A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

Abstract

For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.for the saddle point problem (1), where α is a positive constant and I is the identity matrix.where α ≥ 0, β > 0 and I is the identity matrix. It is easy to see that P SS is a special case of P GSS when α = β. Numerical results in [21,20] confirmed that the GSS preconditioner is superior to the SS preconditioner.

A modified SSOR iterative method for a class of block two-by-two linear systems

“…As a matter of fact, by taking k = 1 in (18) and (19), we can see that ω opt = τ opt , the optimal parameter is reduced to the single ω opt = 1 ± ( 1 + µ 2 min − 1)/µ min as mentioned in Ref. 19.…”

“…In recent years, a variety of iterative techniques are investigated to solve all kinds of linear systems including the complex situation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][21][22][23] , such as saddle-point problem and various generalized forms 16,18,26 . In particular, the SOR, SSOR iterative schemes and their variants are presented for solving some large and sparse linear systems 17 .…”

A shift-splitting preconditioner for asymmetric saddle point problems

“…Based on Lemma 2, the inequalities ( 20) and (21) imply that the roots of the real quadratic equation (19) satisfy |λ| < 1. If t = 0, then Eq.…”

“…On the other hand, combining the shift splitting technique with the matrix splitting technique, some new efficient preconditioners have been developed, such as the modified shiftsplitting preconditioner [34], the generalized modified shift-splitting preconditioner [22], the extended shift-splitting preconditioner [35], a general class of shift-splitting preconditioner [10], the modified generalized shift-splitting preconditioner [21,32], the generalized double shift-splitting preconditioner [16], and so on.…”

A shift-splitting preconditioner for asymmetric saddle point problems

“…Based on the work in Chen and Ma (2015), use the two-parameter shift-splitting preconditioner for the saddle point problems (3) with symmetric positive semidefinite (2, 2)-block, and for the same problem when the symmetry of the (1,1)-block is omitted in , Cao et al (2015) considered the saddle point problems (3) with nonsymmetric positive definite (1, 1)block, Cao and Miao (2016) considered the singular nonsymmetric saddle point problems (3), and so on. On the other hand, combining the shift splitting technique with the matrix splitting technique, some new efficient preconditioners have been developed, such as the modified shift-splitting preconditioner (Zhou et al 2016), the generalized modified shift-splitting preconditioner (Huang et al 2017), the extended shift-splitting preconditioner (Zheng and Lu 2017), a general class of shift-splitting preconditioner (Cao 2019), the modified generalized shift-splitting preconditioner (Huang et al 2018;Salkuyeh and Rahimian 2017), the generalized double shift-splitting preconditioner (Fan et al 2018), and so on.…”