AN INTRODUCTION TO COMPUTATIONAL STOCHASTIC …

1 A N I N T RO D U C T I O N T O C O M P U TAT I O NA LS T O C H A S T I C P D E SThis book gives a comprehensive INTRODUCTION to numerical methods and anal-ysis of STOCHASTIC processes, random fields and STOCHASTIC differential equations,and offers graduate students and researchers powerful tools for understanding un-certainty quantification for risk analysis. Coverage includes traditional stochasticordinary differential equations with white noise forcing, strong and weak approx-imation and the multilevel Monte Carlo method. Later chapters apply the theoryof random fields to the numerical solution of elliptic PDEs with correlated randomdata, discuss the Monte Carlo method and introduce STOCHASTIC Galerkin finite ele-ment methods. Finally, STOCHASTIC parabolic PDEs are little previous exposure to probability and statistics, theory is devel-oped in tandem with state-of-the-art COMPUTATIONAL methods through worked ex-amples, exercises, theorems and proofs.

2 The set of MATLAB codes included (anddownloadable) allows readers to perform computations themselves and solve thetest problems discussed. Practical examples are drawn from finance, mathematicalbiology, neuroscience, fluid flow modelling and materials a professor in the Maxwell Institute, Department of Math-ematics, at Heriot-Watt University. He has worked on STOCHASTIC PDEs and ap-plications for more than ten years. He is the coeditor ofStochastic Methods inNeuroscience(with C. Laing) and has organised a number of international meet-ings in the field. He has served as an Associate Editor for theSIAM Journal onScientific Computingand theSIAM/ASA Journal on Uncertainty a senior lecturer in applied mathematics and numericalanalysis at the University of Manchester. She has worked in the field of stochasticPDEs and uncertainty quantification for more than ten years. Together with TonyShardlow, she initiated the NASPDE (Numerical Analysis of STOCHASTIC PDEs)series of meetings.

3 She has also served as an Associate Editor for theSIAM/ASAJ ournal on Uncertainty SHARDLOWhas been working in the numerical analysis group at the Uni-versity of Bath since 2012. Before that, he held appointments at the universities ofManchester, Durham, Oxford and Minnesota. He completed his PhD in scientificcomputing and COMPUTATIONAL mathematics at Stanford University in in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationCambridge Texts in Applied MathematicsAll titles listed below can be obtained from good booksellers or from Cambridge University Press. For acomplete series listing, visit , Deformation and FractureG. I. BARENBLATTThe Mathematics of Signal ProcessingSTEVEN B. DAMELIN AND WILLARD MILLER, Dispersive WavesMARK J. ABLOWITZC omplex Variables: INTRODUCTION and Applications (2nd Edition)MARK J.

4 ABLOWITZ AND ATHANASSIOS S. FOKASS calingG. I. R. BARENBLATTI ntroduction to Symmetry AnalysisBRIAN J. CANTWELLH ydrodynamic InstabilitiesFRANC OIS CHARRUA First Course in Continuum MechanicsOSCAR GONZALEZ AND ANDREW M. STUARTT heory of Vortex SoundM. S. HOWEA pplied Solid MechanicsPETER HOWELL, GREGORY KOZYREFF AND JOHN OCKENDONP ractical Applied Mathematics: Modelling, Analysis, ApproximationSAM HOWISONA First Course in the Numerical Analysis of Differential Equations(2nd Edition)ARIEH ISERLESA First Course in Combinatorial OptimizationJON LEEAn INTRODUCTION to Parallel and Vector Scientific ComputationRONALD W. SHONKWILER AND LEW in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationA N I N T RO D U C T I O NT O C O M P U TAT I O NA LS T O C H A S T I C P D E SG A B R I E L J.

5 L O R DHeriot-Watt University, EdinburghC AT H E R I N E E . P OW E L LUniversity of ManchesterT O N Y S H A R D L OWUniversity of in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore information32 Avenue of the Americas, New York, NY 10013-2473, USAC ambridge University Press is part of the University of furthers the University s mission by disseminating knowledge in the pursuit ofeducation, learning and research at the highest international levels of on this title: Gabriel J. Lord, Catherine E. Powell and Tony Shardlow 2014 This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University published 2014 Printed in the United States of AmericaA catalog record for this publication is available from the British of Congress Cataloging in Publication DataLord, Gabriel J.

