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Multigrid interpretations of the parareal algorithm leading to an overlapping variant and MGRIT

  • Martin J. Gander
  • Felix Kwok
  • Hui Zhang
Original Article
  • 79 Downloads

Abstract

The parareal algorithm is by construction a two level method, and there are several ways to interpret the parareal algorithm to obtain multilevel versions. We first review the three main interpretations of the parareal algorithm as a two-level method, a direct one, one based on geometric multigrid and one based on algebraic multigrid. The algebraic multigrid interpretation leads to the MGRIT algorithm, when using instead of only an F-smoother, a so called \(\textit{FCF}\)-smoother. We show that this can be interpreted at the level of the parareal algorithm as generous overlap in time. This allows us to generalize the method to even larger overlap, corresponding in MGRIT to \(F(\textit{CF})^{\nu }\)-smoothing, \(\nu \ge 1\), and we prove two new convergence results for such algorithms in the fully non-linear setting: convergence in a finite number of steps, becoming smaller when \(\nu \) increases, and a general superlinear convergence estimate for this generalized version of MGRIT. We illustrate our results with numerical experiments, both for linear and non-linear systems of ordinary and partial differential equations. Our results show that overlap only sometimes leads to faster algorithms.

Keywords

Parareal algorithm with overlap MGRIT Geometric and algebraic multigrid in time 

Notes

Acknowledgements

We would like to thank Jacob Schroder, Rob Falgout and Panayot Vassilevski for the very helpful discussions about algebraic multigrid, and for the historical references.

References

  1. 1.
    Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brandt, A., McCormick, S., Ruge, J.: Algebraic multigrid (AMG) for automatic multigrid solutions with application to geodetic computations. Report, Institute for Computational Studies, Fort Collins, Colorado (1982)Google Scholar
  3. 3.
    Buzbee, B.L., Golub, G.H., Nielson, C.W.: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7(4), 627–656 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chaouqui, F., Gander, M.J., Santugini-Repiquet, K.: On nilpotent subdomain iterations. In: Lee, C.O., Cai, X.C., Keyes, D.E., Kim, H.H., Klawonn, A., Park, E.J., Widlund, O.B. (eds.) Domain Decomposition Methods in Science and Engineering, vol. 116. Springer, Berlin (2016)Google Scholar
  5. 5.
    Falgout, R.D., Friedhoff, S., Kolev, T., MacLachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36(6), C635–C661 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Friedhoff, S., Falgout, R., Kolev, T., MacLachlan, S., Schroder, J.B.: A multigrid-in-time algorithm for solving evolution equations in parallel. In: Sixteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, USA (2013)Google Scholar
  7. 7.
    Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition Methods, pp. 69–113. Springer, Berlin (2015)CrossRefGoogle Scholar
  8. 8.
    Gander, M.J., Hairer, E.: Nonlinear convergence analysis for the parareal algorithm. In: Widlund, O.B., Keyes, D.E. (eds.) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol. 60, pp. 45–56. Springer, Berlin (2008)CrossRefGoogle Scholar
  9. 9.
    Gander, M.J., Hairer, E.: Analysis for parareal algorithms applied to Hamiltonian differential equations. J. Comput. Appl. Math. 259, 2–13 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gander, M.J., Halpern, L.: Méthodes de décomposition de domaines–notions de base. Techniques de l’ingénieur, Méthodes numériques, base documentaire: TIB105DUO (ref. article: af1375) (2013)Google Scholar
  11. 11.
    Gander, M.J., Halpern, L., Santugini-Repiquet, K.: Discontinuous coarse spaces for DD-methods with discontinuous iterates. In: Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XXI, pp. 607–615. Springer, Berlin (2014)Google Scholar
  12. 12.
    Gander, M.J., Halpern, L., Santugini-Repiquet, K.: A new coarse grid correction for RAS/AS. In: Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XXI, pp. 275–283. Springer, Berlin (2014)Google Scholar
  13. 13.
    Gander, M.J., Jiang, Y.L., Song, B., Zhang, H.: Analysis of two parareal algorithms for time-periodic problems. SIAM J. Sci. Comput. 35(5), A2393–2415 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gander, M.J., Loneland, A.: SHEM: an optimal coarse space for RAS and its multiscale approximation. In: Lee, C.O., Cai, X.C., Keyes, D.E., Kim, H.H., Klawonn, A., Park, E.J., Widlund, O.B. (eds.) Domain Decomposition Methods in Science and Engineering XXIII. Springer, Berlin (2016)Google Scholar
  15. 15.
    Gander, M.J., Loneland, A., Rahman, T.: Analysis of a new harmonically enriched multiscale coarse space for domain decomposition methods. arXiv preprint arXiv:1512.05285 (2015)
  16. 16.
    Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lions, J.L., Maday, Y., Turinici, G.: A parareal in time discretization of PDEs. C. R. Acad. Sci. Paris Serie I(332), 661–668 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ries, M., Trottenberg, U., Winter, G.: A note on MGR methods. Linear Algebra Appl. 49, 1–26 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ruge, J.W., Stüben, K.: Algebraic multigrid. Multigrid. Methods 3(13), 73–130 (1987)MathSciNetGoogle Scholar
  20. 20.
    Ruprecht, D., Speck, R., Krause, R.: Parareal for diffusion problems with space- and time-dependent coefficients. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T. (eds.) Domain Decomposition Methods in Science and Engineering XXII, pp. 371–378. Springer, Berlin (2016)CrossRefGoogle Scholar
  21. 21.
    Schröder, J.: Zur Lösung von Potentialaufgaben mit Hilfe des Differenzenverfahrens. ZAMM J. Appl. Math. Mech. 34(7), 241–253 (1954)CrossRefzbMATHGoogle Scholar
  22. 22.
    Stüben, K.: Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput. 13(3), 419–451 (1983)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Section de mathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong
  3. 3.Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang ProvinceZhejiang Ocean UniversityZhoushanChina

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