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

Parallel iterative methods are powerful tool for solving large system of linear equations (LEs). The existing parallel computing research results are focussed mainly on sparse system or others with particular structure. And most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods carried efficiently on multiprcessor systems. In this paper, we proposed a novel parallel splitting operator method based on a new approach. In this method, we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequalities problem (VIs) with separable structure. Finally, we propose an inexact parallel splitting augmented Lagrangian method to solve this variational inequalities problem (VIs). We avoid the matrix inverse operator by introducing proper inexact terms in subproblems, such that complexity of each step of this method is O(n2). This is different to a general iterative method with complexity O(n3). In addition, this method does not depend on any special structure of the problem which is to be solved. Convergence of the proposed methods, in dealing with two and three separable operators respectively, is proved. Numerical experiments are provided to show its applicability, especially its robustness with respect to the scale (dimensions) and condition (measured by the condition number of coefficient matrix A) of this method.