Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for saddle-point problems

摘要

By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a parameterized PHSS (PPHSS) iteration method for non-Hermitian positive semidefinite linear saddle-point systems. The PPHSS method is essentially a two-parameter iteration which covers standard PHSS iteration and can extend the possibility to optimize the iterative process. The iterative sequence produced by the PPHSS method is proved to be convergent to the unique solution of the saddle-point problem when the iteration parameters satisfy a proper condition. In addition, for a special case of the PPHSS iteration method, we derive the optimal iteration parameter and the corresponding optimal convergence factor. Numerical experiments demonstrate the effectiveness and robustness of the PPHSS method both used as a solver and as a preconditioner for Krylov subspace methods.