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

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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 W − T or T − W 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.

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References

  1. Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bai, Z.-Z., Ng, M.K.: On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput. 26, 1710–1724 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Betts, J.T.: Practical methods for optimal control using nonlinear programming. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  16. Day, D.D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real reformulations. SIAM J. Sci. Comput. 23, 480–498 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  18. Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics, vol. 43, pp 1105–1115. Springer, Berlin (2000)

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Strauss, W.A.: Partial differential equations. New York (1992)

  22. Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32, 271–298 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Li-Dan Liao & Guo-Feng Zhang

  2. Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu Province, People’s Republic of China

    Guo-Feng Zhang

Authors
  1. Li-Dan Liao
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  2. Guo-Feng Zhang
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Corresponding author

Correspondence to Guo-Feng Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11771193) and Fundamental Research Funds for the Central Universities (No. lzujbky-2017-it56).

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Liao, LD., Zhang, GF. A note on block diagonal and block triangular preconditioners for complex symmetric linear systems. Numer Algor 80, 1143–1154 (2019). https://doi.org/10.1007/s11075-018-0520-4

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  • DOI: https://doi.org/10.1007/s11075-018-0520-4

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