A Class of Shift-Splitting Preconditioners for Solving the General Block Two-by-Two Linear Systems

Abstract

This paper further extends the shift-splitting (SS) and local shift-splitting (LSS) preconditioners to solve the general block two-by-two linear systems. We demonstrate that the eigenvalues of the corresponding preconditioned matrices cluster tightly around 2 by detailed spectral property analysis. Numerical experiments not only validate the theoretical results but also show the effectiveness and superiority of the SS and LSS preconditioners by comparing them with some existing preconditioners applied to the generalized minimal residual (GMRES) method for solving the block two-by-two linear systems.

Keywords

Block two-by-two linear systems / Shift-splitting (SS) / Eigenvalues / Preconditioner / Spectral properties / 65F10 / 65F08

References

[1] BaiZ-Z. Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput., 2006, 75: 791-815

[2] BaiZ-Z. Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math., 2015, 283: 71-78

[3] BaiZ-Z. Quasi-HSS iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. Numer. Linear Algebra Appl., 2018, 25e2116

[4] BaiZ-Z, BenziM. Regularized HSS iteration methods for saddle-point linear systems. BIT Numer. Math., 2017, 57: 287-311

[5] Bai, Z.-Z., Duff, I.S., Wathen, A.J.: A class of incomplete orthogonal factorization methods. I: methods and theories. BIT Numer. Math. 41, 53–70 (2001)

[6] BaiZ-Z, DuffIS, YinJ-F. Numerical study on incomplete orthogonal factorization preconditioners. J. Comput. Appl. Math., 2009, 226: 22-41

[7] BaiZ-Z, GolubGH. Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal., 2007, 27: 1-23

[8] BaiZ-Z, GolubGH, LiC-K. Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput., 2006, 28: 583-603

[9] BaiZ-Z, GolubGH, NgMK. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 2003, 24: 603-626

[10] BaiZ-Z, GolubGH, PanJ-Y. Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math., 2004, 98: 1-32

[11] BaiZ-Z, NgMK, WangZ-Q. Constraint preconditioners for symmetric indefinite matrices. SIAM J. Matrix Anal. Appl., 2009, 31: 410-433

[12] BaiZ-Z, PanJ-YMatrix Analysis and Computations, 2021, Philadelphia. SIAM.

[13] BaiZ-Z, ParlettBN, WangZ-Q. On generalized successive overrelaxation methods for augmented linear systems. Numer. Math., 2005, 102: 1-38

[14] BaiZ-Z, WangZ-Q. On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl., 2008, 428: 2900-2932

[15] BaiZ-Z, YinJ-F, SuY-F. A shift-splitting preconditioner for non-Hermitian positive definite matrices. J. Comput. Math., 2006, 24: 539-552

[16] BenziM, GolubGH. A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl., 2004, 26: 20-41

[17] BettsJTPractical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2010, Philadelphia. SIAM.

[18] BjorckANumerical Methods for Least Squares Problems, 1996, Philadelphia. SIAM.

[19] CaoY, DuJ, NiuQ. Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math., 2014, 272: 239-250

[20] CaoY, MiaoS-X, RenZ-R. On preconditioned generalized shift-splitting iteration methods for saddle point problems. Comput. Math. Appl., 2017, 74: 859-872

[21] CaoY, YiS-C. A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems. Appl. Math. Comput., 2016, 275: 41-49

[22] CaoZ-H. Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks. Numer. Linear Algebra Appl., 2008, 15: 515-533

[23] CaoZ-H. Block triangular Schur complement preconditioners for saddle point problems and application to the Oseen equations. Appl. Numer. Math., 2010, 60: 193-207

[24] DollarHS, WathenAJ. Approximate factorization constraint preconditioners for saddle-point matrices. SIAM J. Sci. Comput., 2006, 27: 1555-1572

[25] ElmanHC. Preconditioners for saddle point problems arising in computational fluid dynamics. Appl. Numer. Math., 2002, 43: 75-89

[26] ElmanHC, GolubGH. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal., 1994, 31: 1645-1661

[27] FortinM, BrezziFMixed and Hybrid Finite Element Methods, 1991, New York. Springer-Verlag.

[28] GaoW-L, LiX-A, LuX-M. On quasi shift-splitting iteration method for a class of saddle point problems. Comput. Math. Appl., 2020, 79: 2912-2923

[29] GolubGH, GreifC, VarahJM. An algebraic analysis of a block diagonal preconditioner for saddle point systems. SIAM J. Matrix Anal. Appl., 2005, 27: 779-792

[30] GolubGH, WuX, YuanJ-Y. SOR-like methods for augmented systems. BIT Numer. Math., 2001, 41: 71-85

[31] HuangZ-G, WangL-G, XuZ, CuiJ-J. The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems. Appl. Math. Comput., 2017, 299: 95-118

[32] HuangZ-H, SuH. A modified shift-splitting method for nonsymmetric saddle point problems. J. Comput. Appl. Math., 2017, 317: 535-546

[33] LiangZ-Z, ZhangG-F. On block-diagonally preconditioned accelerated parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput., 2013, 221: 89-101

[34] MurphyMF, GolubGH, WathenAJ. A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput., 2000, 21: 1969-1972

[35] NjeruPN, GuoX-P. Accelerated SOR-like method for augmented linear systems. BIT Numer. Math., 2016, 56: 557-571

[36] PanJ-Y, NgMK, BaiZ-Z. New preconditioners for saddle point problems. Appl. Math. Comput., 2006, 172: 762-771

[37] SongS-Z, HuangZ-D. A two-parameter shift-splitting preconditioner for saddle point problems. Comput. Math. Appl., 2022, 124: 7-20

[38] WathenAJ. Preconditioning. Acta Numer., 2015, 24: 329-376

[39] Zhu, M.-Z., Zhang, G.-F., Liang, Z.-Z.: On generalized local Hermitian and skew-Hermitian splitting iterative method for block two-by-two linear systems. Appl. Math. Comput. 250, 463–478 (2015)

Funding

Scientific Research Foundation of Higher Education Institutions of Ningxia(2023BSB03036)

Education Department of Ningxia Hui Autonomous Region(NYG2024055)

RIGHTS & PERMISSIONS

Shanghai University