A Parallel Multilevel Spectral Galerkin Solver for Linear Systems with Uncertain Parameters

We introduce a parallel multilevel method for spectral Galerkin projected linear systems with uncertain parameters using Polynomial Chaos expansions. We utilize the hierarchical multilevel structure with respect to the polynomial degree and carry out the smoothing of high mode errors by employing the mean based preconditioner. The multilevel scheme only requires solutions of deterministic systems based on the mean operator, which makes the approach feasible for use with existing code for deterministic models. We develop an efficient load balancing strategy for the parallel computation of the matrix vector product using distributed memory, which allows for a decoupled application of the restriction and prolongation operators in the multilevel scheme. The parallel efficiency and convergence properties of the numerical method are evaluated on a Poisson benchmark problem with uncertain parameters of varying stochastic complexity.