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.

翻译：本文研究无奇异问题中GMRES方法的收敛性。将GMRES解释为GCR方法，可简化收敛性估计的证明过程。本文探讨了预处理与GMRES中的加权范数，旨在提供一种选取预处理子及GMRES范数以确保快速收敛的方案。文章主要关注Hermitian预处理（即使针对非Hermitian问题）。我们提出选取Hermitian预处理子H，并在由H诱导的内积中应用GMRES方法。若问题矩阵A是正定的，则可证明一个新的收敛界，该收敛界仅取决于H对A的Hermitian部分的预处理效果以及A的非Hermitian程度。特别地，若已知A的Hermitian部分具有可扩展的预处理子，则所提议方法同样具有可扩展性。数值实验验证了该结论。