Description: This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of the art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modeling and materials science.
Brief description: Catherine Powell is a Senior Lecturer in Applied Mathematics and Numerical Analysis at the University of Manchester. She has worked in the field of stochastic PDEs and uncertainty quantification for ten years. She has co-organised several conferences on the subject, and together with Tony Shardlow, initialised the annual NASPDE series of meetings (now in its sixth year). Currently, she is the principal investigator on an Engineering and Physical Sciences Research Council funded project on the 'Numerical Analysis of PDEs with Random Data'. She is a member of the Society for Industrial and Applied Mathematics and an Associate Editor for the SIAM/ASA Journal on Uncertainty Quantification.
Review Quotes: "This book gives both accessible and extensive coverage on stochastic partial differential equations and their numerical solutions. It offers a well-elaborated background needed for solving numerically stochastic PDEs, both parabolic and elliptic. For the numerical solutions it presents not only proofs of convergence results of different numerical methods but also actual implementations, here in Matlab, with technical details included ... With numerical implementations hard to find elsewhere in the literature, and a nice presentation of new research findings together with rich references, the book is a welcome companion for anyone working on numerical solutions of stochastic PDEs, and may also be suitable for use in a course on computational stochastic PDEs."
Roger Pettersson, Mathematical Reviews