A comparison of approaches for the construction of reduced basis for stochastic Galerkin matrix equations

Abstract

We examine different approaches to an efficient solution of the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with different, uncertain coefficients in apriori known subdomains. The solution of the SG system of equations is usually a very challenging task. A relatively new approach to the solution of the SG matrix equations is the reduced basis (RB) solver, which looks for a low-rank representation of the solution. The construction of the RB is usually done iteratively and consists of multiple solutions of systems of equations. We examine multiple approaches and their modifications to the construction of the RB, namely the reduced rational Krylov subspace method and Monte Carlo sampling approach. We also aim at speeding up the process using the deflated conjugate gradients (DCG). We test and compare these methods on a set of problems with a varying random behavior of the material on subdomains as well as different geometries of subdomains.