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Low-rank tensor Krylov subspace methods for parametrized linear systems. (English) Zbl 1237.65034

Summary: 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 partial differential equations (PDEs) with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.

MSC:

65F10 Iterative numerical methods for linear systems
15A69 Multilinear algebra, tensor calculus
65F08 Preconditioners for iterative methods
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
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