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© 2017

Numerical Methods for Stochastic Partial Differential Equations with White Noise

Benefits

  • Includes both theoretical and computational exercises, allowing for use with mixed-level classes

  • Provides Matlab codes for examples

  • The first book to emphasizes the Wong-Zakai approximation

  • Offers an approach to stochastic modeling other than the common Monte Carlo methods

Book

Part of the Applied Mathematical Sciences book series (AMS, volume 196)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Zhongqiang Zhang, George Em Karniadakis
    Pages 1-9
  3. Zhongqiang Zhang, George Em Karniadakis
    Pages 11-51
  4. Zhongqiang Zhang, George Em Karniadakis
    Pages 53-97
  5. Numerical Stochastic Ordinary Differential Equations

    1. Front Matter
      Pages 99-101
    2. Zhongqiang Zhang, George Em Karniadakis
      Pages 103-133
    3. Zhongqiang Zhang, George Em Karniadakis
      Pages 135-160
  6. Temporal White Noise

    1. Front Matter
      Pages 161-163
    2. Zhongqiang Zhang, George Em Karniadakis
      Pages 165-189
    3. Zhongqiang Zhang, George Em Karniadakis
      Pages 191-214
    4. Zhongqiang Zhang, George Em Karniadakis
      Pages 215-246
    5. Zhongqiang Zhang, George Em Karniadakis
      Pages 247-262
  7. Spatial White Noise

    1. Front Matter
      Pages 263-265
    2. Zhongqiang Zhang, George Em Karniadakis
      Pages 267-292
    3. Zhongqiang Zhang, George Em Karniadakis
      Pages 293-329
    4. Zhongqiang Zhang, George Em Karniadakis
      Pages 331-335
  8. Back Matter
    Pages 337-394

About this book

Introduction

This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical comparison with other integration methods in random space is made. Part III covers spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic equations as well as other equations with additive noise. Numerical methods for SPDEs with multiplicative noise are also discussed using the Wiener chaos expansion method. In addition, some SPDEs driven by non-Gaussian white noise are discussed and some model reduction methods (based on Wick-Malliavin calculus) are presented for generalized polynomial chaos expansion methods. Powerful techniques are provided for solving stochastic partial differential equations.

This book can be considered as self-contained. Necessary background knowledge is presented in the appendices. Basic knowledge of probability theory and stochastic calculus is presented in Appendix A. In Appendix B some semi-analytical methods for SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided. In Appendix D, all the conclusions which are needed for proofs are presented, and in Appendix E a method to compute the convergence rate empirically is included.

In addition, the authors provide a thorough review of the topics, both theoretical and computational exercises in the book with practical discussion of the effectiveness of the methods. Supporting Matlab files are made available to help illustrate some of the concepts further. Bibliographic notes are included at the end of each chapter. This book serves as a reference for graduate students and researchers in the mathematical sciences who would like to understand state-of-the-art numerical methods for stochastic partial differential equations with white noise.

Keywords

Nonlinear Stochastic Differential Equations Wong-Zakai Approximation Deterministic Integration Methods Long-time Integration Wick-Malliavin Approximation Space and Time White Noise Ito and Stratonovich Calculus

Authors and affiliations

  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

Bibliographic information

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Energy, Utilities & Environment
Engineering

Reviews

“Zhang and Karniadakis’ book may be used as a textbook, but it may also be considered as a reference for the state of the art concerning the numerical solution of stochastic differential equations involving white noise/Wiener processes/ Brownian motion. … Bibliographic notes address the state of the art in the field. Appendices give the necessary background in probability, stochastic calculus, semi-analytical approximation methods for stochastics differential equation, Gauss quadrature … . “ (José Eduardo Souze de Cursi, Mathematical Reviews, September, 2018)


“It is an interesting book on numerical methods for stochastic partial differential equations with white noise through the framework of Wong-Zakai approximation. ... . It is to be noted that the authors provide a thorough review of topics both theoretical and computational exercises to justify the effectiveness of the developed methods. Further, the MATLAB files are made available to the researchers and readers to understand the state of art of numerical methods for stochastic partial differential equations.” (Prabhat Kumar Mahanti, zbMATH 1380.65021, 2018)