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Strong and Weak Approximation of Semilinear Stochastic Evolution Equations

  • Raphael Kruse

Part of the Lecture Notes in Mathematics book series (LNM, volume 2093)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Raphael Kruse
    Pages 1-10
  3. Raphael Kruse
    Pages 129-153
  4. Back Matter
    Pages 155-180

About this book

Introduction

In this book we analyze the error caused by numerical schemes for the approximation of semilinear stochastic evolution equations (SEEq) in a Hilbert space-valued setting. The numerical schemes considered combine Galerkin finite element methods with Euler-type temporal approximations. Starting from a precise analysis of the spatio-temporal regularity of the mild solution to the SEEq, we derive and prove optimal error estimates of the strong error of convergence in the first part of the book.

The second part deals with a new approach to the so-called weak error of convergence, which measures the distance between the law of the numerical solution and the law of the exact solution. This approach is based on Bismut’s integration by parts formula and the Malliavin calculus for infinite dimensional stochastic processes. These techniques are developed and explained in a separate chapter, before the weak convergence is proven for linear SEEq.

Keywords

65C30,60H15,65M60,60H07,35B65 Galerkin finite element methods Malliavin Calculus SPDE Spatio-temporal regularity Strong and weak convergence

Authors and affiliations

  • Raphael Kruse
    • 1
  1. 1.Dept. of MathematicsBielefeld UniversityBielefeldGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-02231-4
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-02230-7
  • Online ISBN 978-3-319-02231-4
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site