  In this study, we consider the numerical solution of large systems of linear
equations obtained from the stochastic Galerkin formulation of stochastic
partial differential equations. We propose an iterative algorithm that exploits
the Kronecker product structure of the linear systems. The proposed algorithm
efficiently approximates the solutions in low-rank tensor format. Using
standard Krylov subspace methods for the data in tensor format is
computationally prohibitive due to the rapid growth of tensor ranks during the
iterations. To keep tensor ranks low over the entire iteration process, we
devise a rank-reduction scheme that can be combined with the iterative
algorithm. The proposed rank-reduction scheme identifies an important subspace
in the stochastic domain and compresses tensors of high rank on-the-fly during
the iterations. The proposed reduction scheme is a multilevel method in that
the important subspace can be identified inexpensively in a coarse spatial grid
setting. The efficiency of the proposed method is illustrated by numerical
experiments on benchmark problems.
