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

Mike Espig, Wolfgang Hackbusch, Alexander Litvinenko, Hermann G. Matthies, and Philipp Wähnert

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Submission date: 16. May. 2012 (revised version: May 2012)
Pages: 20
published in: Computers and mathematics with applications, 67 (2014) 4, p. 818-829 
DOI number (of the published article): 10.1016/j.camwa.2012.10.008
Bibtex
MSC-Numbers: 65N30, 35R60, 60H15, 15A69
Keywords and phrases: tensor approximation, stochastic PDEs, Galerkin matrix, Tensor formats
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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.