Fields Institute Communications 79

The Fields Institute for Research in Mathematical Sciences

Recent Progress and

Modern Challenges

in Applied

Mathematics,

Modeling and

Computational

Science

Roderick Melnik

Roman Makarov

Jacques Belair Editors

Fields Institute Communications

Vo l u m e 7 9

Fields Institute Editorial Board:

Ian Hambleton, Director of the Institute

Huaxiong Huang, Deputy Director of the Institute

James G. Arthur, University of Toronto

Kenneth R. Davidson, University of Waterloo

Lisa Jeffrey, University of Toronto

Barbara Lee Keyﬁtz, Ohio State University

Thomas S. Salisbury, York University

Noriko Yui, Queen’s University

The Communications series features conference proceedings, surveys, and lecture

notes generated from the activities at the Fields Institute for Research in the

Mathematical Sciences. The publications evolve from each year’s main program and

conferences. Many volumes are interdisciplinary in nature, covering applications of

mathematics in science, engineering, medicine, industry, and ﬁnance.

More information about this series at http://www.springer.com/series/10503

Roderick Melnik • Roman Makarov • Jacques Belair

Editors

Recent Progress and Modern

Challenges in Applied

Mathematics, Modeling

and Computational Science

The Fields Institute for Research

in the Mathematical Sciences

123

Editors

Roderick Melnik

Department of Mathematics

and MS2Discovery Interdisciplinary

Research Institute

Wilfrid Laurier University

Waterloo, ON, Canada

Jacques Belair

Departement de Mathematiques

Universite de Montreal

Montreal, QC, Canada

Roman Makarov

Department of Mathematics

and MS2Discovery Interdisciplinary

Research Institute

Wilfrid Laurier University

Waterloo, ON, Canada

ISSN 1069-5265 ISSN 2194-1564 (electronic)

Fields Institute Communications

ISBN 978-1-4939-6968-5 ISBN 978-1-4939-6969-2 (eBook)

DOI 10.1007/978-1-4939-6969-2

Library of Congress Control Number: 2017943956

Mathematics Subject Classiﬁcation (2010): 15-XX, 34-XX, 35-XX, 37-XX, 60-XX, 45-XX, 49-XX,

62-XX, 65-XX, 68-XX, 74-XX, 76-XX, 91-XX, 92-XX

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Preface

The application of mathematics and statistics in the age of computational science

and engineering has transformed our society and has revolutionized the world we

live in. Being some of the oldest cultural achievements of mankind, nowadays these

disciplines are intrinsic part of our daily life through our activities and technologies,

ranging from our banking and investment systems to new sophisticated electronic

devices, and to our civil infrastructure and environment.

These disciplines continuously grow at their frontiers though new areas of

applications, new theories, and new tools provided by mathematical and statistical

models. As a result, they continue representing the core of human knowledge critical

for new discoveries and innovation, our well-being, and our economic prosperity.

There has been a long and rich interplay between mathematics and statistics on

the one hand and other disciplines on the other, resulting in their fruitful enrich-

ments. With ever-expanding interdisciplinary horizons of applied mathematics and

statistics, we see new progress and modern challenges in their development. This

book is about such progress and challenges in applied mathematics, modelling, and

computational science.

Today, mathematical and statistical models are applied in natural and social

sciences, industry and technology, medicine and ﬁnance. They are at the heart of

a multitude of human activities, allowing connecting such activities in a modern

world, where our communication gets better, faster, and cheaper also due to

mathematics-based models. They substantially contribute to our better understand-

ing of complex systems and networks whose components interact in a dynamic

manner. Furthermore, mathematics-based computational technologies enable us

detailed simulations of complex systems in the areas where the knowledge about

such systems has been limited until very recently. Many such systems are functioned

in a competitive, and often uncertain, environment. Therefore, the development of

mathematical and statistical based methodologies of uncertainty quantiﬁcation, as

well as addressing other associated challenges, is essential.

Along with more traditional applications of mathematics and statistics in physics

and engineering, we are witnessing now substantial contributions of these disci-

plines to new breakthroughs in biology and medicine, ﬁnance, and social sciences.

vi Preface

Equally important, mathematical and statistical models allow us to develop new

important insight and better understanding of environmental and ecological sustain-

ability in our dynamic and complex world.

This book provides details on recent progress and challenges in selected areas of

applied mathematics, modelling, and computational science. It contains 14 chapters

which open to the reader details on state-of-the-art achievements in these selected

areas. The book provides a balance between fundamental theoretical and applied

developments, emphasizing interdisciplinary nature of modern trends in these areas.

Written by 27 experts in their respective ﬁelds, the book is aimed at researchers

in academia, practitioners, and graduate students. It can serve as a reference in

the diverse selected areas of applied mathematics, modelling, and computational

science. The book promotes interdisciplinary collaborations in addressing new

challenges in these areas.

We are thankful to the referees of this volume for their invaluable help and

suggestions. We are also very grateful to the Springer editorial team, and in

particular to Dahlia Fisch, for their highly professional support.

Waterloo, ON, Canada Roderick Melnik

Waterloo, ON, Canada Roman Makarov

Montreal, QC, Canada Jacques Belair

Contents

Part I Modern Challenges and Interdisciplinary Interactions

via Mathematical, Statistical, and Computational Models

Modern Challenges and Interdisciplinary Interactions

via Mathematical, Statistical, and Computational Models .................. 3

Roderick Melnik, Roman Makarov, and Jacques Belair

Part II Large Deviation Theory and Random Perturbations

of Dynamical Systems with Applications

Long Term Effects of Small Random Perturbations on Dynamical

Systems: Theoretical and Computational Tools .............................. 17

Tobias Grafke, Tobias Schäfer, and Eric Vanden-Eijnden

Part III Nonlinear Waves, Hyperbolic Problems, and their

Applications

Long Time Dynamics and Coherent States in Nonlinear

Wave Equations .................................................................. 59

E. Kirr

About Non Linear Stabilization for Scalar Hyperbolic Problems .......... 89

Rémi Abgrall

Part IV Group-Theoretical Approaches to Conservation Laws

and Their Applications

Generalization of Noether’s Theorem in Modern Form

to Non-variational Partial Differential Equations ............................ 119

Stephen C. Anco

vii

viii Contents

Part V Materials Science, Engineering, and New Technologies

Adaptive Simulation Selection for the Discovery of the Ground

State Line of Binary Alloys with a Limited Computational Budget ........ 185

Jesper Kristensen, Ilias Bilionis, and Nicholas Zabaras

Recent Developments in Spectral Element Simulations

of Moving-Domain Problems ................................................... 213

Paul Fischer, Martin Schmitt, and Ananias Tomboulides

Eight Great Reasons to Do Mathematics ...................................... 245

Chris Budd

Part VI Finance and Systemic Risk

Calculation of Exposure Proﬁles and Sensitivities of Options under

the Heston and the Heston Hull-White Models ............................... 265

Q. Feng and C.W. Oosterlee

A Note on Independence Copula for Conditional Markov Chains ......... 303

Tomasz R. Bielecki, Jacek Jakubowski, and Mariusz Niew˛egłowski

The Construction and Properties of Assortative Conﬁguration Graphs... 323

T.R. Hurd

Part VII Life and Environmental Sciences

Coexistence in the Face of Uncertainty ........................................ 349

Sebastian J. Schreiber

Part VIII Number Theory and Algebraic Geometry

in Cryptography and Other Applications

Computing Elliptic Curves over Q: Bad Reduction at One Prime ......... 387

Michael A. Bennett and Andrew Rechnitzer

Part IX Sustainability and Cooperation

Sustainability of Cooperation in Dynamic Games Played over

Event Trees........................................................................ 419

Georges Zaccour

Index ............................................................................... 439

Part I

Modern Challenges and Interdisciplinary

Interactions via Mathematical, Statistical,

and Computational Models

Modern Challenges and Interdisciplinary

Interactions via Mathematical, Statistical,

and Computational Models

Roderick Melnik, Roman Makarov, and Jacques Belair

Abstract We live in an incredible age. Due to extraordinary advances in sciences

and engineering, we better understand the world around us. At the same time, we

witness profound changes in the technology, environment, societal organization,and

economic well-being. We face new challenges never experienced by humans before.

To efﬁciently address these challenges, the role of interdisciplinary interactions

will continue to increase, as well as the role of mathematical, statistical, and

computational models, providing a central link for such interactions.

1 The Role of Mathematical and Statistical Models

Since the dawn of human civilizations, technological innovations have been devel-

oping hand in hand with progress in mathematical and statistical sciences. Inter-

actions and interdependence of mathematics, physics, engineering, and biology

have been well elucidated in the literature with a number of excellent reviews and

historical accounts (e.g., [12, 13, 19] and references therein). In the heart of these

interactions and interdependence are mathematical and statistical models. Their role

will continue to increase rapidly in both traditional (e.g., physics and engineering)

and many emerging (e.g., health and life sciences) areas of their applications

(e.g., [10, 15, 18] and references therein). Moreover, we are witnessing a dramatic

increase in computing power and breathtaking advances in computational science

and engineering which assist further in developing this trend.

Today, many other disciplines are catching up with this trend too. Indeed,

mathematical and statistical models can be used to describe complex phenomena

and systems such as stock markets, the internet trafﬁc, logistics, supply, and demand

R. Melnik ()•R.Makarov

Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid

Laurier University, Waterloo, ON, Canada, N2L 3C5

e-mail: rmel[email protected]

J. Belair

Departement de Mathematiques, Universite de Montreal, Montreal, QC, Canada, H3C 3J7

R. Melnik et al. (eds.), Recent Progress and Modern Challenges in Applied

Mathematics, Modeling and Computational Science, Fields Institute

Communications 79, DOI 10.1007/978-1-4939-6969-2_1