Numerical methods for stochastic partial differential equations with white noise
 Responsibility
 Zhongqiang Zhang, George Em Karniadakis.
 Publication
 Cham : Springer, [2017]
 Physical description
 xv, 394 pages : illustrations (chiefly color) ; 25 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 196.
Access
Available online
Science Library (Li and Ma)
Stacks
Call number  Status 

QA274.25 .Z43 2017  Unknown 
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Creators/Contributors
 Author/Creator
 Zhang, Zhongqiang, author.
 Contributor
 Karniadakis, George, author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 359390) and index.
 Contents

 Preface
 Prologue
 Why random and brownian motion (white noise)?
 Modeling with SPDEs
 Specific topics of this book
 Brownian motion and stochastic calculus
 Gaussian processes and their representations
 Brownian motion and white noise
 Some properties of brownian motion
 Approximation of brownian motion
 Brownian motion and stochastic calculus
 Stochastic chain rule : ito formula
 Integration methods in random space
 Monte Carlo method and its variants
 QuasiMonte Carlo methods
 Wiener chaos expansion method
 Stochastic collocation method
 Application to SODEs
 Bibliographic notes
 Suggested practice
 Numerical methods for stochastic differential equations
 Basic aspects of SODEs
 Existence and uniqueness of strong solutions
 Solution methods
 Numerical methods for SODEs
 Derivation of numerical methods based on numerical integration
 Strong convergence
 Weak convergence
 Linear stability
 Summary of numerical SODEs
 Basic aspects of SPDEs
 Functional spaces
 Solutions in different senses
 Solutions to SPDEs in explicit form
 Linear stochastic advectiondiffusionreaction equations
 Existence and uniqueness
 Conversion between Ito and stratonovich formulation
 Numerical methods for SPDEs
 Direct semidiscretization methods for parabolic SPDEs
 WongZakai approximation for parabolic SPDEs
 Preprocessing methods for parabolic SPDEs
 What could go wrong? examples of stochastic Burgers and NavierStokes equations
 Stability and convergence of existing numerical methods
 Summary of numerical SPDEs
 Summary and bibliographic notes
 Suggested practice
 Numerical stochastic ordinary differential equations
 Numerical schemes for SDEs with time delay using the WongZakai approximation
 WongZakai approximation for SODEs
 WongZakai approximation for SDDEs
 Derivation of numerical schemes
 A predictorcorrector scheme
 The midpoint scheme
 A Milsteinlike scheme
 Linear stability of some schemes
 Numerical results
 Summary and bibliographic notes
 Suggested practice
 Balanced numerical schemes for SDEs with nonLipschitz coefficients
 A motivating example
 Fundamental theorem
 On application of theorem 5.2.3
 Proof of the fundamental theorem
 A balanced Euler scheme
 Numerical examples
 Some numerical schemes
 Numerical results
 Summary and bibliographic notes
 Suggested practice
 Temporal white noise
 Wiener chaos methods for linear stochastic advectiondiffusionreaction equations
 Description of methods
 Multistage WCE method
 Algorithm for computing moments
 Examples in one dimension
 Numerical results for onedimensional advectiondiffusionreaction equations
 Comparison of the WCE algorithm and Monte Carlo type algorithms
 A twodimensional passive scalar equation
 A Monte Carlo method based on the method of characteristics
 Comparison between recursive WCE and Monte Carlo methods
 Summary and bibliographic notes
 Suggested practice
 Stochastic collocation methods for differential equations with white noise
 Introduction
 Isotropic sparse grid for weak integration of SDE
 Probabilistic interpretation of SCM
 Illustrative examples
 Recursive collocation algorithm for linear SPDEs
 Numerical results
 Summary and bibliographic notes
 Suggested practice
 Comparison between Wiener chaos methods and stochastic collocation methods
 Introduction
 Review of wiener chaos and stochastic collocation
 Wiener chaos expansion (WCE)
 Stochastic collocation method (SCM)
 Error estimates
 Error estimates for WCE
 Error estimate for SCM
 Numerical results
 Summary and bibliographic notes
 Suggested practice
 Application of collocation method to stochastic conservation laws
 Introduction
 Theoretical background
 Stochastic Euler equations
 Verification of the Stratonovich and ItoEuler equations
 A splitting method for stochastic Euler equations
 StratonovichEuler equations versus firstorder perturbation analysis
 StratonovichEuler equations versus ItoEuler equations
 Applying the stochastic collocation method
 Summary and bibliographic notes
 Suggested practice
 Spatial White Noise
 Semilinear elliptic equations with additive noise
 Introduction
 Assumptions and schemes
 Error estimates for strong and weak convergence order
 Examples of other PDEs
 Proofs of the strong convergence order
 Weak convergence order
 Error estimates for finite element approximation
 Numerical results
 Summary and bibliographic notes
 Suggested practice
 Multiplicative white noise : the WickMalliavin approximation
 Introduction
 Approximation using the WickMalliavin expansion
 Lognormal coefficient
 Onedimensional example
 White noise as coefficient
 Error estimates
 Numerical results
 Application of WickMalliavin approximation to nonlinear SPDEs
 WickMalliavin approximation : extensions for nonGaussian white noise
 Numerical results
 Malliavin derivatives for poisson noises
 Summary and bibliographic notes
 Suggested practice
 Epilogue
 A review of this work
 Some open problems
 Appendices
 Basics of probability
 Probability space
 Random variable
 Conditional expectation
 Properties of conditional expectation
 Filtration and martingales
 Continuous time stochastic process
 Semianalytical methods for SPDEs
 Gauss quadrature
 Gauss quadrature
 GaussHermite quadrature
 Some useful inequalities and lemmas
 Computation of convergence rate
 References
 Index.
 Publisher's Summary
 This book covers numerical methods for stochastic partial differential equations with white noise using the framework of WongZakai 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 WongZakai 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 advectiondiffusionreaction 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 nonGaussian white noise are discussed and some model reduction methods (based on WickMalliavin 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 selfcontained. 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 semianalytical 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 stateoftheart numerical methods for stochastic partial differential equations with white noise.
(source: Nielsen Book Data)9783319575100 20180115
Subjects
Bibliographic information
 Publication date
 2017
 Series
 Applied mathematical sciences, 00665452 ; volume 196
 ISBN
 9783319575100 (hd.bd.)
 9783319575117 (eBook)
 3319575104 (paperback)