Hermitian Preconditioning for a class of Non-Hermitian Linear Systems

2023/04/07 by Nicole Spillane, Spillane, Nicole · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #65F10 #65Y05 #68W40 #Advanced Optimization Algorithms Research #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2304.03546

openalex publication_date 2023/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Do you precondition on the left or on the right?

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