Truncation Preconditioners for Stochastic Galerkin Finite Element Discretizations
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Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we introduce and analyze a new class of preconditioners for SGFEM that we call truncation preconditioners. Extending the idea of the mean-based preconditioner, the new preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare our preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones.
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