A New Type of Variable Preconditioning for a Generalized Minimum Residual Scheme

Abstract

Krylov linear equation solvers have been widely used on distributed-memory-type supercomputers. Krylov-type linear equation solvers are frequently applied using preconditioning techniques to achieve fast and robust computations. Flexible Krylov solvers that accept variable preconditioning have recently been developed, whereas conventional ones are invariable throughout the Krylov solver iterations. The generalized minimal residual recursive method (GMRESR) is outstandingly unique because it is combined with a preconditioning update scheme based on Eirola and Nevanlinna’s rank-one update scheme. Moreover, the Broyden–Fletcher–Goldfarb–Shanno scheme (BFGS) provides another means to update preconditioning matrices. In this study, we applied BFGS to GMRESR as an alternative means of updating the preconditioning matrices and then measured the performance. As a result, the newly developed method exhibited an approximately 10-fold better convergence than Eirola and Nevanlinna’s approach.