sharp in the sense that, for any given values of lim inft→∞∫τ(t)tp(s)ds∈[0,1e] and ε>0, one cannot improve the condition to

lim supt→∞∫τ(t)tp(s)ds>κ(lim inft→∞∫τ(t)tp(s)ds)-ε.

In particular, κ(0)=1,κ(1e)=1e. Our focus is two-fold. Firstly, we describe the intuitions and methods applied to oscillation. Secondly, we examine the overall advancements and improvements upon the criteria.

Keywords

Delay equationOscillationFirst-orderLambert functionCharacteristic equation

1 Introduction

We discuss the history of the oscillation criterion for first-order scalar nonautonomous delay differential equations with positive coefficient

x′(t)+p(t)x(τ(t))=0.

(1)

where p:R→[0,+∞) is locally Lebesgue integrable, τ:R→R is Lebesgue measurable, τ(t)≤t, limt→∞τ(t)=∞. The pioneering work in the study of oscillation of (1) was that of Myshkis in 1951 [31, 32]. It was in fact in this work that the necessity of the condition limt→∞τ(t)=∞ was proven. We will briefly introduce this equation and analyze the problems and challenges concerning the oscillation of (1). Next, we will trace the evolution of the ideas and methods applied to those problems, over a series of papers obtaining successively improved results, including a relatively recent sharp condition [35].

A solution of (1) is a locally absolutely continuous function x:[inf{τ(s),s≥t0},+∞)→R, which satisfies (1) on [t0,+∞). We note that unique solutions exist under weaker assumptions on the initial data (e.g. Borel or Lebesgue measurability), and that the solutions would become eventually absolutely continuous. It thus suffices for the present asymptotic discussion to shift time so that the solutions considered are continuous or even locally absolutely continuous on some right half-line. On existence/uniqueness and the regularity of the initial data, see e.g. [1, Appendix B], [18, Chap. 2] and the recent discussion in [33].

In the following, oscillation of a solution as an asymptotic property will mean the existence of a sequence of roots of the solution which tend to infinity (arbitrarily large zeros). The Eq. (1) is called oscillatory if all its solutions are oscillatory. Thus oscillation of (1) is equivalent to the nonexistence of a positive solution.

2 Notable Results and Techniques

In the autonomous case, for example x′(t)+x(t-τ0)=0, there exists a spectral decomposition of solutions into a countably infinite family of eigensolutions

x(t)=∑n=0∞cn(t)exp-λnt,

(2)

where

λn=expλnτ0∈C,

and cn(t) are complex polynomials. This decomposition is one of the seminal results of the oscillation theory of delay equations, described already in [2] by Fourier-Laplace type transformations and later in greater generality using methods of functional analysis [13, 18, 19]. The intuitive idea, that existence of a positive solution of (1) is equivalent to existence of a real eigenvalue λ, was proven to be true in [36] and independently in [28]. See some generalizations of this result to more general types of (non)oscillation in [14, 34].

The following result was first proven in [31, 32, Theorem 41] (see a generalization of this method to measurable parameters in [16]) and later by different methods in [24]. A more general result for higher-order equations was obtained in [25].

Theorem 1

The oscillation of

x′(t)+p(t)x(τ(t))≤0

is equivalent to the oscillation of (1).

As a consequence of the aforementioned results, the following criterion was obtained in by Myshkis [32, Chap. VI]:

Corollary 1

(I) Assume

p(t)≥1,andt-τ(t)>const>1e,t>0.

Then (1) oscillates.

(II) Assume

p(t)≤1,andt-τ(t)≤1e,t>0.

Then (1) is nonoscillatory.

It was later generalized to various integral criteria, such as the following, obtained in [22].

Theorem 2

Assume

∫τ(t)tp(s)ds>const>1e,t>0.

Then (1) oscillates.

(II) Assume

∫τ(t)tp(s)ds≤1e,t>0.

Then (1) is nonoscillatory.

We also note an earlier result in the same direction obtained in [26], as well as the further generalization in [4, 24].

