Spatial approximation of stochastic convolutions
Journal article, 2011

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.

Error estimate

Finite element

Wiener process


Stochastic heat equation

Stochastic wave equation

Additive noise


Mihaly Kovacs

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Stig Larsson

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Fredrik Lindgren

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Journal of Computational and Applied Mathematics

0377-0427 (ISSN)

Vol. 235 12 3554-3570

Subject Categories

Computational Mathematics


Basic sciences



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