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GMRES方法在非奇异问题中的收敛性。将GMRES方法视为GCR方法，可简洁证明收敛性估计。本文探讨了预处理技术及GMRES中的加权范数，旨在提供一种选择预处理器与GMRES范数的策略，以确保快速收敛。文章核心关注埃尔米特预处理（即使针对非埃尔米特问题）。提出选择埃尔米特预处理矩阵H，并在由H诱导的内积空间中应用GMRES方法。进一步地，若问题矩阵A为正定矩阵，则证明了一个新的收敛界，该收敛界仅取决于H对A的埃尔米特部分的预处理效果以及A的非埃尔米特性程度。特别地，若已知A的埃尔米特部分具有可扩展预处理器，则所提方法亦具备可扩展性。数值结果验证了这一结论。