A power Schur complement Low-Rank correction preconditioner for general sparse linear systems
A power Schur complement Low-Rank correction preconditioner for general sparse linear systems
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DOI：
10.1137/20m1316445
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
Q. Zheng;Yuanzhe Xi;Y. Saad
中科院分区：
文献类型：
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作者：
Q. Zheng;Yuanzhe Xi;Y. Saad
为一般的大型稀疏线性系统提出了有效的基于功率的平行预处理。预处理结合了一种功率系列扩展方法与一些低级校正技术，其中使用了Sherman-Morrison-Woodbury公式。提出了Schur补体的矩阵分裂，以扩大功率系列。功率系列扩展中使用的术语数可以控制预调节器与Schur补体逆的近似精度。为了构建预调节器，调用图形分区以重新排序原始系数矩阵，从而导致了一个特殊的块两乘二矩阵，其两个偏高的子矩阵是块对角线。通过求解系数矩阵是Schur补体的线性系统，可以获得与界面变量相对应的变量。对于与内部变量相关的变量，只需要求解块对角线线性系统即可。这可以在并联上有效地执行。提供了各种数值示例，以说明拟议的预处理的效率。
An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury formula is utilized. A matrix splitting of the Schur complement is proposed to expand the power series. The number of terms used in the power series expansion can control the approximation accuracy of the preconditioner to the inverse of the Schur complement. To construct the preconditioner, graph partitioning is invoked to reorder the original coefficient matrix, leading to a special block two-by-two matrix whose two off-diagonal submatrices are block diagonal. Variables corresponding to interface variables are obtained by solving a linear system with the coeffcient matrix being the Schur complement. For the variables related to the interior variables, one only needs to solve a block diagonal linear system. This can be performed efficiently in parallel. Various numerical examples are provided to illustrate that the efficiency of the proposed preconditioner.