It is clear that Theorems 1 and 2 are equivalent via a change of variables allowing one to assume p(t)≡1, which first appeared in the stability study of [28], and has subsequently been used in various forms e.g. [3, 6, 17]. The result of [22] was obtained by independent methods. Most methods subsequently inspired by [22] were essentially unchanged under (independent of) the assumption p(t)≡1.

The study of the oscillation of (1) took a new direction with the lim sup criterion of [27].

Theorem 3

Assume τ is nondecreasing and

lim supt→+∞∫τ(t)tp(s)ds>1.

(3)

Then (1) oscillates.

The above result is sharp in the sense that (3) may not in general be replaced by

lim supt→+∞∫τ(t)tp(s)ds>1-ε,

for any given ε>0, see [36]. It should also be noted the monotonicity assumption on the delayed argument τ(t) is necessary, see e.g. [5, Theorem 1]. It was subsequently discovered that the constant in (4) may be improved once a lower bound on ∫τ(t)tp(s)ds is known [8, 11, 12]. These studies applied certain methods of bounding the ratio x(τ(t))x(t), first developed in [22]. Consider for example the following result.

Theorem 4

Assume τ is nondecreasing and

lim supt→+∞∫τ(t)tp(s)ds>1-(lim inft→+∞∫τ(t)tp(s)ds)22.

(4)

Then (1) oscillates.

As it was obviously the joint and compound contribution of lim sup and lim inf which determined oscillation, the following problem arose.

Problem 1

Find the optimal (minimal) function κ:[0,1/e]→[1/e,1] such that, under nondecreasing delay τ(t),

lim supt→∞∫τ(t)tp(s)ds>κlim inft→∞∫τ(t)tp(s)ds

(5)

implies oscillation of (1).

Such a function was the object of much research for decades, and the a priori optimal value

κ(1e)=1e

(6)

was a folklore conjecture. We refer the reader to some notable results on this problem in [21, 24, 37, 38], as well as important generalizations in [15, 23]. It should be noted that in [23], Theorem 2 is obtained through iterative applications of Theorem 1.

3 Note on Intuitions and Framework

The idea behind the methods employed in many of the above results was that existence of positive solutions is equivalent to the following equation (which describes positive solutions of (1)) being licit

x(τ(t))x(t)=exp∫τ(t)tp(s)x(τ(s))x(s)ds,

(7)

as the denominator being nonzero is equivalent to the existence of a function f such that

f(t)=exp∫τ(t)tp(s)f(s)ds.

(8)

On certain early uses of (7), (8) (for both constant and nonautonomous parameters), see e.g. [22, 28, 29, 36]. In [20] the criterion

lim inft→∞[p(t)(τ(t)-t)]>1e

was interpreted as the nonexistence of solutions to another form of the generalized characteristic equation:

infλ>0,t>01λp(t)exp(λ(t-τ(t)))>1.

The equivalence between oscillation and nonexistence of solutions to this form of quasicharacteristic equation for multiple delays (for bounded parameters) was also conjectured by Hunt and Yorke [20], fact later proven by Kwong [24]. On further oscillation criteria related to the quasicharacteristic equation, see e.g. [1, Chap. 2]. Conversely, oscillation was considered equivalent to

x(τ(t))x(t)=∞,

(9)

where equality holds in the sense of lim inf or lim sup, see [22, 28, 29, 36] and the discussion in [24]. Even (9) holding at finite time was used to prove oscillation in [20]. Given the above intuitions, successive improvements on the bounds for x(τ(t))x(t) led to improved versions of criterion (4).

4 Alternative Perspective

An alternative perspective on these methods is as follows [35]. Assume in the following an nondecreasing delayed argument τ(t). For a fixed small positive real ε, consider an interval [τ(t),t] such that ∫τ(t)tp=lim sups→+∞∫τ(s)sp(w)dw-ε. On this interval one can calculate the following bounds for any positive solution:

x(s)≤x(τ(t))exp(-λ∫τ(t)sp(w)dw),fors∈[τ(t),ξ]

(10)

x(s)≤x(ξ)(1-λ∫ξsp(w)dw),∗slashspace∗fors∈[ξ,t],

(11)

where ξ satisfies the threshold condition

∫τ(t)ξp(w)dw=lim infs→∞∫τ(s)sp(w)dw-ε

and

λ=expλlim infs→∞∫τ(s)sp(w)dw-ε∈[1,e].

