Skip to main content
SpringerLink
Log in

A modified parameterized shift-splitting preconditioner for saddle point problems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Inspired by the idea of the parameterized shift-splitting (PSS), this paper presents a modified PSS (MPSS) preconditioner for the generalized nonsymmetric saddle point-type linear systems. All eigenvalues of the corresponding preconditioned matrix are proved to be clustered around 1. Meanwhile, the unconditional convergence of the MPSS iteration method is shown under certain conditions. The proposed method overcomes the shortfalls of the PSS iteration method in Zhang et al. (Math Biosci Eng 16:1021–1033, 2019) for a special case of the considered system. Numerical experiments are provided to validate the theoretical results and illustrate the effectiveness of the proposed preconditioner.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Bai Z-Z (2000) Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl Math Comput 109:273–285

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z (2006) Structured preconditioners for nonsingular matrices of block two-by-two structures. Math Comput 75:791–815

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z (2009) Optimal parameters in the HSS-like methods for saddle-point problems. Numer Linear Algebra Appl 16:447–479

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z (2012) Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models. Numer Linear Algebra Appl 19:914–936

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z (2015) Motivations and realizations of Krylov subspace methods for large sparse linear systems. J Comput Appl Math 283:71–78

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z (2018) Quasi-HSS iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. Numer Linear Algebra Appl 25(e2116):1–19

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Benzi M (2017) Regularized HSS iteration methods for saddle-point linear systems. BIT Numer Math 57:287–311

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Golub GH (2007) Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J Numer Anal 27:1–23

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Wang Z-Q (2008) On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl 428:2900–2932

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Golub GH, Pan J-Y (2004) Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer Math 98:1–32

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Golub GH, Lu L-Z, Yin J-F (2005a) Block triangular and skew-Herimitian splitting methods for positive-definite linear systems. SIAM J Sci Comput 26:844–863

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Parlett BN, Wang Z-Q (2005b) On generalized successive overrelaxation methods for augmented linear systems. Numer Math 102:1–38

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Yin J-F, Su Y-F (2006) A shift-splitting preconditioner for non-Hermitian positive definite matrices. J Comput Math 24:539–552

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Ng MK, Wang Z-Q (2009) Constraint preconditioners for symmetric indefinite matrices. SIAM J Matrix Anal Appl 31:410–433

    MathSciNet  MATH  Google Scholar 

  • Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137

    MathSciNet  MATH  Google Scholar 

  • Benzi M, Olshanskii MA, Wang Z (2011) Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 66:486–508

    MathSciNet  MATH  Google Scholar 

  • Cao Z-H (2006) A note on block diagonal and constraint preconditioners for non-symmetric indefinite linear systems. Int J Comput Math 83:383–395

    MathSciNet  MATH  Google Scholar 

  • Cao Z-H (2008) Augmentation block preconditioners for saddle point-type matrices with singular (1, 1) blocks. Numer Linear Algebra Appl 15:515–533

    MathSciNet  MATH  Google Scholar 

  • Cao Z-H (2009) Constraint Schur complement preconditioners for nonsymmetric saddle point problems. Appl Numer Math 59:151–169

    MathSciNet  MATH  Google Scholar 

  • Cao Z-H (2016) A class of constraint preconditioners for nonsymmetric saddle point matrices. Numer Math 103:47–61

    MathSciNet  MATH  Google Scholar 

  • Cao Y (2019) A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models. Comput Math Appl 77:1124–1143

    MathSciNet  MATH  Google Scholar 

  • Cao Y, Du J, Niu Q (2014) Shift-splitting preconditioners for saddle point problems. J Comput Appl Math 272:239–250

    MathSciNet  MATH  Google Scholar 

  • Chen C-R, Ma C-F (2015) A generalized shift-splitting preconditioner for saddle point problems. Appl Math Lett 43:49–55

    MathSciNet  MATH  Google Scholar 

  • Dou Q-Y, Yin J-F, Liao Z-Y (2017) A fast shift-splitting iteration method for nonsymmetric saddle point problems. East Asian J Appl Math 7:172–191

    MathSciNet  MATH  Google Scholar 

  • Golub GH, Wu X, Yuan J-Y (2001) SOR-like methods for augmented systems. BIT Numer Math 41:71–85

