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

Abstract

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.

Keywords:

• saddle-point problem
• Hermitian and skew-Hermitian splitting
• preconditioning
• iteration method
• convergence analysis
• optimal parameter

2010 AMS Subject Classifications:

• 65F10
• 65F08
• 65N22

Acknowledgements

The authors are very much indebted to the referees for providing very useful comments and suggestions, which greatly improved the original manuscript of this paper.

Funding

This work is partially supported by the National Basic Research (973) Program of China under [grant number 2011CB706903], the National Natural Science Foundation of China [grant number 11271173] and the Mathematical Tianyuan Foundation of China [grant number 11026064.]