Preconditioned Chebyshev BiCG for parameterized linear systems

TL;DR

This paper introduces a preconditioned Chebyshev BiCG method for efficiently solving parameterized large sparse linear systems by linearizing the problem and applying Krylov subspace techniques, enabling simultaneous solutions for many parameter values.

Contribution

It develops a novel short-term recurrence algorithm using Chebyshev interpolation and preconditioned BiCG for parameterized systems, including both exact and inexact preconditioning strategies.

Findings

Effective for large-scale Helmholtz problems with parameterized coefficients.

Achieves simultaneous solutions for multiple parameter values efficiently.

Software implementation is publicly available for reproducibility.

Abstract

We consider the problem of approximating the solution to $A(\mu) x(\mu) = b$ for many different values of the parameter $\mu$. Here we assume $A(\mu)$ is large, sparse, and nonsingular with a nonlinear dependence on $\mu$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(\mu)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation to $x(\mu)$ for many different values of the parameter $\mu \in [-a,a]$ simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.