SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1248-1267, January 2021.
This paper proposes a class of polynomial preconditioners for solving non-Hermitian linear systems of equations. The polynomial is obtained from a least-squares approximation in polynomial space instead of a standard Krylov subspace. The process for building the polynomial relies on an Arnoldi-like procedure in a small dimensional polynomial space and is equivalent to performing GMRES in polynomial space. It is inexpensive and produces the desired polynomial in a numerically stable way. A few improvements to the basic scheme are discussed including the development of a short-term recurrence and the use of compound preconditioners. Numerical experiments, including a test with challenging nonnormal three-dimensional Helmholtz equations and a few publicly available sparse matrices, are provided to illustrate the performance of the proposed preconditioners.

中文翻译：

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Proxy-GMRES：在多项式空间中通过 GMRES 进行预处理

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 3 期，第 1248-1267 页，2021 年 1 月。
本文提出了一类用于求解非厄米线性方程组的多项式预处理器。多项式是从多项式空间中的最小二乘近似而不是标准 Krylov 子空间中获得的。构建多项式的过程依赖于小维多项式空间中的 Arnoldi-like 过程，相当于在多项式空间中执行 GMRES。它很便宜并且以数值稳定的方式产生所需的多项式。讨论了对基本方案的一些改进，包括短期复发的发展和复合预处理器的使用。数值实验，包括具有挑战性的非正态三维亥姆霍兹方程和一些公开可用的稀疏矩阵的测试，