An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations Article Swipe
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· 2017
· Open Access
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· DOI: https://doi.org/10.1137/15m1032399
· OA: W990331194
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. Our working assumptions are that the number of random variables<br/>characterizing the random inputs is modest, in the order of a few tens, and that the dependence on these variables is linear, so that it is sufficient to seek only a reduction in the complexity associated with the spatial component of the<br/>approximation space. 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<br/>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.
