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Robust additive block triangular preconditioners for block two-by-two linear systems

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Abstract

In this paper, a class of additive block triangular preconditioners are constructed for solving block two-by-two linear systems with symmetric positive (semi-)definite sub-matrices. Convergence analysis of the related splitting iteration method shows that it is almost unconditionally convergent and behaves problem independent with a convergence rate less than 0.5 under a practical parameter choice. Optimization of the preconditioned matrices, which have real and tight eigenvalue distributions, shows that it can result in an upper bound less than 2 for the condition number of the preconditioned matrices. Moreover, we also give a special consideration about the feasibility of the proposed preconditioner for solving more general problems with indefinite sub-matrices. Numerical experiments based on examples arising from complex symmetric linear systems and PDE-constrained optimization problems are presented to show the robustness and effectiveness of the proposed preconditioners compared with some other existing preconditioners.

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References

  1. Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical and computational efficiency of solvers for two-phase problems. Comput. Math. Appl. 65(3), 301–314 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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(3), 631–663 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems Stokes control. Numer. Algorithm. 74(1), 19–37 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7(4), 197–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithm. 66(4), 811–841 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93(1), 41–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. 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(1), 343–369 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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. Algorithm. 62, 655–675 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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(3), 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28(3), 598–618 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bosch, J., Stoll, M.: Preconditioning for vector-valued Cahn-Hilliard equations. SIAM J. Sci. Comput. 37(5), s216–s243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Boyanova, P., Neytcheva, M.: Efficient numerical solution of discrete multi-component Cahn-Hilliard systems. Comput. Math. Appl. 67(1), 106–121 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cao, S.-M., Feng, W., Wang, Z.-Q.: On a type of matrix splitting preconditioners for a class of block two-by-two linear systems. Appl. Math. Lett. 79, 205–210 (2018)

    Article  MathSciNet  Google Scholar 

  20. Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23(2), 480–498 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Elman, H.C., Ramage, A., Silvester, D.J.: Algorithm 866 IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33(2), 14 (2007)

    Article  MATH  Google Scholar 

  22. Elman, H.C., Ramage, A., Silvester, D.J.: IFISS A computational laboratory for investigating incompressible flow problems. SIAM Rev. 56(2), 261–273 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hezari, D., Edalatpour, V., Salkuyeh, D.K: Preconditioned GSOR Iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebera Appl. 22(4), 761–776 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hezari, D., Salkuyeh, D.K., Edalatpour, V.: A new iterative method for solving a class of complex symmetric system of linear equations. Numer. Algorithm. 73(4), 927–955 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Krendl, W., Simoncini, V., Zulehner, W.: Stability estimates and structural spectral properties of saddle point problems. Numer. Math. 124(1), 183–213 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lang, C., Ren, Z.-R.: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. 93(1), 87–98 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liang, Z.-Z., Axelsson, O., Neytcheva, M.: A robust structured preconditioner for time-harmonic parabolic optimal control problems. Numer. Algorithm. 79(2), 575–596 (2017)

  29. Liang, Z.-Z., Zhang, G.-F.: On SSOR iteration method for a class of block two-by-two linear systems. Numer. Algorithm. 71(3), 655–671 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liao, L.-D., Zhang, G.-F., Li, R.-X.: Optimizing and improving of the C-to-R method for solving complex symmetric linear systems. Appl. Math. Lett. 336, 281–296 (2018)

    MathSciNet  Google Scholar 

  31. Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), A1079–A1109 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Notay, Y: AGMG software and documentation; see http://agmg.eu/

  33. Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pearson, J.W., Wathen, A.J.: A new approximation of the schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 19(5), 816–829 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Praetorius, S., Voigt, A.: Development and analysis of a block-preconditioner for the phase-field crystal equation. SIAM J. Sci. Comput. 37(3), s425–s451 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ren, Z.-R., Cao, Y., Zhang, L.-L.: On preconditioned MHSS real-valued iteration methods for a class of complex symmetric indefinite linear systems. East Asian J. Appl. Math. 6(2), 192–210 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  38. 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(4), 802–815 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sommerfeld, A.: Partial Differential Equations. Academic Press, New York (1949)

    MATH  Google Scholar 

  40. Van Rienen, U.: Numerical methods in computational electrodynamic: linear systems in practical applications. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  41. Wang, T., Lu, L.-Z.: Alternating-directional PMHSS Iteration method for a class of two-by-two block linear systems. Appl. Math. Lett. 58, 159–164 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu, W.-W.: A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems. Appl. Math. Comput. 219, 10510–10517 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Yan, H.-Y., Huang, Y.-M.: Splitting-based block preconditioning methods for block two-by-two matrices of real square blocks. Appl. Math. Comput. 243, 825–837 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Zeng, M.-L., Zhang, G.-F.: Parameterized rotated block preconditioning techniques for block two-by-two systems with application to complex linear systems. Comput. Math. Appl. 70(12), 2946–2957 (2015)

    Article  MathSciNet  Google Scholar 

  45. Zheng, Q.-Q., Ma, C.-F.: Accelerated PMHSS Iteration methods for complex symmetric linear systems. Numer. Algorithm. 73(2), 501–516 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to express our sincere thanks to the unknown reviewer for his careful reading of the manuscript. His useful comments and valuable suggestions greatly improve the quality of the paper.

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Authors and Affiliations

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

    Zhao-Zheng Liang & Guo-Feng Zhang

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  1. Zhao-Zheng Liang
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  2. Guo-Feng Zhang
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Correspondence to Zhao-Zheng Liang or Guo-Feng Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11801242 and 11771193) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2018-31).

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Liang, ZZ., Zhang, GF. Robust additive block triangular preconditioners for block two-by-two linear systems. Numer Algor 82, 503–537 (2019). https://doi.org/10.1007/s11075-018-0611-2

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  • DOI: https://doi.org/10.1007/s11075-018-0611-2

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