Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems

• Authors: Daniel Kressner;Christine Tobler
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2011

We consider linear systems $A(\alpha) x(\alpha) = b(\alpha)$ depending on possibly many parameters $\alpha = (\alpha_1,\ldots,\alpha_p)$. Solving these systems simultaneously for a standard discretization of the parameter range would require a computational effort growing drastically with the number of parameters. We show that a much lower computational effort can be achieved for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that $x(\alpha)$ can be well approximated by a tensor of low rank. In particular, low-rank tensor variants of short-recurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.