求解鞍点问题的一种Uzawa-AO R方法

摘要

Saddle point linear system is a symmetric and indefinite linear system,which comes from the optimization problem,the least square problem and so on.In practical applications,the system is usually large,and the coefficient matrix is sparse,so we should use the iterative method to solve the problem.Uza-wa algorithm is an effective method for solving the saddle point problem,the algorithm is simple,but it is slow in convergence.In this paper,it is of great interest to develop an efficient iterative method for solving the saddle point problems.Based on iterative scheme,we present a new Uzawa-AOR method and prove the convergence of the proposed method.Actually,the new method can be considered as an inexact itera-tion method with the Uzawa as the outer iteration and the AOR as the inner iteration.Numerical examples are used to illustrate the efficiency of the new iteration method.%鞍点线性系统是一类对称不定的线性系统，它来源于最优化问题、最小二乘问题等研究领域。实际应用中，这类系统通常都是大规模的，并且系数矩阵具有稀疏性，因此应采用迭代法进行求解。Uzawa算法是求解鞍点问题的有效方法，该算法格式简单，但收敛速度较慢。为了快速有效地求解鞍点问题，在迭代算法的基础上，提出了一种新的Uzawa－AOR算法并证明了该算法的收敛性。新的算法是将 Uzawa算法作为外迭代，以AOR算法作为内迭代构造了一种求解鞍点问题的迭代算法。数值例子用来说明新迭代法的效率。