Skip to main content
SpringerLink
Log in

A Look-Back-type restart for the restarted Krylov subspace methods for solving non-Hermitian linear systems

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the restarted Krylov subspace methods, as typified by the GMRES(m) method and the FOM(m) method, for solving non-Hermitian linear systems. We have recently focused on the restart of the GMRES(m) method and proposed the extension of the GMRES(m) method based on the error equations. The main purpose of this paper is to apply the extension to other restarted Krylov subspace methods, and propose a specific restart technique for the restarted Krylov subspace method. The specific restart technique is named as the Look-Back-type restart, and is based on an implicit residual polynomial reconstruction via the initial guess. The comparison analysis based on the residual polynomials and some numerical experiments indicate that the Look-Back-type restart achieves more efficient convergence results than the traditional restarted Krylov subspace methods.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Baglama, J., Calvetti, D., Golub, G.H., Reichel, L.: Adaptively preconditioned GMRES algorithms. SIAM J. Sci. Comput. 20, 17–29 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Baglama, J., Reichel, L.: Augmented GMRES-type methods. Numer. Linear Algebra Appl. 14, 337–350 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, A.H., Jessup, E.R., Kolev, T.V.: A simple strategy for varying the parameter in GMRES(\(m\)). J. Comput. Appl. Math. 230, 751–761 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baker, A.H., Jessup, E.R., Manteuffel, T.: A technique for accelerating the convergence of restarted GMRES. SIAM J. Math. Anal. Appl. 26, 962–984 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bayliss, A., Goldstein, C.I., Turkel, E.: An iterative method for the Helmholtz equation. J. Comput. Phys. 49, 443–457 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burrage, K., Erhel, J.: On the performance of various adaptive preconditioned GMRES strategies. Numer. Linear Algebra Appl. 5, 101–121 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erhel, J., Burrage, K., Pohl, B.: Restarted GMRES preconditioned by deflation. J. Comput. Appl. Math. 69, 303–318 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Imakura, A., Sogabe, T., Zhang, S.-L.: An efficient variant of the GMRES(\(m\)) method based on the error equations. EAJAM 2, 1–22 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric \(M\)-matrix. Math. Comput. 31, 148–162 (1977)

    MathSciNet  MATH  Google Scholar 

  10. Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16, 1154–1171 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Morgan, R.B.: Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 21, 1112–1135 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Morgan, R.B.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24, 20–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Moriya, K., Nodera, T.: The DEFLATED-GMRES(\(m, k\)) method with switching the restart frequency dynamically. Numer. Linear Algebra Appl. 7, 569–584 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  15. Saad, Y.: Krylov subspace methods for solving large unsymmetric linear systesms. Math. Comput. 37, 105–126 (1981)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Sleijpen, G.L.G., Fokkema, D.R.: BiCGStab(\(l\)) for linear equations involving unsymmetric matrices with complex spectrum. Electron. Trans. Numer. Anal. 1, 11–32 (1993)

    MathSciNet  MATH  Google Scholar 

  18. Sosonkina, M., Watson, L.T., Kapania, R.K., Walker, H.F.: A new adaptive GMRES algorithm for achieving high accuracy. Numer. Linear Algebra Appl. 5, 275–297 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. SuiteSparse Matrix Collection. https://sparse.tamu.edu/. Accessed 1 Oct 2017

  20. van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, New York (2003)

    Book  MATH  Google Scholar 

  21. van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. van der Vorst, H.A., Vuik, C.: GMRESR: a family of nested GMRES methods. Numer. Linear Algebra Appl. 1, 369–386 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, L., Nodera, T.: A new adaptive restart for GMRES(\(m\)) method. ANZIAM J. 46, 409–426 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to anonymous referees for useful comments. This research was supported partly by JST/ACT-I (no. JPMJPR16U6), JSPS KAKENHI (no. 17K12690).

Author information

Authors and Affiliations

  1. Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8573, Japan

    Akira Imakura

  2. Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-Cho, Chikusa-Ku, Nagoya, 464-8603, Japan

    Tomohiro Sogabe & Shao-Liang Zhang

Authors
  1. Akira Imakura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomohiro Sogabe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shao-Liang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akira Imakura.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Imakura, A., Sogabe, T. & Zhang, SL. A Look-Back-type restart for the restarted Krylov subspace methods for solving non-Hermitian linear systems. Japan J. Indust. Appl. Math. 35, 835–859 (2018). https://doi.org/10.1007/s13160-018-0308-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-018-0308-x

Keywords

Mathematics Subject Classification

Search

Navigation