Compound Krylov subspace methods for parametric linear systems

TL;DR

This paper introduces a reduced basis method utilizing compound Krylov subspaces for efficiently solving parametric linear systems with symmetric positive definite matrices, demonstrated through PDE applications.

Contribution

It presents a novel approach combining compound Krylov subspaces with reduced basis methods for parametric linear systems, including error estimates and numerical validation.

Findings

Effective reduction in computational cost for parametric PDEs

Error bounds established for subspace solutions

Numerical experiments confirm theoretical results

Abstract

In this work, we propose a reduced basis method for efficient solution of parametric linear systems. The coefficient matrix is assumed to be a linear matrix-valued function that is symmetric and positive definite for admissible values of the parameter $\mathbf{\sigma}\in \mathbb{R}^s$. We propose a solution strategy where one first computes a basis for the appropriate compound Krylov subspace and then uses this basis to compute a subspace solution for multiple $\mathbf{\sigma}$. Three kinds of compound Krylov subspaces are discussed. Error estimate is given for the subspace solution from each of these spaces. Theoretical results are demonstrated by numerical examples related to solving parameter dependent elliptic PDEs using the finite element method (FEM).