    MathSciNet  MATH  Google Scholar 

  • He J, Huang T-Z (2011) New preconditioners for nonsymmetric and indefinite saddle point linear systems. Miskolc Math Notes 12:41–49

    MathSciNet  MATH  Google Scholar 

  • Huang ZH, Su H (2017) A modified shift-splitting method for nonsymmetric saddle point problems. J Comput Appl Math 317:535–546

    MathSciNet  MATH  Google Scholar 

  • Huang Z-G, Wang L-G, Xu Z, Cui J-J (2017) The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems. Appl Math Comput 299:95–118

    MathSciNet  MATH  Google Scholar 

  • Li J-L, Huang T-Z, Li L (2010) The spectral properties of the preconditioned matrix for nonsymmetric saddle point problems. J Comput Appl Math 235:270–285

    MathSciNet  MATH  Google Scholar 

  • Pan J-Y, Ng MK, Bai Z-Z (2006) New preconditioners for saddle point problems. Appl Math Comput 172:762–771

    MathSciNet  MATH  Google Scholar 

  • Ren Z-R, Cao Y, Niu Q (2017) Spectral analysis of the generalized shift-splitting preconditioned saddle point problem. J Comput Appl Math 311:539–550

    MathSciNet  MATH  Google Scholar 

  • Rozložník M (2018) Saddle-point problems and their iterative solution. Birkhäuser, Basel

    MATH  Google Scholar 

  • Salkuyeh DK, Rahimian M (2017) A modification of the generalized shift-splitting method for singular saddle point problems. Comput Math Appl 74:2940–2949

    MathSciNet  MATH  Google Scholar 

  • Salkuyeh DK, Masoudi M, Hezari D (2015) On the generalized shift-splitting preconditioner for saddle point problems. Appl Math Lett 48:55–61

    MathSciNet  MATH  Google Scholar 

  • Wang N-N, Li J-C (2019) A class of new extended shift-splitting preconditioners for saddle point problems. J Comput Appl Math 357:123–145

    MathSciNet  MATH  Google Scholar 

  • Wathen AJ (2015) Preconditioning. Acta Numer 24:329–376

    MathSciNet  MATH  Google Scholar 

  • Wu S-L, Salkuyeh DK (2020) A shift-splitting preconditioner for asymmetric saddle point problems. Comput Appl Math 39:314. https://doi.org/10.1007/s40314-020-01364-8

    Article  MathSciNet  Google Scholar 

  • Young DM (1971) Iterative solution of large linear systems. Academic Press, New York

    MATH  Google Scholar 

  • Zhang J-L, Gu C-Q (2016) A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems. BIT Numer Math 56:587–604

    MathSciNet  MATH  Google Scholar 

  • Zhang J-H, Chen X-P, Zhao J (2019a) Generalized fast shift-splitting preconditioner for nonsymmetric saddle-point problems. Comput Appl Math 38:1–24

    MathSciNet  MATH  Google Scholar 

  • Zhang L-T, Li C-Q, Li Y-T (2019b) A parameterized shift-splitting preconditioner for saddle point problems. Math Biosci Eng 16:1021–1033

    MathSciNet  Google Scholar 

  • Zheng Q-Q, Lu L-Z (2017) Extended shift-splitting preconditioners for saddle point problems. J Comput Appl Math 313:70–81

    MathSciNet  MATH  Google Scholar 

  • Zhou S-W, Yang A-L, Dou Y, Wu Y-J (2016) The modified shift-splitting preconditioners for nonsymmetric saddle point problems. Appl Math Lett 59:109–114

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very much indebted to Prof. Davod Khojasteh Salkuyeh for providing the codes of numerical experiments. We are also thankful to the reviewers for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, Shaanxi, China

    Bo Wu & Xing-Bao Gao

Authors
  1. Bo Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xing-Bao Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing-Bao Gao.

Additional information

Communicated by Joerg Fliege.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China under Grants 61273311 and 61803247.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, B., Gao, XB. A modified parameterized shift-splitting preconditioner for saddle point problems. Comp. Appl. Math. 40, 1 (2021). https://doi.org/10.1007/s40314-020-01383-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01383-5

Keywords

Mathematics Subject Classification

Search

Navigation