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

TL;DR

This paper investigates preconditioning techniques for solving stochastic diffusion equations with lognormal coefficients, demonstrating that field-split preconditioners outperform existing methods in efficiency and parameter dependence.

Contribution

It introduces a novel field-splitting preconditioning strategy for stochastic PDEs, inspired by deterministic physics-based preconditioners, and evaluates its effectiveness.

Findings

Field-split preconditioners outperform other strategies in computational efficiency.

The approach reduces dependence on stochastic parameters.

Numerical results confirm improved convergence properties.

Abstract

We present a computational study of several preconditioning techniques for the GMRES algorithm applied to the stochastic diffusion equation with a lognormal coefficient discretized with the stochastic Galerkin method. The clear block structure of the system matrix arising from this type of discretization motivates the analysis of preconditioners designed according to a field-splitting strategy of the stochastic variables. This approach is inspired by a similar procedure used within the framework of physics based preconditioners for deterministic problems, and its application to stochastic PDEs represents the main novelty of this work. Our numerical investigation highlights the superior properties of the field-split type preconditioners over other existing strategies in terms of computational time and stochastic parameter dependence.