Skip to main content
SpringerLink
Log in

Efficient HSS-based preconditioners for generalized saddle point problems

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

Abstract

A fast iteration method based on HSS is proposed for solving the nonsymmetric generalized saddle point problem. It converges to the unique solution of the generalized saddle point problem unconditionally. We devise a new preconditioner induced by the new iteration method. We analyze the spectrum of the preconditioned coefficient matrix, and reveal the relation between the theoretically required number of iteration steps and the dimension of the preconditioned Krylov subspace. Furthermore, some practical inexact variants of the new preconditioner have been developed to reduce the computational overhead. Numerical experiments validate the effectiveness of the proposed preconditioners.

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
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. In this case, EVHSS-preconditioned GMRES uses two full restarts (each of 10 inner steps) and a restart with just 1 inner step. Thus, the total number of inner steps is \((3-1)\times 10+1=21\).

References

  • Axelsson O (2019) Optimality properties of a square block matrix preconditioner with applications. Comput Math Appl. https://doi.org/10.1016/j.camwa.2019.09.024

    Article  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, 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, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24:603–626

    MathSciNet  MATH  Google Scholar 

  • Bai Z-Z, Golub GH, Lu L-Z (2005a) Block triangular and skew-Hermitian 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, Golub GH, Ng MK (2008) On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl 428:413–440

    MathSciNet  MATH  Google Scholar 

  • Beik FPA, Benzi M, Chaparpordi SHA (2017) On block diagonal and block triangular iterative schemes and preconditioners for stabilized saddle point problems. J Comput Appl Math 326:15–30

    MathSciNet  MATH  Google Scholar 

  • Benzi M (2002) Preconditioning techniques for large linear systems: a survey. J Comput Phys 182:418–477

    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 

  • Betts JT (2001) Practical methods for optimal control using nonlinear programming. SIAM, Philadelphia

    MATH  Google Scholar 

  • Cao Y, Yi S-C (2016) A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian sddle point problems. Appl Math Comput 275:41–49

    MathSciNet  MATH  Google Scholar 

  • Cao Y, Yao L-Q, Jiang M-Q (2013) A modified dimensional split preconditioner for generalized saddle point problems. J Comput Appl Math 250:70–82

    MathSciNet  MATH  Google Scholar 

  • Cao Y, Dong J-L, Wang Y-M (2015) A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation. J Comput Appl Math 273:41–60

    MathSciNet  MATH  Google Scholar 

  • Cao Y, Ren Z-R, Shi Q (2016) A simplified HSS preconditioner for generalized saddle point problems. BIT Numer Math 56:423–439

    MathSciNet  MATH  Google Scholar 

  • Cao Y, Ren Z, Yao L (2019) Improved relaxed positive-definite and skew-Hermitian splitting preconditioners for saddle point problems. J Comput Math 37:95–111

    MathSciNet  MATH  Google Scholar 

  • Chen F (2018) On convergence of EVHSS iteration method for solving generalized saddle-point linear systems. Appl Math Lett 86:30–35

    MathSciNet  MATH  Google Scholar 

  • de Sturler E, Liesen J (2005) Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. Part I: theory. SIAM J Sci Comput 26:1598–1619

    MathSciNet  MATH  Google Scholar 

  • Deuring P (2009) Eigenvalue bounds for the Schur complement with a pressure convection-diffusion preconditioner in incompressible flow computations. J Comput Appl Math 228:444–457

    MathSciNet  MATH  Google Scholar 

  • Elman HC, Ramage A, Silvester DJ (2014) IFISS: a computational laboratory for investigating incompressible flow problems. SIAM Rev 56:261–273

    MathSciNet  MATH  Google Scholar 

  • Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Huang T-Z, Wu S-L, Li C-X (2009) The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems. J Comput Appl Math 229:37–46

    MathSciNet  MATH  Google Scholar 

  • Kay D, Loghin D, Wathen AJ (2002) A preconditioner for the steady-state Navier–Stokes equations. SIAM J Sci Comput 24:237–256

    MathSciNet  MATH  Google Scholar 

  • Ke Y-F, Ma C-F (2017) The dimensional splitting iteration methods for solving saddle point problems arising from time-harmonic eddy current models. Appl Math Comput 303:146–164

    MathSciNet  MATH  Google Scholar 

  • Keller C, Gould NIM, Wathen AJ (2000) Constraint preconditioning for indefinite linear systems. SIAM J Matrix Anal Appl 21:1300–1317

    MathSciNet  MATH  Google Scholar 

  • Li C-X, Wu S-L (2015) A single-step HSS method for non-Hermitian positive definite linear systems. Appl Math Lett 44:26–29

    MathSciNet  MATH  Google Scholar 

  • Liang Z-Z, Zhang G-F (2016) Two new variants of the HSS preconditioner for regularized saddle point problems. Comput Math Appl 72:603–619

