Preconditioned infinite GMRES for parameterized linear systems

TL;DR

This paper introduces a preconditioned infinite GMRES method for efficiently solving large, sparse, parameterized linear systems, providing a cheap-to-evaluate approximate solution function across parameters with improved performance over existing methods.

Contribution

The paper develops a novel approach combining companion linearization and flexible GMRES, allowing inexact preconditioning and efficient evaluation of solutions for many parameter values.

Findings

Method improves computational efficiency over infinite GMRES

Error estimation accounts for parameter magnitude and preconditioning inexactness

Numerical results demonstrate competitiveness on Helmholtz equation problems

Abstract

We are interested in obtaining approximate solutions to parameterized linear systems of the form $A(\mu) x(\mu) = b$ for many values of the parameter $\mu$. Here $A(\mu)$ is large, sparse, and nonsingular, with a nonlinear analytic dependence on $\mu$. Our approach is based on a companion linearization for parameterized linear systems. The companion matrix is similar to the operator in the infinite Arnoldi method, and we use this to adapt the flexible GMRES setting. In this way, our method returns a function $\tilde{x}(\mu)$ which is cheap to evaluate for different $\mu$, and the preconditioner is applied only approximately. This novel approach leads to increased freedom to carry out the action of the operation inexactly, which provides performance improvement over the method infinite GMRES, without a loss of accuracy in general. We show that the error of our method is estimated based on the magnitude of the parameter $\mu$, the inexactness of the preconditioning, and the spectrum of the linear companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The simulations are reproducible and publicly available online.