三类有效预处理子的关系及其优化

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.

RELATIONSHIP AND IMPROVEMENTS OF THREE EFFICIENT BLOCK PRECONDITIONERS

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.