Stochastic Galerkin and Monte Carlo Methods for Parabolic Problems: Numerical Performance of Variational Matrix‐Free Approximations

ABSTRACT Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block‐preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space‐time subsystems. Monte Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte Carlo approach is built on the space‐time finite element formulation together with the GMRES‐GMG solver technology. All algorithms have been implemented in a unified matrix‐free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed.