An introduction to computational stochastic PDEs (Q2925391)

This book introduces the reader to the numerical solution of stochastic partial differential equations by devoting considerable space to stating and discussing pertinent concepts, definitions, and theorems from several branches of mathematics that form the foundation of the subject. Chapter 1 presents ideas from real, functional, and Fourier analysis. In Chapter 2, spectral Galerkin and finite element solution of deterministic two point boundary value problems, and of elliptical partial differential equations are discussed. Chapter 3 delves into numerical methods for deterministic systems of first-order ordinary differential equations, especially the Euler and backward Euler methods, and for deterministic partial differential equations, especially Galerkin methods. Chapter 4 lists some basic ideas of probability theory, featuring convergence of sequences of random variables, random number generation, and Monte Carlo simulation. In Chapter 5, stochastic processes are introduced with special attention to Gaussian processes and their Karhunen-Loeve expansion. Chapter 6 is devoted to stationary Gaussian processes and sampling them. Chapter 7 turns to random fields, both stationary and isotropic, and their approximation. In Chapter 8, stochastic ordinary differential equations are presented with main emphasis on the Itō type and their numerical solution.NEWLINENEWLINE Having covered the key ideas from the multiple areas that comprise the prerequisite background, the numerical solution of stochastic partial differential equations is addressed in the last two chapters. Chapter 9 concerns elliptic partial differential equations with random data. Galerkin finite element approximation and its use with Monte Carlo simulation is described. Discussion of numerical results and error analysis follows. A brief glimpse at stochastic collocation is given. Chapter 10 deals with semilinear stochastic partial differential equations and introduces \(Q\)-Wiener processes. Itō semilinear evolution equations are discussed and approximated using the finite difference method and using Galerkin approximation together with semi-implicit Euler and Milstein numerical methods.NEWLINENEWLINE The book contains many helpful examples, code for implementing some of the numerical schemes, and a lengthy list of references. Each chapter ends with around two dozen exercises that can engage the reader more deeply into the chapter content. This book is a valuable addition to the literature both as an introduction to the subject and as a useful reference.