AN INTRODUCTION TO COMPUTATIONAL STOCHASTIC PDES
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationAN I N T RO D U C T I O N TO COMPUTATIONALS TO C H A S T I C PDESThis book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations,and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochasticordinary differential equations with white noise forcing, strong and weak approximation 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 element methods. Finally, stochastic parabolic PDEs are developed.Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of-the-art computational methods through worked examples, exercises, theorems and proofs. 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 science.GABRIEL J . LORDis a professor in the Maxwell Institute, Department of Mathematics, at Heriot-Watt University. He has worked on stochastic PDEs and applications for more than ten years. He is the coeditor of Stochastic Methods inNeuroscience (with C. Laing) and has organised a number of international meetings in the field. He has served as an Associate Editor for the SIAM Journal onScientific Computing and the SIAM/ASA Journal on Uncertainty Quantification.CATHERINE E . POWELL is 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. She has also served as an Associate Editor for the SIAM/ASAJournal on Uncertainty Quantification.has been working in the numerical analysis group at the University 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 1997.TONY SHARDLOW in this web service Cambridge University Presswww.cambridge.org
Cambridge 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 www.cambridge.org/mathematics.Flow, Deformation and FractureG. I. BARENBLATTThe Mathematics of Signal ProcessingSTEVEN B. DAMELIN AND WILLARD MILLER, JR.Nonlinear Dispersive WavesMARK J. ABLOWITZComplex Variables: Introduction and Applications (2nd Edition)MARK J. ABLOWITZ AND ATHANASSIOS S. FOKASScalingG. I. R. BARENBLATTIntroduction to Symmetry AnalysisBRIAN J. CANTWELLHydrodynamic InstabilitiesFRANÇOIS CHARRUA First Course in Continuum MechanicsOSCAR GONZALEZ AND ANDREW M. STUARTTheory of Vortex SoundM. S. HOWEApplied Solid MechanicsPETER HOWELL, GREGORY KOZYREFF AND JOHN OCKENDONPractical 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 LEFTON in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationA N INTRO D UCTIONTO COMPU TATIONALS TO CHA S T IC PDESG A B R I E L J. LORDHeriot-Watt University, EdinburghC AT H E R I N E E. POWELLUniversity of ManchesterTO N Y S H A RDLOWUniversity of Bath in this web service Cambridge University Presswww.cambridge.org
Cambridge 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, USACambridge University Press is part of the University of Cambridge.It furthers the University’s mission by disseminating knowledge in the pursuit ofeducation, learning and research at the highest international levels of excellence.www.cambridge.orgInformation on this title: www.cambridge.org/9780521728522c Gabriel J. Lord, Catherine E. Powell and Tony Shardlow 2014This 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 Press.First published 2014Printed in the United States of AmericaA catalog record for this publication is available from the British Library.Library of Congress Cataloging in Publication DataLord, Gabriel J., author.An introduction to computational stochastic PDEs / Gabriel J. Lord, Heriot-Watt University,Edinburgh, Catherine E. Powell, University of Manchester, Tony Shardlow, University of Bath.pages cm – (Cambridge texts in applied mathematics; 50)Includes bibliographical references and index.ISBN 978-0-521-89990-1 (hardback) – ISBN 978-0-521-72852-2 (paperback)1. Stochastic partial differential equations. I. Powell, Catherine E., author.II. Shardlow, Tony, author. III. Title.QA274.25.L67 2014519.20 2–dc232014005535ISBN 978-0-521-89990-1 HardbackISBN 978-0-521-72852-2 PaperbackAdditional resources for this publication at www.cambridge.org/9780521728522Cambridge 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 appropriate. in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationContentspage i xPrefacePART ONE1DETERMINISTIC DIFFERENTIAL EQUATIONS1Linear Analysis1.1Banach spaces Cr and L p1.2Hilbert spaces L 2 and H r1.3Linear operators and spectral theory1.4Fourier analysis1.5NotesExercises119152835362Galerkin Approximation and Finite Elements2.1Two-point boundary-value problems2.2Variational formulation of elliptic PDEs2.3The Galerkin finite element method for elliptic PDEs2.4NotesExercises4040586680833Time-dependent Di erential Equations3.1Initial-value problems for ODEs3.2Semigroups of linear operators3.3Semilinear evolution equations3.4Method of lines and finite di erences for semilinear PDEs3.5Galerkin methods for semilinear PDEs3.6Finite elements for reaction–di usion equations3.7Non-smooth error 34PART TWO1374STOCHASTIC PROCESSES AND RANDOM FIELDSProbability Theory4.1Probability spaces and random variables4.2Least-squares approximation and conditional expectation in this web service Cambridge University Press137137152www.