三类有效预处理子的关系及其优化

doi:10.12286/jssx.j2021-0835

计算数学 ›› 2022, Vol. 44 ›› Issue (4): 545-560.DOI: 10.12286/jssx.j2021-0835

三类有效预处理子的关系及其优化

廖丽丹^1,2, 张国凤²

1. 南昌大学数学系, 南昌 330031;
2. 兰州大学数学与统计学院, 兰州 730000

收稿日期:2021-07-07 出版日期:2022-11-14 发布日期:2022-11-08
基金资助:
国家自然科学基金（11901278，11771193）和江西省自然科学基金（20202BAB211002）资助.

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廖丽丹, 张国凤. 三类有效预处理子的关系及其优化[J]. 计算数学, 2022, 44(4): 545-560.

Liao Lidan, Zhang Guofeng. RELATIONSHIP AND IMPROVEMENTS OF THREE EFFICIENT BLOCK PRECONDITIONERS[J]. Mathematica Numerica Sinica, 2022, 44(4): 545-560.

RELATIONSHIP AND IMPROVEMENTS OF THREE EFFICIENT BLOCK PRECONDITIONERS

Liao Lidan^1,2, Zhang Guofeng²

1. Department of mathematics, Nanchang University, Nanchang 330031, China;
2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received:2021-07-07 Online:2022-11-14 Published:2022-11-08

针对一类由时谐抛物方程约束的最优控制问题导出的分块$2\times2$复线性方程组,进一步研究了三类有效的块预处理子,推导了这三类预处理子间的关系,结论表明三个预处理矩阵的特征值由同一个矩阵确定.通过分析预处理矩阵的谱性质,获得了有效的参数选择策略,可以进一步改进和优化现有结果,同时获得了预处理矩阵的精确特征值分布,并证明了此结果是目前文献中最优结果.最后,给出实例,不仅验证了优化的预处理子和迭代方法的有效性,而且说明了理论结果是令人信服的.

关键词： 时谐抛物方程, 最优控制, 参数选择, 预处理子, 复线性方程组

For a class of block two-by-two complex linear systems arising from the optimal control problems constrained by time-harmonic parabolic equations, we further study the three existing block preconditioners. The relationships between the three preconditioners are derived, i.e. the eigenvalues of the three precondioned matrices are determined by a same matrix. By analyzing the spectral properties of the preconditioned matrices, we obtain an efficient parameter selection strategy that can further improve and optimize the existing results, and at the same time obtain the accurate eigenvalue distribution of the preconditioned matrices, which are better than many current results. Finally, examples are given, which not only verify the effectiveness of the optimized preconditioners and iterative method, but also show that the theoretical result is convincing.

Key words: time-harmonic parabolic equation, optimal control, parameter selection, preconditioner, complex linear systems

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三类有效预处理子的关系及其优化

RELATIONSHIP AND IMPROVEMENTS OF THREE EFFICIENT BLOCK PRECONDITIONERS

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