Powell, C. E.; Silvester, D.; Simoncini, V. An efficient reduced basis solver for stochastic Galerkin matrix equations. (English) Zbl 1381.35257 SIAM J. Sci. Comput. 39, No. 1, A141-A163 (2017). Summary: 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 multiterm 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 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 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 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. Cited in 22 Documents MSC: 35R60 PDEs with randomness, stochastic partial differential equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs Keywords:generalized matrix equations; PDEs with random data; stochastic finite elements; iterative solvers; rational Krylov subspace methods Software:redbKIT; ALEA PDFBibTeX XMLCite \textit{C. E. Powell} et al., SIAM J. Sci. Comput. 39, No. 1, A141--A163 (2017; Zbl 1381.35257) Full Text: DOI References: [1] I. Babuška, R. Tempone, and G.E. Zouraris, {\it Galerkin finite element approximations of stochastic elliptic partial differential equations}, SIAM J. Numer. Anal., 42 (2004), pp. 800-825. · Zbl 1080.65003 [2] J. Ballani and L. 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