鞍点问题的等价模型及其求解  

The Equivalent Model of Large Sparse Saddle Point Problems and its Solving Strategy

作　　者：张秀梅[1] 王川龙[2] 

机构地区：[1]太原理工大学数学学院,太原030024 [2]太原师范学院数学系,太原030012

出　　处：《工程数学学报》2014年第1期75-83,共9页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11071184);山西省自然科学基金(2010011006;2012011015-6)~~

摘　　要：本文将鞍点问题转化为一个具有对称正定系数矩阵的等价模型.在同等条件下,将求解鞍点问题的SOR-like方法与等价模型的SOR方法进行了对比,发现等价模型效果更好.此外,我们还提出了一种新的修正Chebyshev加速迭代方法,它的参数是由优化模型而不是Chebyshev多项式产生,并讨论了修正的Chebyshev加速迭代方法的收敛性.最后,通过数值例子比较各种算法的收敛速度和迭代次数,验证了修正的Chebyshev加速迭代方法的收敛性优势.In this paper, we equivalently transform the saddle point problem into a new model whose coefficient matrix is a symmetric positive definite matrix. Under similar settings, we compare the proposed SOR method for the equivalent model with the traditional SOR-like method, which are used to solve the saddle point problem, and find that the equivalent model has a better performance. In addition, we propose a modified Chebyshev accelerative iterative method, whose parameter is obtained from an optimization model while not the conventional Chebyshev polynomial. The convergence of the modified Chebyshev accelerative iterative method is also studied. Finally, a numerical example is given to compare the convergence rate and the iteration number of all kinds of algorithms. The results have showed that the modified Chebyshev accelerative iterative method has a better convergence rate than other accelerative iterative methods.

关 键 词：鞍点问题 SOR方法 SOR-like方法 Chebyshev加速方法 修正的Chebyshev加速 

分 类 号：O241.6[理学—计算数学]