SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 659-682, January 2021.
A parallel preconditioner is proposed for general large sparse linear systems that combines a power series expansion method with low-rank correction techniques. To enhance convergence, a power series expansion is added to a basic Schur complement iterative scheme by exploiting a standard matrix splitting of the Schur complement. One of the goals of the power series approach is to improve the eigenvalue separation of the preconditioner thus allowing an effective application of a low-rank correction technique. Experiments indicate that this combination can be quite robust when solving highly indefinite linear systems. The preconditioner exploits a domain-decomposition approach, and its construction starts with the use of a graph partitioner to reorder the original coefficient matrix. In this framework, unknowns corresponding to interface variables are obtained by solving a linear system whose coefficient matrix is the Schur complement. Unknowns associated with the interior variables are obtained by solving a block diagonal linear system where parallelism can be easily exploited. Numerical examples are provided to illustrate the effectiveness of the proposed preconditioner, with an emphasis on highlighting its robustness properties in the indefinite case.

中文翻译：

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适用于一般稀疏线性系统的 Power Schur 补充低阶校正预处理器

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 659-682 页，2021 年 1 月。
针对一般大型稀疏线性系统提出了一种并行预处理器，该系统将幂级数展开方法与低秩校正技术相结合。为了增强收敛性，通过利用 Schur 补集的标准矩阵分裂，将幂级数展开添加到基本 Schur 补集迭代方案中。幂级数方法的目标之一是改善预处理器的特征值分离，从而允许低秩校正技术的有效应用。实验表明，在求解高度不确定的线性系统时，这种组合非常稳健。预处理器利用域分解方法，其构建开始于使用图分割器对原始系数矩阵进行重新排序。在这个框架中，对应于界面变量的未知数是通过求解系数矩阵为 Schur 补码的线性系统获得的。与内部变量相关的未知数是通过求解块对角线性系统获得的，其中可以轻松利用并行性。提供了数值例子来说明所提出的预处理器的有效性，重点是突出其在不确定情况下的鲁棒性。