It is noteworthy that equality (at least up to negligible error) in (10), (11), is possible, and in that sense, (10), (11) are sharp or optimal. Subsequently, using Lemma 1, we may demand that the bound obtained by assuming equality in (10), (11), when considered as initial data for a delay equation with

∫τ(r)rp(w)dw=lim infs→∞∫τ(s)sp(w)dw-ε,∀r>t,

(12)

leads to a positive solution. The delay equation satisfying (12) is easily seen to be equivalent to an equation with constant parameters,

x′(t)+x(t-τ0)=0,

where

τ0=liminfs→∞∫τ(s)sp(w)dw-ε<1e.

(13)

So the initial oscillation problem is simplified to determining which initial conditions give rise to positive solutions of constant delay equations. One can readily calculate precisely which initial conditions correspond to positive solutions by considering (2) (cf. [10]). In virtue of (13), the coefficients cn in (2) are constants and can be easily determined explicitly:

cn=x(t)expλnt-λn∫t-τ0tx(s)exp(λns)ds1-τ0λn.

Translating c0>0 into a condition for (1), one obtains a continuous, κ function, which in fact satisfies the conjecture (6). The explicit expression is the following:

κ(a)=1fora=0,[10pt]2a+2λ(a)-1fora∈0,ln22,[10pt]2a-2λ(a)-1λ(a)W-1-λ(a)e2fora∈ln22,1e,

(14)

where

λ(a)=expaλ(a)=W0(-a)-a∈[1,e],fora∈(0,1/e]

and W0(·),W-1(·) are the 0-th and (-1)-th branch of the Lambert function (on which, see e.g. [9]). Finally, the following result was obtained in [35].

Theorem 5

The function κ(a), as in (14), yields a solution to Problem 1. Under nondecreasing delay τ(t), the inequality

lim supt→∞∫τ(t)tp(s)ds>κlim inft→∞∫τ(t)tp(s)ds

(15)

implies oscillation of (1). The function κ is sharp, i.e. inequality (15) may not in general be replaced by

lim supt→∞∫τ(t)tp(s)ds>κlim inft→∞∫τ(t)tp(s)ds-ε

for any given values of lim inft→∞∫τ(t)tp(s)ds∈[0,1e] and ε>0.

Acknowledgements

We thank the referees and the editors for valuable suggestions which greatly refined the presentation and discussion of the results. We acknowledge support of Ariel University and Georgia Institute of Technology.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

D. M. Kilgour et al. (eds.)Addressing Modern Challenges in the Mathematical, Statistical, and Computational SciencesSpringer Proceedings in Mathematics & Statistics494https://doi.org/10.1007/978-3-031-84869-8_46

Observer and Command Filter-Based Prescribed Performance Control for a Class of Nonlinear Systems with Mismatched Disturbances

Min Wang1  , Xinzhi Liu2   and Jinsheng Sun3  

(1)

Nanjing University of Science and Technology, Jiangsu, China

(2)

University of Waterloo, Waterloo, Canada

(3)

Nanjing University of Science and Technology, Jiangsu, China

Min Wang (Corresponding author)

Email: wangm0811@njust.edu.cn

Xinzhi Liu

Email: xzliu@uwaterloo.ca

Jinsheng Sun

Email: jssun67@163.com

Abstract

In this paper, a prescribed performance controller based on a command filter is designed for a class of nonlinear systems subject to mismatched disturbances. In the design of intermediate control signal, disturbance estimations are introduced to compensate the effects caused by external disturbances. The so-called “explosion of complexity” problem in traditional backstepping procedure is solved by command filter technique. To enhance the control performance further, a prescribed performance function (PPF) characterizing transient and steady-state performance is used for the tracking error transformation. The proposed controller can ensure that the output tracking error converges to a predefined arbitrarily small residual set, and all the signals of the closed-loop system can be bounded. Theoretical results are validated and illustrated via simulation example.

