Hermitian Preconditioning for a Class of Non-Hermitian Linear Systems

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1903-A1922, June 2024.
Abstract. This work considers the convergence of GMRES for nonsingular problems. GMRES is interpreted as the generalized conjugate residual 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 ensures 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 [math] and to apply GMRES in the inner product induced by [math]. If, moreover, the problem matrix [math] is positive definite, then a new convergence bound is proved that depends only on how well [math] preconditions the Hermitian part of [math], and on how non-Hermitian [math] is. In particular, if a scalable preconditioner is known for the Hermitian part of [math], then the proposed method is also scalable. This result is illustrated numerically.