Advertisement

A note on block diagonal and block triangular preconditioners for complex symmetric linear systems

  • Li-Dan Liao
  • Guo-Feng Zhang
Original Paper
  • 68 Downloads

Abstract

In this note, the additive block diagonal preconditioner (Bai et al., Numer. Algorithms 62, 655–675 2013) and the block triangular preconditioner (Pearson and Wathen, Numer. Linear Algebra Appl. 19, 816–829 2012) are further studied and optimized, respectively. The eigenvalue properties of these two preconditioned matrices are analyzed by new way and an expression of the quasi-optimal parameter is derived. Particularly, when WT or TW is symmetric positive semidefinite, the exact eigenvalue bounds are obtained which are tighter than the state of the art. At last, numerical experiments are presented to show the effectiveness of the two proposed optimized preconditioners.

Keywords

Preconditioner Complex symmetric problem Optimal parameter Condition number 

Mathematics Subject Classification (2010)

65F10 65F50 65W05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors are very thankful to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.

References

  1. 1.
    Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Axelsson, O., Neytcheva, M.G., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66, 811–841 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bai, Z.-Z.: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109, 273–285 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bai, Z.-Z., Chen, F., Wang, Z.-Q.: Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices. Numer. Algorithms 62, 655–675 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bai, Z.-Z., Ng, M.K.: On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput. 26, 1710–1724 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Betts, J.T.: Practical methods for optimal control using nonlinear programming. SIAM, Philadelphia (2001)zbMATHGoogle Scholar
  15. 15.
    Dijk, W.V., Toyama, F.M.: Accurate numerical solutions of the time-dependent Schrödinger equation. Phys. Rev. E 75, 036707-1-036707-10 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Day, D.D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real reformulations. SIAM J. Sci. Comput. 23, 480–498 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Feriani, A., Perotti, F., Simoncini, V.: Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 1719–1739 (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics, vol. 43, pp 1105–1115. Springer, Berlin (2000)CrossRefGoogle Scholar
  19. 19.
    Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 19, 816–829 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lass, O., Vallejos, M., Borzi, A., Douglas, C.C.: Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84, 27–48 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Strauss, W.A.: Partial differential equations. New York (1992)Google Scholar
  22. 22.
    Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32, 271–298 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zheng, Z., Zhang, G.-F.: A note on preconditioners for complex linear systems arising from PDE-constrained optimization problems. Appl. Math. Lett. 61, 114–121 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lang, C., Ren, Z.-R.: Rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. 93, 87–98 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Key Laboratory of Applied Mathematics and Complex SystemsLanzhouPeople’s Republic of China

Personalised recommendations