Strong and Weak Approximation of Semilinear Stochastic Evolution Equations

Nonfiction, Science & Nature, Mathematics, Number Systems, Statistics
Cover of the book Strong and Weak Approximation of Semilinear Stochastic Evolution Equations by Raphael Kruse, Springer International Publishing
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Author: Raphael Kruse ISBN: 9783319022314
Publisher: Springer International Publishing Publication: November 18, 2013
Imprint: Springer Language: English
Author: Raphael Kruse
ISBN: 9783319022314
Publisher: Springer International Publishing
Publication: November 18, 2013
Imprint: Springer
Language: English

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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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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