基于一类块2×2结构矩阵的Schur补矩阵的预处理技术优化研究

OPTIMIZATION OF THE SCHUR COMPLEMENT MATRIX BASED PRECONDITIONING TECHNIQUES ON A CLASS OF BLOCK 2×2 STRUCTURE MATRICES

本文针对一类块2×2结构的线性方程组,利用其系数矩阵的结构性质以及Schur补近似矩阵的匹配技巧,讨论了两类Schur补矩阵的近似矩阵以及它们之间的关系,提出了一个新的结构约束预处理子,并且给出了该预处理子理论推导和算法优势.通过极小化预处理矩阵的谱聚集程度,得到了优化这两类Schur补矩阵的参数选择策略及特征值分布,并证明了在满足一定特殊条件下,可以进一步改进和优化基于Schur补近似的预处理技术.同时比较了这两类Schur补近似矩阵的效果及其适用范围,最后总结得到一类通用可靠且有效的预处理技术,并运用在目前最有效的三类预处理子上.我们通过几个数值实验例子证明理论分析是可信服的,也验证了优化的预处理子的有效性.

In this paper,for a class of linear equations with block 2×2 structure,the preconditioning techniques of two kinds of Schur complement matrices and their relations are discussed.We also get a new structure-constrained preconditioner in the derivation process,which possesses both theoretical advantages and computing advantages.By minimizing the spectral clustering of the preconditioned matrices,we obtained two kinds of effective parameter selection strategies and exact eigenvalue distribution of the preconditioned matrices.Also,we proved that under certain special conditions,the preconditioned technologies based on Schur’s complement approximation can be further improved and optimized.At the same time,the effects of these two kinds of Schur approximate matrices and their applications are compared.Finally,a general,reliable and effective preprocessing technique is summarized,which is applied to the three most effective preconditioners at present.Several numerical examples show that the theoretical analysis is convincing,and the effectiveness of the optimized preconditioners are also verified.