Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

TL;DR

This paper introduces a hierarchical Schur complement preconditioner for stochastic Galerkin finite element methods that efficiently solves large linear systems by exploiting the recursive matrix structure without explicitly forming the matrices.

Contribution

It proposes a novel preconditioner leveraging the recursive hierarchy of the global matrices, combining Schur complement ideas with Krylov iterations, improving efficiency for stochastic finite element systems.

Findings

The preconditioner reduces the condition number of the system.

Numerical experiments demonstrate improved convergence rates.

The method avoids explicit matrix formation, saving computational resources.

Abstract

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each one of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix, nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem and the performance of the method is illustrated by numerical experiments.