Keywords

Disturbance observerPrescribed performance controlCommand filters

1 Introduction

In practical applications, most control systems are affected by various uncertainties, such as load disturbances, sensor noise, and unknown nonlinear friction, which inevitably degrade system performance. Consequently, disturbance attenuation has become a prominent topic in control theory [1]. When disturbances are measurable, feedforward compensation strategies can attenuate or eliminate the effects of disturbances. In many instances, external disturbances are unmeasurable or not measurable due to their cost [2]. A straightforward approach for this issue is to estimate disturbances from measurable variables. Then, a controller can be designed based on the estimate to compensate for the effects of disturbances. This is the main idea of disturbance observer-based control (DOBC) technology. Because of simple structure and strong robustness, DOBC is widely employed in various control systems, including hydraulic systems [3], small UAV systems [4], robot systems [5], smart grid systems [6], and so on.

The majority of aforementioned works focus on the steady-state performance of systems. The transient performance (e.g., convergence rate and maximum overshoot) is equally significant in practical systems. For instance, by imposing constraints on steady-state values and overshoot, it can enhance ride comfort in autonomous ground vehicles, and prevent vehicle collisions or rollovers as discussed in [7]. Prescribed performance control proposed in [8, 9] can simultaneously achieve constraints on both the transient and steady-state performance of the system. The main idea of PPC is to convert the constrained tracking error into an equivalent unconstrained variable. The PPC method has been further utilized in [10–13] to resolve a variety of different control problems.

Backstepping is a highly classical method for constructing nonlinear system controllers. It is worth noting that the partial derivative of the virtual control signal needs to be repeatedly calculated during the control design process, which may lead to the problem of “complexity explosion”. The command filters presented in [14] take virtual control signals as the input, and the filters’ output can be used in place of the derivative of the control signal, thereby reducing the computational complexity of backstepping design. On the other hand, this technique reduces errors caused by the filters by providing an error compensation mechanism. Due to the advantages of command filter, it has been extended to control design for various nonlinear systems [15–19].

Based on the above discussions, this paper designs a prescribed performance tracking controller combined with command filters and disturbance observer for disturbed nonlinear systems. The proposed method guarantees that all signals in the closed-loop system are uniformly ultimately bounded (UUB). Furthermore, throughout the evolution process, the tracking error remains within the specified performance bounds. The designed composite controller reduces computational complexity, compensates for mismatched disturbances, and further enhances the disturbance attenuation capability of the control system.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Sect. 2. Controller design and stability analysis are presented in Sect. 3. The numerical simulation example is given in Sect. 4. Finally, a conclusion is drawn in Sect. 5.

2 Problem Formulation and Preliminaries

2.1 Problem Formulation

The following notations will be used throughout the paper. For a given vector x=[x1,…,xn]T, x¯i=[x1,…,xi]T,i=1,…,n. Euclidean space with dimension n is represented by Rn, the set of all nonnegative real numbers is denoted by R+. The arguments of functions are sometimes simplified, a function f(x(t)) can be denoted by f(x) or f.

Consider a general strict-feedback nonlinear system of the form

x˙i(t)=xi+1(t)+fi(x¯i)+di(t),i=1,2,⋯,n-1,x˙n(t)=u(t)+fn(x¯n)+dn(t),y(t)=x1(t),

(1)

where x(t)=[x1(t),…,xn(t)]T∈Rn denotes system state, u(t) denotes control input and y(t) denotes system output, fi, i=1,…,n are continuous nonlinear functions, di(t), i=1,…,n are mismatched external disturbances.

The objective of this paper is to design a composite controller for system (1) such that: (i) all the states in the corresponding closed-loop system are UUB; and (ii) achieve prescribed transient/steady-state behavioral bounds -γ(t)<e(t)<γ(t),∀t≥0 on the tracking error e(t)=y(t)-yd(t), where yd is a given reference signal, γ(t) is a pre-defined function. The following assumptions are needed.

Assumption 1

di, i=1,…,n are nonvanishing but bounded, and the derivatives of di are also bounded.

Assumption 2

The desired trajectory yd and its i-order derivatives are known and bounded.