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationviContents4.3Convergence of random variables4.4Random number generation4.5NotesExercises1571641771785Stochastic Processes5.1Introduction and Brownian motion5.2Gaussian processes and the covariance function5.3Brownian bridge, fractional Brownian motion, and white noise5.4The Karhunen–Loève expansion5.5Regularity of stochastic 6Stationary Gaussian Processes6.1Real-valued stationary processes6.2Complex-valued random variables and stochastic processes6.3Stochastic integrals6.4Sampling by quadrature6.5Sampling by circulant 7Random Fields7.1Second-order random fields7.2Circulant embedding in two dimensions7.3Turning bands method7.4Karhunen–Loève expansion of random fields7.5Sample path continuity for Gaussian random T THREE314STOCHASTIC DIFFERENTIAL EQUATIONS8Stochastic Ordinary Di erential Equations8.1Examples of SODEs8.2Itô integral8.3Itô SODEs8.4Numerical methods for Itô SODEs8.5Strong approximation8.6Weak approximation8.7Stratonovich integrals and 689Elliptic PDEs with Random Data9.1Variational formulation on D372374 in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationContents10vii9.2Monte Carlo FEM9.3Variational formulation on D9.4Variational formulation on D9.5Stochastic Galerkin FEM on D9.6Stochastic collocation FEM on D9.7NotesExercises380386393396421423428Semilinear Stochastic PDEs10.1 Examples of semilinear SPDEs10.2 Q-Wiener process10.3 Itô stochastic integrals10.4 Semilinear evolution equations in a Hilbert space10.5 Finite di erence method10.6 Galerkin and semi-implicit Euler approximation10.7 Spectral Galerkin method10.8 Galerkin finite element method10.9 endix AReferencesIndex482489499 in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore information in this web service Cambridge University Presswww.cambridge.org
Cambridge 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 di erential equations traditionally used by appliedmathematicians to model phenomena such as fluid flow, neural dynamics, electromagneticscattering, tumour growth, telecommunications, phase transitions, etc. are now mature.Parameters within those models (e.g., 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 excuses.The 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 di erential 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. It is perhaps even more so for establishedresearchers who have not received any training in probability theory and stochastic processes.We hope this book bridges this gap and will provide training for a new generation ofresearchers — that is, you.This 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 solvingdi erential equations. Our treatment of model problems includes analysis, appropriatenumerical methods and a discussion of practical implementation. Matlab is a convenientcomputer environment for numerical scientific computing and is used throughout the bookto solve examples that illustrate key concepts. We provide code to implement the algorithmson model problems, and sample code is available from the authors’ or the publisher’swebsite . Each chapter concludes with exercises, to help the reader study and become morehttp://www.cambridge.org/9780521728522 in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationxPrefacefamiliar with the concepts involved, and a section of notes, which contains pointers andreferences to the latest research directions and results.The book is divided into three parts, as follows.Part One: Deterministic Di erential Equations We start with a deterministic or nonrandom outlook and introduce preliminary background material on functional analysis,numerical analysis, and di erential equations. Chapter 1 reviews linear analysis andintroduces Banach and Hilbert spaces, as well as the Fourier transform and other key toolsfrom Fourier analysis. Chapter 2 treats