New convergence bounds are presented for weighted, preconditioned, and deflated GMRES for the solution of large, sparse, non-Hermitian linear systems. These bounds are given for the case when the Hermitian part of the coefficient matrix is positive definite, the preconditioner is Hermitian positive definite, and the weight is equal to the preconditioner. The new bounds are a novel contribution in and of themselves. In addition, they are sufficiently explicit to indicate how to choose the preconditioner and the deflation space to accelerate the convergence. One such choice of deflating space is presented, and numerical experiments illustrate the effectiveness of such space.

翻译：针对大规模稀疏非厄米线性系统的求解，本文提出了加权、预处理及收缩GMRES方法的新收敛界。该收敛界适用于系数矩阵的厄米部分正定、预处理子为厄米正定且加权矩阵等于预处理子的情形。新收敛界本身即构成一项创新贡献。此外，这些界具有足够的显式特征，能够指导预处理子与收缩空间的选取以加速收敛。文中提出了一种具体的收缩空间选择方案，并通过数值实验验证了该空间的有效性。