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On the residual norms, the Ritz values and the harmonic Ritz values that can be generated by restarted GMRES

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Abstract

The paper gives a characterization of all linear systems such that when restarted GMRES is applied, prescribed admissible residual norms and (harmonic) Ritz values for all iterations inside the individual cycles are generated. Additionally, the system matrices can have any nonzero eigenvalues. The total number of GMRES iterations inside all cycles considered is assumed to be smaller than the system size. It is shown that stagnation at the end of a restart cycle must be mirrored at the beginning of the next cycle and that this is the only restriction for prescribed residual norms of restarted GMRES. The relation between prescribed residual norms of restarted GMRES and those of the corresponding full GMRES process is studied and linear systems are given where full and restarted GMRES give the same convergence history.

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References

  1. Arioli, M., Pták, V., Strakoš, Z.: Krylov sequences of maximal length and convergence of GMRES. BIT 38(4), 636–643 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Brown, P.N.: A theoretical comparison of the Arnoldi and GMRES algorithms. SIAM J. Sci. Statist. Comput. 12(1), 58–78 (1991)

    MathSciNet  MATH  Google Scholar 

  3. Chapman, A., Saad, Y.: Deflated and augmented Krylov subspace techniques. Numer. Linear Algebra Appl. 4(1), 43–66 (1997)

    MathSciNet  MATH  Google Scholar 

  4. De Sturler, E.: Truncation strategies for optimal Krylov subspace methods. SIAM J. Numer. Anal. 36(3), 864–889 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Du, K., Duintjer Tebbens, J., Meurant, G.: Any admissible harmonic Ritz value set is possible for GMRES. Electron. Trans. Numer. Anal. 47(SI), 37–56 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Duintjer Tebbens, J., Meurant, G.: Any Ritz value behavior is possible for Arnoldi and for GMRES. SIAM J. Matrix Anal. Appl. 33(3), 958–978 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Duintjer Tebbens, J., Meurant, G.: Prescribing the behavior of early terminating GMRES and Arnoldi iterations. Num. Algor. 65(1), 69–90 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Duintjer Tebbens, J., Meurant, G.: On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems. BIT 56(1), 77–97 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Duintjer Tebbens, J., Meurant, G., Sadok, H., Strakoš, Z.: On investigating GMRES convergence using unitary matrices. Lin. Alg. Appl. 450, 83–107 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Eiermann, M., Ernst, O.G.: Geometric aspects of the theory of Krylov subspace methods. Acta Numer. 10, 251–312 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Eiermann, M., Ernst, O.G., Schneider, O.: Analysis of acceleration strategies for restarted minimal residual methods. J. Comput. Appl. Math. 123(1–2), 261–292 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Elman, H.: Iterative Methods for Large Sparse Nonsymmetric Systems of Linear Equations. PhD thesis. Department of Computer Science. Yale University, New Haven (1982)

    Google Scholar 

  13. Embree, M.: The tortoise and the hare restart GMRES. SIAM Rev. 45(2), 259–266 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Faber, V., Liesen, J., Tichý, P.: The Faber-Manteuffel theorem for linear operators. SIAM J. Numer. Anal. 46(3), 1323–1337 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Faber, V., Manteuffel, T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21(2), 352–362 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Freund, R.W., Nachtigal, N.M.: QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60(3), 315–339 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Gaul, A., Gutknecht, M.H., Liesen, J., Nabben, R.: A framework for deflated and augmented Krylov subspace methods. SIAM J. Matrix Anal. Appl. 34 (2), 495–518 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Greenbaum, A., Pták, V., Strakoš, Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17(3), 465–469 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Greenbaum, A., Strakoš, Z.: Matrices that generate the same Krylov residual spaces. In: Recent Advances in Iterative Methods, IMA Vol. Math. Appl., vol. 60, pp 95–118. Springer, New York (1994)

  20. Liesen, J., Strakoš, Z.: GMRES convergence analysis for a convection-diffusion model problem. SIAM J. Sci. Comput. 26(6), 1989–2009 (2005). (electronic)

