一种求解鞍点问题的改进Uzawa-PSS方法

主要针对非Hermitian鞍点问题，在已有Uzawa-PSS方法基础上构建了一种改进的Uzawa-PSS迭代法，其主要求解思想是在Uzawa-PSS方法的每一步迭代中需求解系数矩阵αI+P和αI+S的两个线性子系统.第一个子系统可用CG方法求解，但第二个子系统求解很困难.改进算法采用单步PSS迭代法逼近x_k+1，然后用新方法分别求解了非奇异和奇异鞍点问题，并给出了相应的收敛性分析.数值仿真实验验证了改进Uzawa-PSS迭代法在迭代步数、占用CPU时间和相对残差上都有明显的优势.

An Improved Uzawa-PSS Method for to Solve Point Problems

Aiming at the non-Hermitian saddle point problem， an improved Uzawa-PSS iteration method is constructed based on the existing Uzawa-PSS method. The main idea of the new method is to solve two linear subsystems in each iteration step of Uzawa-PSS method， whose coefficient matrices are αI+P and αI+S， respectively. The first subsystem can be solved by CG method， but the second subsystem is very difficult to solve. The improved algorithm uses the single-step PSS iteration method to approximate the problem. Then the new method is used to solve the non-singular and singular saddle point problems respectively， and the corresponding convergence analysis is given. The numerical simulation also proves that the improved Uzawa-PSS iteration method has obvious advantages in iteration steps， CPU time and relative residuals.