Strong And Weak Approximation Of Semilinear Stochastic Evolution Equations
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Strong and Weak Approximation of Semilinear Stochastic Evolution Equations
In this book we analyze the error caused by numerical schemes for the approximation of semilinear stochastic evolution equations (SEEq) in a Hilbert space-valued setting. The numerical schemes considered combine Galerkin finite element methods with Euler-type temporal approximations. Starting from a precise analysis of the spatio-temporal regularity of the mild solution to the SEEq, we derive and prove optimal error estimates of the strong error of convergence in the first part of the book. The second part deals with a new approach to the so-called weak error of convergence, which measures the distance between the law of the numerical solution and the law of the exact solution. This approach is based on Bismut’s integration by parts formula and the Malliavin calculus for infinite dimensional stochastic processes. These techniques are developed and explained in a separate chapter, before the weak convergence is proven for linear SEEq.
Numerical Approximations of Stochastic Maxwell Equations
The stochastic Maxwell equations play an essential role in many fields, including fluctuational electrodynamics, statistical radiophysics, integrated circuits, and stochastic inverse problems. This book provides some recent advances in the investigation of numerical approximations of the stochastic Maxwell equations via structure-preserving algorithms. It presents an accessible overview of the construction and analysis of structure-preserving algorithms with an emphasis on the preservation of geometric structures, physical properties, and asymptotic behaviors of the stochastic Maxwell equations. A friendly introduction to the simulation of the stochastic Maxwell equations with some structure-preserving algorithms is provided using MATLAB for the reader’s convenience. The objects considered in this book are related to several fascinating mathematical fields: numerical analysis, stochastic analysis, (multi-)symplectic geometry, large deviations principle, ergodic theory, partial differential equation, probability theory, etc. This book will appeal to researchers who are interested in these topics.
Numerical Methods for Stochastic Partial Differential Equations with White Noise
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.