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Low-Rank Solution of Unsteady Diffusion Equations with Stochastic Coefficients

We study the solution of linear systems resulting from the discretization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure, and thus solving them can be both time- and computer memory-consuming. Under certain assumptions, we show that the solution of such linear systems can be approximated with a vector of low tensor rank. We then solve the linear systems using low-rank preconditioned iterative solvers. Numerical experiments demonstrate that these low-rank preconditioned solvers are effective, especially when the fluctuations in the random data are not too large relative to their mean values.