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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (5): 808-824.doi: 10.3969/j.issn.1005-3085.2024.05.002

• • 上一篇    下一篇

求解$3\times3$块鞍点问题的广义SOR方法

高  翔,   温瑞萍,   王川龙   

  1. 太原师范学院智能优化计算与区块链技术山西省重点实验室，晋中 030619
  • 收稿日期:2022-01-10 接受日期:2022-09-30 出版日期:2024-10-15
  • 通讯作者: 温瑞萍 E-mail: wenrp@163.com
  • 基金资助:
    国家自然科学基金 (12371381)；山西省自然科学基金 (201901D211423).

Generalized SOR Method for the Three-order Block Saddle Point Problems

GAO Xiang,  WEN Ruiping,  WANG Chuanlong   

  1. Shanxi Key Laboratory for Intelligent Optimization Computing and Block-chain Technology, Taiyuan Normal University, Jinzhong 030619
  • Received:2022-01-10 Accepted:2022-09-30 Online:2024-10-15
  • Contact: R. Wen. E-mail address: wenrp@163.com
  • Supported by:
    The National Natural Science Foundation of China (12371381); the Natural Science Foundation of Shanxi Province (201901D211423).

摘要：

$3\times3$块鞍点问题作为一类特殊的线性方程组，其迭代方法的研究极具挑战性。基于经典的广义逐次超松弛(Generalized Successive Over Relaxation, GSOR)方法，针对$3\times3$块大型稀疏鞍点问题，提出了三参数的中心预处理GSOR方法并讨论了其收敛性。同时，通过数值实验验证了新方法在计算花费方面优于中心预处理的Uzawa-Low方法。进一步地，还将新方法拓展到$i\times i$块鞍点问题，提出了相应的GSOR类迭代框架，通过数值实验和数据分析，给出了选择较优$i$的初步建议。

关键词: 鞍点问题, $3\times3$块鞍点问题, SOR方法, GSOR方法, 中心预处理方法

Abstract:

As a special kind of linear system, the three-order block saddle point problem has challenging to study its iterative solution. Based on the classical generalized successive over relaxation (GSOR) method, the centered preconditioned GSOR method with three parameters for a class of three-order block large sparse saddle point problem is established and the convergence condition is discussed in this paper. Moreover, experimental results show that the new method has an advantage of computational cost over the centered preconditioned Uzawa-Low method. In addition, an extended one of the new method is provided, implementation details and analyses of corresponding framework about $i$-order block systems are shown, the blocking for saddle point problems are preliminarily proposed by some numerical results.

Key words: saddle point problem, three-order block saddle point problem, SOR method, \mbox{GSOR} method, centered preconditioned method

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