An inexact parallel splitting augmented Lagrangian method for large system of linear equations

作者：Peng, Zheng^*; Wu, DongHua

来源：Applied Mathematics and Computation, 2010, 216(5): 1624-1636.

DOI：10.1016/j.amc.2010.03.018

• System of linear equations
• Parallel iteration
• Matrix decomposition
• Augmented Lagrangian method
• Inexact terms

Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is O(n(2)). In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods.

• 出版日期2010-5-1
• 单位上海大学; 南京大学