Numerical Methods for Stochastic Partial Differential Equations with White Noise
This book covers numerical methods for shastic 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 shastic 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 shastic collocation methods are presented for linear shastic advection-diffusion-reaction equations. In addition, shastic Euler equations are exploited as an application of shastic 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 shastic 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 shastic 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 shastic partial differential equations with white noise.

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Numerical Methods for Stochastic Partial Differential Equations with White Noise
This book covers numerical methods for shastic 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 shastic 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 shastic collocation methods are presented for linear shastic advection-diffusion-reaction equations. In addition, shastic Euler equations are exploited as an application of shastic 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 shastic 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 shastic 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 shastic partial differential equations with white noise.

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Numerical Methods for Stochastic Partial Differential Equations with White Noise

Numerical Methods for Stochastic Partial Differential Equations with White Noise

Numerical Methods for Stochastic Partial Differential Equations with White Noise
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Numerical Methods for Stochastic Partial Differential Equations with White Noise

Numerical Methods for Stochastic Partial Differential Equations with White Noise

Overview

This book covers numerical methods for shastic 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 shastic 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 shastic collocation methods are presented for linear shastic advection-diffusion-reaction equations. In addition, shastic Euler equations are exploited as an application of shastic 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 shastic 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 shastic 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 shastic partial differential equations with white noise.

Product Details

ISBN-13: 9783319575100
Publisher: Springer International Publishing
Publication date: 09/02/2017
Series: Applied Mathematical Sciences , #196
Edition description: 1st ed. 2017
Pages: 394
Product dimensions: 6.10(w) x 9.25(h) x (d)

Table of Contents

Preface.- Prologue.- Brownian Motion and Shastic Calculus.- Numerical Methods for Shastic Differential Equations.- Part I Shastic Ordinary Differential Equations.- Numerical Schemes for SDEs with Time Delay Using the Wong-Zakai Approximation.- Balanced Numerical Schemes for SDEs with non-Lipschitz Coefficients.- Part II Temporal White Noise.- Wiener Chaos Methods for Linear Shastic Advection-Diffusion-Reaction Equations.- Shastic Collocation Methods for Differential Equations with White Noise.- Comparison Between Wiener Chaos Methods and Shastic Collocation Methods.- Application of Collocation Method to Shastic Conservation Laws.- Part III Spatial White Noise.- Semilinear Elliptic Equations with Additive Noise.- Multiplicative White Noise: The Wick-Malliavin Approximation.- Epilogue.- Appendices.- A. Basics of Probability.- B. Semi-analytical Methods for SPDEs.- C. Gauss Quadrature.- D. Some Useful Inequalities and Lemmas.- E. Computation of Convergence Rate.
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