An Introduction to Computational Stochastic PDEs

This book offers a practical presentation of stochastic partial differential equations arising in physical applications and their numerical approximation.

Gabriel J. Lord (Author), Catherine E. Powell (Author), Tony Shardlow (Author)

9780521728522, Cambridge University Press

Paperback, published 11 August 2014

520 pages, 107 b/w illus. 16 colour illus. 222 exercises
24.7 x 17.5 x 2.2 cm, 1.02 kg

'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

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 modelling and materials science.

Part I. Deterministic Differential Equations: 1. Linear analysis
2. Galerkin approximation and finite elements
3. Time-dependent differential equations
Part II. Stochastic Processes and Random Fields: 4. Probability theory
5. Stochastic processes
6. Stationary Gaussian processes
7. Random fields
Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs)
9. Elliptic PDEs with random data
10. Semilinear stochastic PDEs.

Subject Areas: Mathematical theory of computation [UYA], Stochastics [PBWL], Probability & statistics [PBT], Numerical analysis [PBKS], Differential calculus & equations [PBKJ], Mathematics [PB], Finance [KFF], Risk assessment [GPQD]