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

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

Mike Espig, Wolfgang Hackbusch, Alexander Litvinenko, Hermann G. Matthies, Philipp Wähnert, Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats. Special Issue of CAMWA "High-Order Finite Element Approximation for PDEs", Volume 67 (Issue 4), March 2014
Mike Espig, Wolfgang Hackbusch, Alexander Litvinenko, Hermann G. Matthies, Philipp Wähnert
Tensor approximation; Stochastic PDEs; Stochastic Galerkin matrix; Low-rank tensor formats; Uncertainty quantification
2014
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.
DOI: 10.1016/j.camwa.2012.10.008