In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.
Singular saddle point problem Deteriorated PSS iteration method Semi-convergence Preconditioning
Mathematics Subject Classification (2010)
This is a preview of subscription content, log in to check access.
We would like to express our sincere thanks to the unknown reviewers for their careful reading of the manuscript. Their useful comments and valuable suggestions greatly improve the quality of the paper.
Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27(1), 1–23 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Golub, G.H., Li, C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28(2), 583–603 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76(257), 287–298 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24(3), 603–626 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Golub, G.H., Ng, M.K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14(4), 319–335 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Golub, G.H., Ng, M.K.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428(2), 413–440 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98(1), 1–32 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
Bai, Z.-Z., Li, G.-Q.: Restrictively preconditioned conjugate gradient methods for systems of linear equations. IMA J. Numer. Anal. 23(4), 561–580 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
Cao, Y., Dong, J.-L., Wang, Y.-M.: A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes equation. J. Comput. Appl. Math. 273, 41–60 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
Cao, Y., Miao, S.-X.: On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems. Comput. Math. Appl. 71(7), 1503–1511 (2016)MathSciNetCrossRefGoogle Scholar
Cao, Z.-H.: Comparison of performance of iterative methods for singular and nonsingular saddle point linear systems arising from Navier–Stokes equations. Appl. Math Comput. 174(1), 630–642 (2006)MathSciNetzbMATHGoogle Scholar
Chao, Z., Chen, G.-L.: A note on semi-convergence of generalized parameterized inexact Uzawa method for singular saddle point problems. Numer. Algor. 68(1), 95–105 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
Wang, R.-R., Niu, Q., Ma, F., Lu, L.-Z.: Spectral properties of a class of matrix splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 298, 138–151 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
Yin, J.-F., Bai, Z.-Z.: The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems. J. Comput. Math. 26(2), 240–249 (2008)MathSciNetzbMATHGoogle Scholar
Zheng, B., Bai, Z.-Z., Yang, X.: On semi-convergence of parameterized Uzawa methods for singular saddle point problems. Linear Algebra Appl. 431(5–7), 808–817 (2009)MathSciNetzbMATHCrossRefGoogle Scholar