Truncation Preconditioners for Stochastic Galerkin Finite Element Discretizations

by   Alex Bespalov, et al.

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.



There are no comments yet.


page 1

page 2

page 3

page 4


A stabilizer free weak Galerkin element method with supercloseness of order two

The weak Galerkin (WG) finite element method is an effective and flexibl...

On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion

Convergence of an adaptive collocation method for the stationary paramet...

Parameter-robust Stochastic Galerkin mixed approximation for linear poroelasticity with uncertain inputs

Linear poroelasticity models have a number of important applications in ...

An efficient iterative method for solving parameter-dependent and random diffusion problems

This paper develops and analyzes a general iterative framework for solvi...

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

This paper analyses the following question: let A_j, j=1,2, be the Galer...

A computational study of preconditioning techniques for the stochastic diffusion equation with lognormal coefficient

We present a computational study of several preconditioning techniques f...

A priori error analysis of a numerical stochastic homogenization method

This paper provides an a priori error analysis of a localized orthogonal...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.