Remark 1

Assumption 1 is widely used in the field of disturbance estimation [1–4]; the derivative of the disturbance affects the convergence of the error dynamics equation, which is an important premise for analyzing the stability of disturbance estimation error. It is worth noting that Assumption 2 is a standard assumption for output tracking control of nonlinear systems and similar assumptions can also be found in the literature [9, 10].

2.2 Prescribed Performance

In order to ensure the transient/steady-state performance of tracking error e(t), a positive smooth decreasing function γ(t):R+→R+ with limt→∞γ(t)=γ∞>0 is selected as the PPF. Without losing generality, we can represent γ(t) as

γ(t)=(γ0-γ∞)exp(-σt)+γ∞,

(2)

where σ and γ0>γ∞ are positive parameters to be designed. Following [8, 9], the prescribed performance can be ensured by satisfying

-γ(t)<e(t)<γ(t),∀t≥0.

(3)

To address the control problem in (3), consider transforming the constrained tracking error e(t) into an equivalent unconstrained error μ(t), To achieve this, a strictly increasing smooth function H(μ(t)) satisfying the following conditions will be defined.

(i)

-1<H(μ(t))<1,∀μ(t)∈L∞,

(ii)

limμ(t)→+∞H(μ(t))=1, limμ(t)→-∞H(μ(t))=-1.

From the properties of H(μ(t)), inequality (3) can be transformed as

e(t)=γ(t)H(μ(t)).

(4)

Because H(μ(t)) is strictly monotonically increasing and γ(t)≥γ∞>0, the inverse function can be derived as

μ(t)=H-1e(t)γ(t).

(5)

It is evident that if μ(t) is bounded, then the prescribed performance (3) holds. To construct a controller for μ(t), we select the function H(μ(t))=exp(μ(t))-exp(-μ(t))exp(μ(t))+exp(-μ(t)). From (5), μ(t) can be deduced as

μ(t)=12lnH(μ(t))+11-H(μ(t)).

(6)

2.3 Disturbance Observer Design

v˙i=xi+1+fi+d^i,d^i=ai(xi-vi),i=1,⋯,n,

(7)

where xn+1=u, ai>0 is an observer gain, d^i>0 denotes the estimation of di, and vi denotes an auxiliary vector of the observer. The dynamics of d^i is described as

d^˙i=ai(x˙i-vi)=ai(di-d^i),i=1,⋯,n,

(8)

The disturbance estimation error system can be described by

ω˙i=-aiωi+d˙i,i=1,⋯,n,

(9)

where ωi=di-d^i.

3 Controller Design and Stability Analysis

3.1 Controller Design

The tracking errors are defined as

z1=μ,zi=xi-xi,d,i=2,…,n,

(10)

where xi,d is generated by the command filter. The command filters are described as

α˙i,1∗=pnαi,2∗,α˙i,2∗=-2ςpnαi,2∗-pn(αi,1∗-αi),

(11)

where i=1,…,n-1, xi,d=αi-1,1∗, αi is the control input of the filter, αi,1∗(0)=αi(0), αi,2∗(0)=0. The effect of xi,d-αi-1 is eliminated by the error compensation signals which are designed as

β˙i=-ciβi+hiβi+1+hi(xi+1,d-αi),β˙n=-cnβn,

(12)

where ci>0 is a given constant and β˙i(0)=0. We can get |βi| is bounded by [7].

Define the compensated tracking errors as ξi=zi-βi,i=1,…,n. In the following, a prescribed performance controller based on observer and command filters is designed.

Step 1. Consider the candidate Lyapunov function

V1=12ξ12+12ω12,

(13)

The derivative of V1 with respect to t can be calculated as

V˙1=ξ1(Γ(z2+x2,d+f1+d1-y˙d-eγ˙γ)-β˙1)+ω1(-a1ω1+d˙1)≤ξ1(Γ(f1+d1-y˙d-eγ˙γ+ξ2+α1)+c1(z1-ξ1))-a1ω12+ω1d˙1.

(14)

From Young’s inequality, we have ξ1Γω1≤Γ24ε1ξ12+ε1ω12. The virtual control law α1 is chosen as

α1=-(c1Γz1+f1+d^1+Γ4ε1ξ1-y˙d-eγ˙γ).

