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In this study, we consider the numerical solution of large systems of linearequations obtained from the stochastic Galerkin formulation of stochasticpartial differential equations. We propose an iterative algorithm that exploitsthe Kronecker product structure of the linear systems. The proposed algorithmefficiently approximates the solutions in low-rank tensor format. Usingstandard Krylov subspace methods for the data in tensor format iscomputationally prohibitive due to the rapid growth of tensor ranks during theiterations. To keep tensor ranks low over the entire iteration process, wedevise a rank-reduction scheme that can be combined with the iterativealgorithm. The proposed rank-reduction scheme identifies an important subspacein the stochastic domain and compresses tensors of high rank on-the-fly duringthe iterations. The proposed reduction scheme is a multilevel method in thatthe important subspace can be identified inexpensively in a coarse spatial gridsetting. The efficiency of the proposed method is illustrated by numericalexperiments on benchmark problems.

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations