求解特定鞍点问题的改进SOR-Like方法

鞍点问题广泛出现在众多的工程研究领域,如流体力学、电磁学、最优化问题、最小二乘问题、椭圆偏微分方程问题等.以SOR类方法为基础,结合HS分裂思想,将经典鞍点问题的求解方法推广到特殊鞍点问题的求解上.给出一种具有新型分裂迭代格式的MSOR-Like方法,用以求解一类含有非对称块的鞍点系统,给出了相应的收敛性分析以及最优松弛参数选取方法.数值算例验证了对于不同的预优矩阵,MSORLike方法只有收敛速度的分别,没有收敛性能的影响,且在相同计算精度下,该方法解决特殊鞍点问题的迭代效果优于常规方法解决经典鞍点问题.

Modified SOR-Like Method for Saddle Point Problems

Saddle point problems exist in many engineering research areas such as fluid mechanics, electromagnetism, optimization problems, the least squares problems, elliptic partial differential equations, and etc. Based on SOR-Like methods in combination of the concept of HS splitting, a new iteration splitting improvement method was presented so as to apply the classic saddle point solutions to special saddle point problems. Then, the MSOR-Like method was proposed to handle the above special saddle point system containing asymmetric blocks, and the convergence analysis as well as the selection of optimal relaxation parameters were also given. Finally, a numerical example was given to verify different optimal matrix of the modified SOR method, and it was found that the only difference is in the convergence rate while there is no difference in the convergence effect. Furthermore, under the same calculation accuracy, the modified SOR method for solving the special saddle point problems is better than the conventional methods in solving the classical saddle point problems.