On the restrictively preconditioned conjugate gradient method for solving saddle point problems

Abstract

Bai et al. [2003, IMA J Numer. Anal. 23, 561–580] proposed the restrictively preconditioned conjugate gradient (RPCG) method. In this paper, based on the special structure of saddle point systems, we consider the RPCG method and propose a new format. This new format can be obtained by applying the classical PCG method to a simpler system instead of the original format, which greatly reduces computational cost. The new format of the RPCG method can often attain almost the same convergence rate as the original one. In particular, for some practical problems, the former converges faster than the latter. Numerical experiments show the efficiency of the proposed format.

Keywords:

• Saddle point problems
• restrictive preconditioners
• conjugate gradient method
• spectral condition number
• convergence rate

2000 AMS Subject Classifications:

• 65F10
• 65F15

Acknowledgements

The authors are very grateful to the anonymous referee for part of the MATLAB codes and numerous valuable comments and suggestions. Thanks also to the associated editor for his/her encouraging comments. The authors would like to thank Prof. Zhong-Zhi Bai for his many helpful suggestions in both mathematics and English. The first author would like to thank Prof. Zhilin Li and North Carolina State University since part of the research was done during the author's visit to NCSU.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

The work was supported by National Natural Science Foundation [Grant No. 10971075, 11101164, 11271144]; Research Fund for the Doctoral Program of Higher Education of China [Grant No.20104407110001]; Guangdong provincial Natural Science Foundation [Grant No. S2012010009985, S2013010012530]; Project of Department of Education of Guangdong Province (2013KJCX0053).