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Strong and weak approximation of semilinear stochastic evolution equations

Author: Raphael Kruse
Publisher: Cham [Switzerland] : Springer, [2014] ©2014
Series: Lecture notes in mathematics (Springer-Verlag), 2093.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Kruse, Raphael.
Strong and weak approximation of semilinear stochastic evolution equations.
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Raphael Kruse
ISBN: 9783319022314 3319022318
OCLC Number: 867046875
Notes: Based on the author's thesis (doctoral)--Universität Bielefeld, 2012.
Description: 1 online resource (xiv, 177 pages) : illustrations.
Contents: Introduction --
Stochastic Evolution Equations in Hilbert Spaces --
Optimal Strong Error Estimates for Galerkin Finite Element Methods --
A Short Review of the Malliavin Calculus in Hilbert Spaces --
A Malliavin Calculus Approach to Weak Convergence --
Numerical Experiments --
Some Useful Variations of Gronwall's Lemma --
Results on Semigroups and their Infinitesimal Generators --
A Generalized Version of Lebesgue's Theorem --
References --
Index.
Series Title: Lecture notes in mathematics (Springer-Verlag), 2093.
Responsibility: Raphael Kruse.

Abstract:

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.

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