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Spatial approximation of stochastic convolutions

Journal article
Authors Mihaly Kovacs
Stig Larsson
Fredrik Lindgren
Published in J. Comput. Appl. Math.
Volume 235
Issue 12
Pages 3554-3570
ISSN 0377-0427
Publication year 2011
Published at Department of Mathematical Sciences, Mathematics
Pages 3554-3570
Language en
Links dx.doi.org/10.1016/j.cam.2011.02.01...
Keywords Finite element; Wavelet; Stochastic heat equation; Stochastic wave equation; Wiener process; Additive noise; Error estimate
Subject categories Numerical analysis

Abstract

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process which is driving the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancellation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.

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