Hermitian Preconditioning for a class of Non-Hermitian Linear Systems

TL;DR

This paper introduces a Hermitian preconditioning approach for non-Hermitian linear systems, providing convergence guarantees for GMRES by focusing on the Hermitian part of the matrix and demonstrating scalability with numerical results.

Contribution

It proposes a novel Hermitian preconditioning method for non-Hermitian systems and establishes convergence bounds based on the Hermitian part of the matrix.

Findings

Convergence bounds depend on preconditioning the Hermitian part of A.

The method is scalable if a preconditioner for the Hermitian part is available.

Numerical experiments confirm the effectiveness of the proposed approach.

Abstract

This work considers the convergence of GMRES for non-singular problems. GMRES is interpreted as the GCR method which allows for simple proofs of the convergence estimates. Preconditioning and weighted norms within GMRES are considered. The objective is to provide a way of choosing the preconditioner and GMRES norm that ensure fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems). It is proposed to choose a Hermitian preconditioner H and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on how non-Hermitian A is. In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. This result is illustrated numerically.