    MathSciNet  MATH  Google Scholar 

  21. Liesen, J., Strakoš, Z.: Krylov Subspace Methods, Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)

    MATH  Google Scholar 

  22. Meurant, G.: Necessary and sufficient conditions for GMRES complete and partial stagnation. Submitted

  23. Meurant, G., Duintjer Tebbens, J.: The role eigenvalues play in forming GMRES residual norms with non-normal matrices. Numer. Algor. 68, 143–165 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16(4), 1154–1171 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Morgan, R.B.: Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 21(4), 1112–1135 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Morgan, R.B.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24(1), 20–37 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Nachtigal, N.M., Reichel, L., Trefethen, L.N.: A hybrid GMRES algorithm for nonsymmetric linear systems. SIAM J. Matrix Anal. Appl. 13(3), 796–825 (1992)

    MathSciNet  MATH  Google Scholar 

  28. Parlett, B., Strang, G.: Matrices with prescribed Ritz values. Linear Algebra Appl. 428(7), 1725–1739 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Saad, Y.: Analysis of augmented Krylov subspace methods. SIAM J. Matrix Anal. Appl. 18(2), 435–449 (1997)

    MathSciNet  MATH  Google Scholar 

  30. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856–869 (1986)

    MathSciNet  MATH  Google Scholar 

  31. Schweitzer, M.: Any finite convergence curve is possible in the initial iterations of restarted FOM. Electron. Trans. Numer. Anal. 45, 133–145 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Simoncini, V.: On the convergence of restarted Krylov subspace methods. SIAM J. Matrix Anal. Appl. 22(2), 430–452 (2000)

    MathSciNet  MATH  Google Scholar 

  33. Simoncini, V., Szyld, D.B.: New conditions for non-stagnation of minimal residual methods. Numer. Math. 109(3), 477–487 (2008)

    MathSciNet  MATH  Google Scholar 

  34. de Surler, E.: Personal communication (2013)

  35. Vecharynski, E., Langou, J.: The cycle-convergence of restarted GMRES for normal matrices is sublinear. SIAM J. Sci. Comput. 32(1), 186–196 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Vecharynski, E., Langou, J.: Any admissible cycle-convergence behavior is possible for restarted GMRES at its initial cycles. Num. Lin. Algebr. Appl. 18, 499–511 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Zavorin, I., O’Leary, D.P., Elman, H.: Complete stagnation of GMRES. Linear Algebra Appl. 367, 165–183 (2003)

    MathSciNet  MATH  Google Scholar 

  38. Zhong, B., Morgan, R.B.: Complementary cycles of restarted GMRES. Numer. Linear Algebra Appl. 15(6), 559–571 (2008)

    MathSciNet  MATH  Google Scholar 

  39. Zítko, J.: Generalization of convergence conditions for a restarted GMRES. Numer. Linear Algebra Appl. 7(3), 117–131 (2000)

    MathSciNet  MATH  Google Scholar 

  40. Zítko, J.: Some remarks on the restarted and augmented GMRES method. Electron. Trans. Numer. Anal. 31, 221–227 (2008)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank Eric de Sturler for a discussion that has stimulated the search for some of the results in Section 4.

Funding

The work of J. Duintjer Tebbens was supported by the long-term strategic development financing of the Institute of Computer Science (RVO: 67985807) of the Czech Academy of Sciences.

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 2, 18 207, Prague 8 - Libeň, Czech Republic

    Jurjen Duintjer Tebbens

  2. Department of Biophysics and Physical Chemistry, Faculty of Pharmacy, Charles University, Akademika Heyrovského 1203, 500 05, Hradec Králové, Czech Republic

    Jurjen Duintjer Tebbens

  3. 30 rue du sergent Bauchat, 75012, Paris, France

    Gérard Meurant

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  1. Jurjen Duintjer Tebbens
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  2. Gérard Meurant
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Correspondence to Jurjen Duintjer Tebbens.

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Duintjer Tebbens, J., Meurant, G. On the residual norms, the Ritz values and the harmonic Ritz values that can be generated by restarted GMRES. Numer Algor 84, 1329–1352 (2020). https://doi.org/10.1007/s11075-019-00846-z

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  • DOI: https://doi.org/10.1007/s11075-019-00846-z

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