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An Introduction to Computational Stochastic PDEs


Part of Cambridge Texts in Applied Mathematics

  • Date Published: October 2014
  • availability: Available
  • format: Paperback
  • isbn: 9780521728522

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About the Authors
  • 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
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    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

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    Product details

    • Date Published: October 2014
    • format: Paperback
    • isbn: 9780521728522
    • length: 520 pages
    • dimensions: 247 x 175 x 22 mm
    • weight: 1.02kg
    • contains: 107 b/w illus. 16 colour illus. 222 exercises
    • availability: 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.

  • Resources for

    An Introduction to Computational Stochastic PDEs

    Gabriel J. Lord, Catherine E. Powell, Tony Shardlow

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  • Authors

    Gabriel J. Lord, Heriot-Watt University, Edinburgh
    Gabriel Lord is a Professor in the Maxwell Institute, Department of Mathematics, at Heriot-Watt University, Edinburgh. He has worked on stochastic PDEs and applications for the past ten years. He is the co-editor of Stochastic Methods in Neuroscience with C. Liang, has organised a number of international meetings in the field, and is principal investigator on the porous media processes and mathematics network funded by the Engineering and Physical Sciences Research Council (UK). He is a member of the Society for Industrial and Applied Mathematics, LMS, and EMS, as well as an Associate Editor for the SIAM Journal on Scientific Computing and the SIAM/ASA Journal on Uncertainty Quantification.

    Catherine E. Powell, University of Manchester
    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.

    Tony Shardlow, University of Bath
    Tony Shardlow has been working in the numerical analysis group at the University of Bath since 2012. Before that, he held appointments at the universities of Manchester, Durham, Oxford, and Minnesota. He completed his Ph.D. in Scientific Computing and Computational Mathematics at Stanford University in 1997.

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