The improvements of the generalized shift-splitting preconditioners for non-singular and singular saddle point problems

ABSTRACT To solve the saddle point problems with symmetric positive definite (1,1) parts, the improved generalized shift-splitting (IGSS) preconditioner is established in this paper, which yields the IGSS iteration method. Theoretical analysis shows that the IGSS iteration method is convergent and semi-convergent unconditionally. The choices of the iteration parameters are discussed. Moreover, some spectral properties, including the eigenvalue and eigenvector distributions of the preconditioned matrix are also investigated. Finally, numerical results are presented to verify the robustness and the efficiency of the proposed iteration method and the corresponding preconditioner for solving the non-singular and singular saddle point problems.

[1]  Qingqing Zheng,et al.  Extended shift-splitting preconditioners for saddle point problems , 2017, J. Comput. Appl. Math..

[2]  Guo-Yan Meng.  A practical asymptotical optimal SOR method , 2014, Appl. Math. Comput..

[3]  Jun Zou,et al.  An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems , 2001, SIAM J. Matrix Anal. Appl..

[4]  Naimin Zhang,et al.  On the optimal parameters of GMSSOR method for saddle point problems , 2016, Appl. Math. Lett..

[5]  Peng Guo,et al.  A modified SOR-like method for the augmented systems , 2015, J. Comput. Appl. Math..

[6]  Yimin Wei,et al.  Semi-convergence analysis of Uzawa methods for singular saddle point problems , 2014, J. Comput. Appl. Math..

[7]  Yang Cao,et al.  Shift-splitting preconditioners for saddle point problems , 2014, J. Comput. Appl. Math..

[8]  M. Ng,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[9]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[10]  Davod Khojasteh Salkuyeh,et al.  On the generalized shift-splitting preconditioner for saddle point problems , 2014, Appl. Math. Lett..

[11]  Beresford N. Parlett,et al.  On generalized successive overrelaxation methods for augmented linear systems , 2005, Numerische Mathematik.

[12]  Hong Su,et al.  A modified shift-splitting method for nonsymmetric saddle point problems , 2017, J. Comput. Appl. Math..

[13]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[14]  Charles R. Johnson.  Review: Abraham Berman and Robert J. Plemmons,Nonnegative matrices in the mathematical sciences , 1982 .

[15]  Yang Cao,et al.  A Modified Relaxed Positive-Semidefinite and Skew-Hermitian Splitting Preconditioner for Generalized Saddle Point Problems , 2017 .

[16]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[17]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[18]  Zhong-Zhi Bai,et al.  Structured preconditioners for nonsingular matrices of block two-by-two structures , 2005, Math. Comput..

[19]  Zhong-zhi,et al.  A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES , 2006 .

[20]  Jun-Feng Yin,et al.  A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems , 2017 .

[21]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[22]  Gene H. Golub,et al.  Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems , 2005, SIAM J. Sci. Comput..

[23]  Yu-Jiang Wu,et al.  The modified shift-splitting preconditioners for nonsymmetric saddle-point problems , 2016, Appl. Math. Lett..

[24]  Yang Cao,et al.  On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems , 2016, Comput. Math. Appl..

[25]  Jing Li,et al.  A triple-parameter modified SSOR method for solving singular saddle point problems , 2016 .

[26]  Andrew J. Wathen,et al.  Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations , 2002, Numerische Mathematik.

[27]  Gene H. Golub,et al.  Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems , 2004, Numerische Mathematik.

[28]  Fang Chen,et al.  On choices of iteration parameter in HSS method , 2015, Appl. Math. Comput..

[29]  Sen Li,et al.  A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems , 2015, Appl. Math. Lett..

[30]  Jun Zou,et al.  Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions , 2004, Math. Comput..

[31]  N. SIAMJ.,et al.  A NONOVERLAPPING DOMAIN DECOMPOSITION METHOD FOR MAXWELL ’ S EQUATIONS IN THREE DIMENSIONS , 1998 .

[32]  Zhong Xu,et al.  The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems , 2017, Appl. Math. Comput..

[33]  Maolin Liang,et al.  A new generalized parameterized inexact Uzawa method for solving saddle point problems , 2015, Appl. Math. Comput..

[34]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[35]  Bing Zheng,et al.  On semi-convergence of parameterized Uzawa methods for singular saddle point problems☆ , 2009 .

[36]  Yang Cao,et al.  A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems , 2016, Appl. Math. Comput..

[37]  Yu-Mei Huang,et al.  A practical formula for computing optimal parameters in the HSS iteration methods , 2014, J. Comput. Appl. Math..

[38]  Zeng-Qi Wang,et al.  On parameterized inexact Uzawa methods for generalized saddle point problems , 2008 .

[39]  Changfeng Ma,et al.  A generalized shift-splitting preconditioner for singular saddle point problems , 2015, Appl. Math. Comput..

[40]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[41]  Jun Zou,et al.  Nonlinear Inexact Uzawa Algorithms for Linear and Nonlinear Saddle-point Problems , 2006, SIAM J. Optim..

[42]  Xue-Ping Guo,et al.  Accelerated SOR-like method for augmented linear systems , 2016 .

[43]  M. Benzi,et al.  Regularized HSS iteration methods for saddle-point linear systems , 2017 .

[44]  Qingqing Zheng,et al.  The corrected Uzawa method for solving saddle point problems , 2015, Numer. Linear Algebra Appl..

[45]  Jun Zou,et al.  Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems , 2002, Numerische Mathematik.

[46]  Yang Cao,et al.  Spectral analysis of the generalized shift-splitting preconditioned saddle point problem , 2017, J. Comput. Appl. Math..

[47]  Alfio Quarteroni,et al.  Parameter estimates for the Relaxed Dimensional Factorization preconditioner and application to hemodynamics , 2016 .

[48]  Gene H. Golub,et al.  SOR-like Methods for Augmented Systems , 2001 .

[49]  Gene H. Golub,et al.  Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems , 2007 .