    MathSciNet  MATH  Google Scholar 

  • Liao L-D, Zhang G-F (2019) A generalized variant of simplified HSS preconditioner for generalized saddle point problems. Appl Math Comput 346:790–799

    MathSciNet  MATH  Google Scholar 

  • Ling S-T, Liu Q-B (2017) New local generalized shift-splitting preconditioners for saddle point problems. Appl Math Comput 302:58–67

    MathSciNet  MATH  Google Scholar 

  • Loghin D, Wathen AJ (2002) Schur complement preconditioners for the Navier–Stokes equations. Int J Numer Methods Fluids 40:403–412

    MathSciNet  MATH  Google Scholar 

  • Murphy MF, Golub GH, Wathen AJ (2000) A note on preconditioning for indefinite linear systems. SIAM J Sci Comput 21:1969–1972

    MathSciNet  MATH  Google Scholar 

  • Olshanskii MA, Vassilevski YV (2007) Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J Sci Comput 29:2686–2704

    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 

  • Pearson JW, Wathen AJ (2012) A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer Linear Algebra Appl 19:816–829

    MathSciNet  MATH  Google Scholar 

  • Pearson JW, Wathen AJ (2018) Matching Schur complement approximations for certain saddle-point systems. Contemporary computational mathematics—a celebration of the 80th birthday of Ian Sloan

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

    MATH  Google Scholar 

  • Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia

    MATH  Google Scholar 

  • Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869

    MathSciNet  MATH  Google Scholar 

  • Shen S-Q (2014) A note on PSS preconditioners for generalized saddle point problems. Appl Math Comput 237:723–729

    MathSciNet  MATH  Google Scholar 

  • Shen Q-Q, Shi Q (2016) Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems. Comput Math Appl 72:632–641

    MathSciNet  MATH  Google Scholar 

  • Shen H-L, Wu H-Y, Shao X-H, Song X-D (2019) The PPS method-based constraint preconditioners for generalized saddle point problems. Comput Appl Math 38(1):21

    MathSciNet  MATH  Google Scholar 

  • Simoncini V (2004) Block triangular preconditioners for symmetric saddle-point problems. Appl Numer Math 49:63–80

    MathSciNet  MATH  Google Scholar 

  • Simoncini V, Szyld DB (2003) Flexible inner-outer Krylov subspace methods SIAM. J Numer Anal 40:2219–2239

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Yun JH (2013) Variants of the Uzawa method for saddle point problem. Comput Math Appl 65:1037–1046

    MathSciNet  MATH  Google Scholar 

  • Zeng M-L, Ma C-F (2016) A parameterized SHSS iteration method for a class of complex symmetric system of linear equations. Comput Math Appl 71:2124–2131

    MathSciNet  Google Scholar 

  • Zhang J-L (2018) An efficient variant of HSS preconditioner for generalized saddle point problems. Numer Linear Algebra Appl 25:e2166. https://doi.org/10.1002/nla.2166

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang J, Shang J (2010) A class of Uzawa-SOR methods for saddle point problems. Appl Math Comput 216:2163–2168

    MathSciNet  MATH  Google Scholar 

  • Zhang G-F, Ren Z-R, Zhou Y-Y (2011) On HSS-based constraint preconditioners for generalized saddle-point problems. Numer Algorithms 57:273–287

    MathSciNet  MATH  Google Scholar 

  • Zhang J-L, Gu C-Q, Zhang K (2014) A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems. Appl Math Comput 249:468–479

    MathSciNet  MATH  Google Scholar 

  • Zhang K, Zhang J-L, Gu C-Q (2017) A new relaxed PSS preconditioner for nonsymmetric saddle point problems. Appl Math Comput 308:115–129

    MathSciNet  MATH  Google Scholar 

  • Zhang C-H, Wang X, Tang X-B (2019) Generalized AOR method for solving a class of generalized saddle point problems. J Comput Appl Math 350:69–79

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11601323,11801362) and the Key Discipline Fund (Multivariate Statistical Analysis 2018-2019) of College of Arts and Sciences in Shanghai Maritime University. The authors would like to thank Dr. Ju-Li Zhang of Shanghai University of Engineering Science for helpful discussions. The first author also would like to thank Professor Jun-Feng Yin for gracious invitation and hospitality during his visit to Tongji University.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, People’s Republic of China

    Ke Zhang & Lin-Na Wang

Authors
  1. Ke Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin-Na Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ke Zhang.

Additional information

Communicated by Natasa Krejic.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, K., Wang, LN. Efficient HSS-based preconditioners for generalized saddle point problems. Comp. Appl. Math. 39, 154 (2020). https://doi.org/10.1007/s40314-020-01180-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01180-0

Keywords

Mathematics Subject Classification

Search

Navigation