一种求解分布控制问题的含参预处理最小残量法(英文)

Parameterized preconditioned MINRES method for solving distributed control problems

近年来,提出了一些求解约束优化分布控制问题的方法,其中最常用的方法是先离散偏微分方程,然后求解离散得到的线性方程组.文献中提出了一些Krylov子空间预处理方法用来求解该线性方程组.通过分析张晓莹等提出的块对角预处理矩阵(Zhang X Y,Yan H Y.Huang Y M.On preconditionedMINRES method for solving the distributed control problems.Commun Appl Math Comput,2014,28:128-132.),构造了一个含参数的块对角预处理线性方程组,并运用含参数预处理最小残量方法求解该线性方程组.预处理矩阵的谱分析表明当参数大于1时,含参预处理线性方程组的谱分布更加集中.数值实验结果验证了含参数的预处理最小残量方法对于求解分布式控制问题是有效的.

In recent years, many efforts have been made to numerically solving the constrained optimization distributed control problems, in which the most common one is to discretize the partial differential equation first and then solve the resulting system of linear equations. A number of preconditioned Krylov subspace methods have been constructed to solve the resulting system of linear equations in the literature. In this paper, by analyzing the block-diagonal preconditioner presented by Zhang, et al. （Zhang X Y, Yan H Y, Huang Y M. On preconditioned MINRES method for solving the distributed control problems. Commun Appl Math Comput, 2014, 28： 128-132.）, we propose a parameterized block-diagonally preconditioned linear system where a parameterized preconditioner is utilized and the preconditioned MINRES method is applied to solve the system of linear equations. The spectral analysis of the proposed preconditioned matrix shows that the spectral distribution of the parameterized preconditioning matrix should be much more clustered if the parameter is greater than 1. Numerical Experiments show that the preconditioned MINRES method is efficient for solving the distributed control problems.