elliptic PDEs, starting with a two-point boundaryvalue problem (BVP), and develops Galerkin approximation and the finite element method.Chapter 3 develops numerical methods for initial-value problems for ordinary di erentialequations (ODEs) and a class of semilinear PDEs that includes reaction–di usion equations.We develop finite di erence methods and spectral and finite element Galerkin methods.Chapters 2 and 3 include not only error analysis for selected numerical methods but alsoMatlab implementations for test problems that illustrate numerically the theoretical ordersof convergence.Part Two: Stochastic Processes and Random Fields Here we turn to probability theoryand develop the theory of stochastic processes (one parameter families of random variables)and random fields (multi-parameter families of random variables). Stochastic processes andrandom fields are used to model the uncertain inputs to the di erential equations studiedin Part Three and are also the appropriate way to interpret the corresponding solutions.Chapter 4 starts with elementary probability theory, including random variables, limittheorems, and sampling methods. The Monte Carlo method is introduced and applied toa di erential equation with random initial data. Chapters 5–7 then develop theory andcomputational methods for stochastic processes and random fields. Specific stochasticprocesses discussed include Brownian motion, white noise, the Brownian bridge, andfractional Brownian motion. In Chapters 6 and 7, we pay particular attention to the importantspecial classes of stationary processes and isotropic random fields. Simulation methodsare developed, including a quadrature scheme, the turning bands method, and the highlye cient FFT-based circulant embedding method. The theory of these numerical methods isdeveloped alongside practical implementations in Matlab.Part Three: Stochastic Di erential Equations There are many ways to incorporate stochastic e ects into di erential equations. In the last part of the book, we consider threeclasses of stochastic model problems, each of which can be viewed as an extension to adeterministic model introduced in Part One. These are:Chapter 8 ODE (3.6)Chapter 9 Elliptic BVP (2.1)Chapter 10 Semilinear PDE (3.39) white noise forcing correlated random data space–time noise forcingNote the progression from models for time t and sample variable ! in Chapter 8, to modelsfor space x and ! in Chapter 9, and finally to models for t; x;! in Chapter 10. In eachcase, we adapt the techniques from Chapters 2 and 3 to show that the problems are wellposed and to develop numerical approximation schemes. Matlab implementations are alsodiscussed. Brownian motion is key to developing the time-dependent problems with whitenoise forcing considered in Chapters 8 and 10 using the Itô calculus. It is these types in this web service Cambridge University Presswww.cambridge.org
Cambridge University Press978-0-521-89990-1 - An Introduction to Computational Stochastic PdesGabriel J. Lord, Catherine E. Powell and Tony ShardlowFrontmatterMore informationPrefacexiof di erential equations that are traditionally known as stochastic di erential equations(SODEs and SPDEs). In Chapter 9, however, we consider elliptic BVPs with both a forcingterm and coe cients that are represented by random fields not of white noise type. Manyauthors prefer to reserve the term ‘stochastic PDE’ only for PDEs forced by white noise. Weinterpret it more broadly, however, and the title of this book is intended to incorporate PDEswith data and/or forcing terms described by both white noise (which is uncorrelated) andcorrelated random fields. The analytical tools required to solve these two types of problemsare, of course, very di erent and we give an overview of the key results.Chapter 8 introduces the class of stochastic ordinary di erential equations (SODEs)consisting of ODEs with white noise forcing, discusses existence and uniqueness of solutionsin the sense of Itô calculus, and develops the Euler–Maruyama and Milstein approximationschemes. Strong approximation (of samples of the solution) and weak approximation(of averages) are discussed, as well as the multilevel Monte Carlo method. Chapter 9treats elliptic BVPs with correlated random data on two-dimensional spatial domains.These typically arise in the modelling of fluid flow in porous media. Solutions are alsocorrelated random fields and, here, do not depend on time. To