Spatial approximation of stochastic convolutions

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Error estimate

Finite element

Wiener process

Wavelet

Stochastic heat equation

Stochastic wave equation

Additive noise

Finite element; Wavelet; Stochastic heat equation; Stochastic wave equation; Wiener process; Additive noise; Error estimate

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