6 , INTRODUCTION to COMPUTATIONAL STOCHASTIC PDEs / Gabriel J. Lord, Heriot-Watt University,Edinburgh, Catherine E. Powell, University of Manchester, Tony Shardlow, University of cm (Cambridge texts in applied mathematics; 50)Includes bibliographical references and 978-0-521-89990-1 (hardback) ISBN 978-0-521-72852-2 (paperback)1. STOCHASTIC partial differential equations. I. Powell, Catherine E., Shardlow, Tony, author. III. 2 dc232014005535 ISBN 978-0-521-89990-1 HardbackISBN 978-0-521-72852-2 PaperbackAdditional resources for this publication at University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party Internet websites referred to in this publication,and does not guarantee that any content on such websites is, or will remain,accurate or in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J.

7 Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationContentsPrefacepageiPART ONE DETERMINISTIC DIFFERENTIAL EQUATIONS11 Linear spaces operators and spectral Approximation and Finite boundary-value formulation of elliptic Galerkin finite element method for elliptic Differential problems for of linear evolution of lines and finite differences for semilinear methods for semilinear elements for reaction diffusion error TWO STOCHASTIC PROCESSES AND RANDOM FIELDS1374 Probability spaces and random approximation and conditional in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore of random number and Brownian processes and the covariance bridge, fractional Brownian motion, and white Karhunen Lo ve of STOCHASTIC Gaussian stationary random variables and STOCHASTIC by by circulant random embedding in two bands Lo ve expansion of random path continuity for Gaussian random THREE STOCHASTIC DIFFERENTIAL EQUATIONS3148 STOCHASTIC Ordinary Differential of methods for It integrals and PDEs with Random formulation in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J.

8 Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore Carlo formulation onD formulation onD Galerkin FEM onD collocation FEM onD STOCHASTIC of semilinear STOCHASTIC evolution equations in a Hilbert difference and semi-implicit Euler Galerkin finite element in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore in this web service Cambridge University PressCambridge University Press978-0-521-89990-1 - An INTRODUCTION to COMPUTATIONAL STOCHASTIC PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationPrefaceTechniques for solving many of the differential equations traditionally used by appliedmathematicians to model phenomena such as fluid flow, neural dynamics, electromagneticscattering, tumour growth, telecommunications, phase transitions, etc.

9 Are now within those models ( , material properties, boundary conditions, forcingterms, domain geometries) are often assumed to be known exactly, even when it is clearthat is not the case. In the past, mathematicians were unable to incorporate noise and/oruncertainty into models because they were constrained both by the lack of computationalresources and the lack of research into STOCHASTIC analysis. These are no longer good rapid increase in computing power witnessed in recent decades allows the extra level ofcomplexity induced by uncertainty to be incorporated into numerical simulations. Moreover,there are a growing number of researchers working on STOCHASTIC partial differential equations(PDEs) and their results are continually improving our theoretical understanding of thebehaviour of STOCHASTIC systems. The transition from working with purely deterministicsystems to working with STOCHASTIC systems is understandably daunting for recent graduateswho have majored in applied mathematics.

10 It is perhaps even more so for establishedresearchers who have not received any training in probability theory and STOCHASTIC hope this book bridges this gap and will provide training for a new generation ofresearchers that is, text provides a friendly INTRODUCTION and practical route into the numerical solutionand analysis of STOCHASTIC PDEs. It is suitable for mathematically grounded graduates whowish to learn about STOCHASTIC PDEs and numerical solution methods. The book will alsoserve established researchers who wish to incorporate uncertainty into their mathematicalmodels and seek an INTRODUCTION to the latest numerical techniques. We assume knowledgeof undergraduate-level mathematics, including some basic analysis and linear algebra, butprovide background material on probability theory and numerical methods for solvingdifferential equations. Our treatment of model problems includes analysis, appropriatenumerical methods and a discussion of practical a convenientcomputer environment for numerical scientific computing and is used throughout the bookto solve examples that illustrate key concepts.