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A new version of a preconditioning method for certain two-by-two block matrices with square blocks

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Abstract

Two-by-two block matrices with square matrix blocks arise in many important applications. Since the problems are of large scale, iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is crucial. This paper presents two earlier used such preconditioners followed by a novel preconditioner based on transforming the given matrix to a proper form. Sharp eigenvalue estimates are derived. The condition numbers of each of the three methods are robust with respect to all parameters involved, including the mesh parameter. Therefore, the preconditioners are suitable for a variety of problems where such matrix structures arise. The performance of the methods are also compared numerically on a set of test problems.

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References

  1. Axelsson, O.: On the efficiency of a class of A-stable methods. BIT Numer. Math. 14, 279–287 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  3. Axelsson, O., Blaheta, R.: Low-rank improvements of two-level grid preconditioned matrices. J. Comput. Appl. Math. 340, 432–442 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Poisson and convection-diffusion control. Numer. Algorithm 73, 631–663 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Stokes Control Numer. Algorithm 74, 19–37 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Axelsson, O., Farouq, S., Neytcheva, M.: A preconditioner for optimal control problems constrained by Stokes equation with a time-harmonic control. J. Comput. Appl. Math. 310, 5–18 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Axelsson, O., Lukáš, D.: Preconditioning methods for eddy current optimally controlled time-harmonic electromagnetic problems. J. Numer. Math. (2018). https://doi.org/10.1515/jnma-2017-0064

  9. Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithm 66, 811–841 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithm 56, 93–317 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bai, Z.-Z., Golub, G.H., Ng, M.K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14, 319–335 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102, 1–38 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Blaheta, R., Axelsson, O., Luber, T.: High order time discretization for DAEs with efficient block preconditioners. In: AIP Conference Proceedings, vol. 1863, p. 340002. AIP Publishing (2017)

  19. Bruno, O.P., Kunyansky, L.A.: High-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications. J. Comput. Phys. 169, 80–110 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Butcher, J.C.: Integration processes based on Radau quadrature formulas. Math. Comput. 18, 233–244 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hezari, D., Edalatpour, V., Salkuyeh, D.K.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22, 761–776 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Howle, V.E., Vavasis, S.A.: An iterative method for solving complex-symmetric systems arising in electrical power modeling. SIAM J. Matrix Anal. Appl. 26, 1150–1178 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics. Lecture Notes in Computational Science and Engineering, vol. 15. Springer, Berlin (2000)

  25. Hezari, D., Salkuyeh, D.K., Edalatpour, V.: A new iterative method for solving a class of complex symmetric linear of equations. Numer. Algorithm 73, 927–955 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kormann, K.: Efficient and reliable simulation of quantum molecular dynamics. Ph.D. Thesis, Faculty of Science and Technology, Uppsala University, Uppsala, Sweden (2012)

  27. Langdon, S., Chandler-Wilde, S.N.: A wavenumber independent boundary element method for an acoustic scattering problem. SIAM J. Numer. Anal. 43, 2450–2477 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liang, Z.-Z., Axelsson, O., Neytcheva, M.: A robust structured preconditioner for the time-harmonic parabolic optimal control problem. Numer. Algorithm, 1–22 (2018)

  29. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  30. Rao, S.M.: An iterative method to solve acoustic scattering problems using a boundary integral equation. J. Acoust. Soc. Am. 130, 1792–1798 (2011)

    Article  Google Scholar 

  31. 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 

  32. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Press, New York (1995)

    Google Scholar 

  33. Salkuyeh, D.K.: Two-step scale-splitting method for solving complex symmetric system of linear equations. arXiv:1705.02468

  34. Salkuyeh, D.K., Hezari, D., Edalatpour, V.: Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. Int. J. Comput. Math. 92, 802–815 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Salkuyeh, D.K., Siahkolaei, T.S.: Two-parameter TSCSP method for solving complex symmetric system of linear equations. Calcolo 55, 1–22 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sommerfeld, A.: Partial Differential Equations in Physics. Academic Press, Cambridge (1949)

    MATH  Google Scholar 

  37. van Dijk, W., Toyama, F.M.: Accurate numerical solutions of the time-dependent Schrödinger equation. Phys. Rev. E. 75, 036707 (2007)

    Article  MathSciNet  Google Scholar 

  38. van Rienen, U.: Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  39. Wang, T., Zheng, Q.-Q., Lu, L.-Z.: A new iteration method for a class of complex symmetric linear systems. J. Comput. Appl. Math. 325, 188–197 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions. Several valuable improvements of the text by Maya Neytcheva is also gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Geonics, The Czech Academy of Sciences, Ostrava, Czech Republic

    Owe Axelsson

  2. Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    Davod Khojasteh Salkuyeh

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  1. Owe Axelsson
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  2. Davod Khojasteh Salkuyeh
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Correspondence to Owe Axelsson.

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Communicated by Lothar Reichel.

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The work of Owe Axelsson is supported by The Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II), project “IT4Innovations excellence in science—LQ1602”. The work of Davod Khojasteh Salkuyeh is partially supported by University of Guilan.

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Axelsson, O., Salkuyeh, D.K. A new version of a preconditioning method for certain two-by-two block matrices with square blocks. Bit Numer Math 59, 321–342 (2019). https://doi.org/10.1007/s10543-018-0741-x

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  • DOI: https://doi.org/10.1007/s10543-018-0741-x

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