(15)

Substituting (15) into (14) yields

V˙1≤-c1ξ12+Γξ1ξ2-(a1-ε1)ω12+ω1d˙1.

(16)

Step i. Choose the candidate Lyapunov function

Vi=Vi-1+12ξi2+12ωi2.

(17)

With the help of ξiωi≤14εiξi2+εiωi2 and the virtual control law αi=-(ξi-1+fi+d^i-x˙i,d+cizi+14εiξi), we can obtain

V˙i≤-∑j=1i(cjξj2+(aj-εj)ωj2)+∑j=1iωjd˙j+ξiξi+1.

(18)

Step n. Next, if i=n, we have

Vn=Vn-1+12ξn2+12ωn2.

(19)

The actual control input can be designed as

u=-(fn+d^n-x˙n,d+cnzn+14εnξn+ξn-1).

(20)

Then, we have

V˙n≤-∑j=1n(cjξj2+(aj-εj)ωj2)+∑j=1nωjd˙j.

(21)

3.2 Stability Analysis

Theorem 1

Let Assumption 1 hold. Under controller (20), all signals of the closed-loop system are uniformly ultimately bounded, and the prescribed control performance (3) is preserved.

Proof

Select the Lyapunov function as

Vn=12∑i=1nξi2+12∑i=1nωi2.

(22)

It can be seen from (21)

Vn≤aV+b,

(23)

where a=min{cj,aj-εj},j=1,…,n, b=sup∑j=1nωjd˙j. Based on Assumption 1 and (9), we obtain that b is a constant. Let a>bϱ with ϱ>0, then V˙n<0 on Vn=ϱ. Therefore, Vn≤ϱ is an invariant set. Integrating (23) yields

0≤Vn(t)≤ba+(Vn(0)-ba)e-at,∀t≥0.

(24)

This implies that Vn(t) is eventually bounded by ba, and thus the resulting closed-loop system is UUB. For i=1,…,n, ξi and ωi are bounded. Because |βi| is bounded and zi=ξi+βi, zi is bounded. The boundedness of x(t) and each signal is justified. According to the properties of function H(μ(t)), we can conclude that the tracking error e(t) strictly satisfies prescribed performance (3). This completes the proof.    □

4 Simulation Example

Consider the following uncertain nonlinear system

x˙1=x2+sin(x1)+d1(t),x˙2=u+x2sin(t)+d2(t),y=x1.

(25)

The parameters of PPF are selected as γ0=4, γ∞=0.1, σ=1. The reference signal is selected as yd=sin(t) and the external disturbances are given by

d1=2,t≤20,sin(2t),20<t<30,2,t≥30,

(26)

Fig. 1

The trajectories of e(t) and performance function

Fig. 2

The trajectories of α1 and the output of the command filter

Fig. 3

The trajectories of disturbance errors

Fig. 4

The trajectory of x2

and d2=sin(t). The simulation results are shown in Figs. 1, 2, 3 and 4. It can be seen from Fig. 1 that (i) the tracking error e(t) always stay within the given prescribed performance bounds during the convergence process. (ii) With the help of PPC, e(t) always satisfies -γ(t)<e(t)<γ(t). On the contrary, e(t) will exceed the boundary if there is no PPC. Figure 2 shows the validity of the command filters. The disturbance observer errors are illustrated in Fig. 3. Figure 4 shows the trajectory of x2. From Figs. 1, 2, 3 and 4, we can observe that the composite anti-disturbance controller constructed in this paper achieves a satisfactory disturbance rejection performance, and the control objectives are guaranteed successfully.

5 Conclusion

For nonlinear systems with mismatched external disturbances, a prescribed performance controller based on command filtering and disturbance observation has been proposed. A prescribed performance function is constructed to formulate constraints on the transient and steady-state performance of the tracking error. Subsequently, an error transformation is applied to convert the constrained tracking error into an unconstrained signal. In the backstepping control approach, command filters are introduced to tackle the issue of “complexity explosion”. This controller ensures that all signals in the closed-loop system are UUB.

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