An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations

Powell, Catherine E. and Silvester, David and Simoncini, Valeria (2015) An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations. [MIMS Preprint]

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Abstract

Stochastic Galerkin finite element approximation of PDEs with random inputs leads to linear systems of equations with coefficient matrices that have a characteristic Kronecker product structure. By reformulating the systems as multi-term linear matrix equations, we develop an efficient solution algorithm which generalizes ideas from rational Krylov subspace approximation. The new approach determines a low-rank approximation to the solution matrix by performing a projection onto a low-dimensional space and provides an efficient solution strategy whose convergence rate is independent of the spatial approximation. Moreover, it requires far less memory than the standard preconditioned conjugate gradient method applied to the Kronecker formulation of the linear systems.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	generalized matrix equations, PDEs with random data, stochastic finite elements, iterative solvers, rational Krylov subspace methods
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Dr C.E. Powell
Date Deposited:	27 Jul 2015
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2350

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