begin, we consider lognormal coe cients. After sampling the input data, we study weak solutions to the resultingdeterministic problems and apply the Galerkin finite element method. The Monte Carlomethod is then used to estimate the mean and variance. By approximating the data usingKarhunen–Loève expansions, the stochastic PDE problem may also be converted to adeterministic one on a (possibly) high-dimensional parameter space. After setting up anappropriate weak formulation, the stochastic Galerkin finite element method (SGFEM),which couples finite elements in physical space with global polynomial approximation ona parameter space, is developed in detail. Chapter 10 develops stochastic parabolic PDEs,such as reaction–di usion equations forced by a space–time Wiener process, and we discuss(strong) numerical approximation in space and in time. Model problems arising in the fieldsof neuroscience and fluid dynamics are included.The number of questions that can be asked of stochastic PDEs is large. Broadly speaking,they fall into two categories: forward problems (sampling the solution, determining exittimes, computing moments, etc.) and inverse problems (e.g., fitting a model to a set ofobservations). In this book, we focus on forward problems for specific model problems. Wepay particular attention to elliptic PDEs with coe cients given by correlated random fieldsand reaction–di usion equations with white noise forcing. We also focus on methods tocompute individual samples of solutions and to compute moments (means, variances) offunctionals of the solutions. Many other stochastic
1 Linear Analysis 1 1.1 Banach spaces Cr and Lp 1 1.2 Hilbert spaces L2 and Hr 9 1.3 Linear operators and spectral theory 15 1.4 Fourier analysis 28 1.5 Notes 35 Exercises 36 2 Galerkin Approximation and Finite Elements 40 2.1 Two-point bo
Jul 09, 2010 · Stochastic Calculus of Heston’s Stochastic–Volatility Model Floyd B. Hanson Abstract—The Heston (1993) stochastic–volatility model is a square–root diffusion model for the stochastic–variance. It gives rise to a singular diffusion for the distribution according to Fell
are times when the fast stochastic lines either cross above 80 or below 20, while the slow stochastic lines do not. By slowing the lines, the slow stochastic generates fewer trading signals. INTERPRETATION You can see in the figures that the stochastic oscillator fluctuates between zero and 100. A stochastic value of 50 indicates that the closing
sion analysis on discrete-time stochastic processes. We now turn our focus to the study of continuous-time stochastic pro-cesses. In most cases, it is di cult to exactly describe the probability dis-tribution for continuous-time stochastic processes. This was also di cult for discrete time stochastic processes, but for them, we described the .
(e.g. bu er stocks, schedule slack). This approach has been criticized for its use of a deterministic approximation of a stochastic problem, which is the major motivation for stochastic programming. This dissertation recasts this debate by identifying both deterministic and stochastic approaches as policies for solving a stochastic base model,
Stochastic Programming Stochastic Dynamic Programming Conclusion : which approach should I use ? Objective and constraints Evaluating a solution Presentation Outline 1 Dealing with Uncertainty Objective and constraints Evaluating a solution 2 Stochastic Programming Stochastic Programming Approach Information Framework Toward multistage program
STOCHASTIC CALCULUS AND STOCHASTIC DIFFERENTIAL EQUATIONS 5 In discrete stochastic processes, there are many random times similar to (2.3). They are non-anticipating, i.e., at any time n, we can determine whether the cri-terion for such a random time is met or not solely by the “history” up to time n.
processes 1.Basics of stochastic processes 2.Markov processes and generators 3.Martingale problems 4.Exisence of solutions and forward equations 5.Stochastic integrals for Poisson random measures 6.Weak and strong solutions of stochastic equations 7.Stochastic equations for Markov processes in Rd 8.Convergenc
Step decision rules for multistage stochastic programming: a heuristic approach J. Th eni e J.-Ph.Vial September 27, 2006 Abstract Stochastic programming with step decision rules, SPSDR, is an attempt to over-come the curse of computational complexity of multistage stochastic programming problems. SPSDR combines several techniques.
