Convergence analysis of modified PGSS methods for singular saddle-point problems

Abstract Recently, variants of shift-splitting iteration method have been proposed for solving singular saddle-point problems. However, these methods can only be proved to converge to one of the solutions of the consistent singular linear system, not knowing any further information about this solution. In this work, we consider a modified preconditioned generalized shift-splitting (MPGSS) iteration method for solving both consistent and inconsistent singular saddle-point problems. This method is proved to converge to the best least squares solution. Moreover, based on the iteration form, a preconditioner is obtained to accelerate Krylov subspace methods. Theoretical analysis shows that the preconditioned GMRES method also converges to the best least squares solution of the consistent singular saddle-point problem. In addition, numerical results are presented to show the effectiveness and robustness of the proposed iteration method and preconditioner.

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

[2]  X. Wu,et al.  Conjugate Gradient Method for Rank Deficient Saddle Point Problems , 2004, Numerical Algorithms.

[3]  S. Oliveira,et al.  Algebraic multigrid (AMG) for saddle point systems from meshfree discretizations , 2004, Numer. Linear Algebra Appl..

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

[5]  Apostolos Hadjidimos,et al.  The saddle point problem and the Manteuffel algorithm , 2016 .

[6]  Zhong-Zhi Bai,et al.  On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems , 2010, Computing.

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

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

[9]  Gene H. Golub,et al.  A Preconditioner for Generalized Saddle Point Problems , 2004, SIAM J. Matrix Anal. Appl..

[10]  Gene H. Golub,et al.  An Iteration for Indefinite Systems and Its Application to the Navier-Stokes Equations , 1998, SIAM J. Sci. Comput..

[11]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

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

[13]  Yang Cao,et al.  On preconditioned generalized shift-splitting iteration methods for saddle point problems , 2017, Comput. Math. Appl..

[14]  H. Walker,et al.  GMRES On (Nearly) Singular Systems , 1997, SIAM J. Matrix Anal. Appl..

[15]  Davod Khojasteh Salkuyeh,et al.  A modification of the generalized shift-splitting method for singular saddle point problems , 2017, Comput. Math. Appl..

[16]  Guo-Feng Zhang,et al.  General constraint preconditioning iteration method for singular saddle-point problems , 2015, J. Comput. Appl. Math..

[17]  Naimin Zhang,et al.  Constraint preconditioners for solving singular saddle point problems , 2013, J. Comput. Appl. Math..

[18]  Yu-Jiang Wu,et al.  Modified parameterized inexact Uzawa method for singular saddle-point problems , 2015, Numerical Algorithms.

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

[20]  Apostolos Hadjidimos,et al.  Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix ☆ , 2014 .

[21]  Gene H. Golub,et al.  Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices , 2007, Math. Comput..

[22]  Xu Li,et al.  On semi-convergence of the Uzawa-HSS method for singular saddle-point problems , 2015, Appl. Math. Comput..

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

[24]  Michael K. Ng,et al.  Constraint Preconditioners for Symmetric Indefinite Matrices , 2009, SIAM J. Matrix Anal. Appl..

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

[26]  Tomás Kozubek,et al.  On the Moore–Penrose inverse in solving saddle‐point systems with singular diagonal blocks , 2012, Numer. Linear Algebra Appl..

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

[28]  A. Berman,et al.  Cones and Iterative Methods for Best Least Squares Solutions of Linear Systems , 1974 .

[29]  M. Benzi,et al.  Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems , 2013 .

[30]  Zhong-Zhi Bai,et al.  Optimal parameters in the HSS‐like methods for saddle‐point problems , 2009, Numer. Linear Algebra Appl..

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

[32]  Naimin Zhang.  A note on preconditioned GMRES for solving singular linear systems , 2010 .

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

[34]  Guo-Feng Zhang,et al.  A generalization of parameterized inexact Uzawa method for singular saddle point problems , 2013, Appl. Math. Comput..

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

[36]  Quan Shi,et al.  Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems , 2016, Comput. Math. Appl..

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

[38]  Jinyun Yuan,et al.  Block SOR methods for rank-deficient least-squares problems , 1998 .

[39]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[40]  Xu Li,et al.  On generalized parameterized inexact Uzawa methods for singular saddle-point problems , 2014, Numerical Algorithms.

[41]  Naimin Zhang,et al.  A generalized preconditioned HSS method for singular saddle point problems , 2013, Numerical Algorithms.

[42]  Yang Cao,et al.  A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization , 2013, Journal of Computational Mathematics.