A VARIABLE PRECONDITIONING USING THE SOR METHOD FOR GCR-LIKE METHODS
Kuniyoshi Abe, Shao‐Liang Zhang · 2005
We propose a variant of variable preconditioning for Generalized Conjugate Residual (GCR)-like methods. The preconditioning is carried out by roughly solving Az = v by an iterative method to a certain degree of accu- racy instead of computing Kz = v in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing Az = v is changed at each iteration by establishing a stopping criterion. This enables the use of a stationary iterative method when applying difierent precondition- ers. The proposed procedure is incorporated into GCR, and the mathematical convergence is proved. In numerical experiments, we employ the Successive Over-Relaxation (SOR) method for computing Az = v, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more ro- bust than GCR with an incomplete LU preconditioning, and the FGMRES and GMRESR methods with the variable preconditioning using the General- ized Minimal Residual (GMRES) method. Moreover, we conflrm that difierent preconditioners are applied at each iteration.