An Introduction to Computational Stochastic PDEs
An Introduction to Computational Stochastic PDEs
Catherine E. Powell, University of Manchester
Tony Shardlow, University of Bath
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.
- Assumes little previous exposure to probability and statistics, appealing to traditional applied mathematicians and numerical analysts
- Includes downloadable MATLAB® code and discusses practical implementations, giving a practical route for examining stochastic effects in the reader's own work
- Uses numerical examples throughout to bring theoretical results to life and build intuition
Reviews & endorsements
'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
Product details
August 2014Hardback
9780521899901
516 pages
254 × 180 × 28 mm
1.18kg
107 b/w illus. 16 colour illus. 222 exercises
Available
Table of Contents
- 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.
