Efficient Solution of Stochastic Galerkin Matrix Equations via Reduced Basis and Tensor Train Approximation

This contribution focuses on the development of a computational method to efficiently solve matrix equations arising from stochastic Galerkin (SG) discretization of steady Darcy flow problems with uncertain and separable permeability fields. The proposed method consists of a two-step solution process. Firstly, we construct a reduced basis for the finite element portion of the discretization using the Monte Carlo (MC) method. We consider various sampling techniques for the MC method. Secondly, we use a tensor polynomial basis to handle the stochastic aspect of the problem and employ a tensor-train (TT) approximation to approximate the overall solution of the reduced SG system. To enhance the convergence of the TT approximation, we use an implicitly preconditioned system with a Kronecker-type preconditioner. Moreover, we also develop low-cost error indicators to assess the accuracy of both the reduced basis and the final solution of the reduced system.