Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats

Authors:
Mike Espig;Wolfgang Hackbusch;Alexander Litvinenko;Hermann G. Matthies;Philipp Wähnert
Affiliations:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany;Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany;Technische Universität Braunschweig, Germany;Technische Universität Braunschweig, Germany;Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
Venue:
Computers & Mathematics with Applications
Year:
2014

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Abstract

In this article, we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log-normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials.