Acceleration of the Scheduled Relaxation Jacobi method: Promising strategies for solving large, sparse linear systems

Abstract

The main aim of this paper is to develop two algorithms based on the Scheduled Relaxation Jacobi (SRJ) method (Yang and Mittal (2014) [7]) for solving problems arising from the finite-difference discretization of elliptic partial differential equations on large grids. These two algorithms are the Alternating Anderson-Scheduled Relaxation Jacobi (AASRJ) method by utilizing Anderson mixing after each SRJ iteration cycle and the Minimal Residual Scheduled Relaxation Jacobi (MRSRJ) method by minimizing residual after each SRJ iteration cycle, respectively. Through numerical experiments, we show that AASRJ is competitive with the optimal version of the SRJ method (Adsuara et al. (2017) [9]) in most problems we considered here, and MRSRJ outperforms SRJ in all cases. The properties of AASRJ and MRSRJ are demonstrated. Both of them are promising strategies for solving large, sparse linear systems while maintaining the simplicity of the Jacobi method.