Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems

Abstract

For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods.

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

References

  1. Bai Z-Z (1998) The convergence of the two-stage iterative method for Hermitian positive definite linear systems. Appl Math Lett 11(2):1–5

    MathSciNet  MATH  Article  Google Scholar 

  2. Bai Z-Z (1999) Convergence analysis of the two-stage multisplitting method. Calcolo 36(2):63–74

    MathSciNet  MATH  Article  Google Scholar 

  3. 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(4):e2116

    MathSciNet  MATH  Article  Google Scholar 

  4. Bai Z-Z, Benzi M (2017) Regularized HSS iteration methods for saddle-point linear systems. BIT Numer Math 57(2):287–311

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  6. Bai Z-Z, Wang D-R (1997) The monotone convergence of the two-stage iterative method for solving large sparse systems of linear equations. Appl Math Lett 10(1):113–117

    MathSciNet  MATH  Article  Google Scholar 

  7. 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(3):603–626

    MathSciNet  MATH  Article  Google Scholar 

  8. 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):1–32

    MathSciNet  MATH  Article  Google Scholar 

  9. Benzi M, Golub GH (2004) A preconditioner for generalized saddle point problems. SIAM J Matrix Anal Appl 26(1):20–41

    MathSciNet  MATH  Article  Google Scholar 

  10. Bramble JH, Pasciak JE, Vassilev AT (1997) Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J Numer Anal 34(3):1072–1092

    MathSciNet  MATH  Article  Google Scholar 

  11. Cao Y, Ren Z-R, Shi Q (2016) A simplified HSS preconditioner for generalized saddle point problems. BIT Numer Math 56(2):423–439

    MathSciNet  MATH  Article  Google Scholar 

  12. Elman HC, Golub GH (1994) Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J Numer Anal 31(6):1645–1661

    MathSciNet  MATH  Article  Google Scholar 

  13. Frommer A, Szyld DB (1992) \(H\)-splittings and two-stage iterative methods. Numer Math 63(1):345–356

    MathSciNet  MATH  Article  Google Scholar 

  14. Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore

    Google Scholar 

  15. Golub GH, Wu X, Yuan J-Y (2001) SOR-like methods for augmented systems. BIT Numer Math 41(1):71–85

    MathSciNet  MATH  Article  Google Scholar 

  16. Huang Z-G, Wang L-G, Xu Z, Cui J-J (2016) Improved PPHSS iterative methods for solving nonsingular and singular saddle point problems. Comput Math Appl 72(1):92–109

    MathSciNet  Article  Google Scholar 

  17. Huang Z-G, Wang L-G, Xu Z, Cui J-J (2018) The modified PAHSS-PU and modified PPHSS-SOR iterative methods for saddle point problems. Comput Appl Math 37(5):6076–6107

    MathSciNet  MATH  Article  Google Scholar 

  18. Nichols NK (1973) On the convergence of two-stage iterative processes for solving linear equations. SIAM J Numer Anal 10(3):460–469

    MathSciNet  MATH  Article  Google Scholar 

  19. Wang K, Di J-J, Liu D (2016) Improved PHSS iterative methods for solving saddle point problems. Numer Algorithms 71(4):753–773

    MathSciNet  MATH  Article  Google Scholar 

  20. Yang A-L, An J, Wu Y-J (2010) A generalized preconditioned HSS method for non-Hermitian positive definite linear systems. Appl Math Comput 216(6):1715–1722

    MathSciNet  MATH  Google Scholar 

  21. Zhao J-X (1998) The generalized Cholesky factorization method for saddle point problems. Appl Math Comput 92(1):49–58

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fang Chen.

Additional information

Publisher's Note

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

Supported by The National Natural Science Foundation (no. 11501038), and The Science and Technology Planning Projects of Beijing Municipal Education Commission (no. KM201911232010, no. KM202011232019 and no. KM201811232020), People’s Republic of China.

Communicated by Zhong-Zhi Bai.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, F., Li, T. & Lu, K. Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems. Comp. Appl. Math. 39, 162 (2020). https://doi.org/10.1007/s40314-020-01187-7

Download citation

Keywords

  • Saddle-point linear problem
  • Hermitian and skew-Hermitian splitting
  • Matrix splitting iteration
  • Convergence

Mathematics Subject Classification

  • 65F10
  • 65F15