Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

Zur Robustheit des gedämpftenV-Zyklus bei der FDMGM mit Wavelets

Abstract

The dampedV-cycle of the wavelet variation of the “Frequency decomposition multigrid method” of Hackbusch [Numer. Math.56, pp. 229–245 (1989)] is considered. It is shown that the convergence speed under sufficient damping is not affected by the presence of anisotropy but still depends on the number of levels. Our analysis is based on properties of wavelet packets which are supplied and proved. Numerical approximations to the speed of convergence illustrate the theoretical results.

Zusammenfassung

Wir betrachten den gedämpftenV-Zyklus für die Wavelet-Variante der “Frequenzzerlegungs-Multigridmethode” von Hackbusch [Numer. Math.56, 229–245 (1989)]. Es wird gezeigt, daß die Konvergenzgeschwindigkeit bei hinreichender Dämpfung durch Anisotropie nicht beeinflußt wird, aber noch von der Anzahl des Niveaus abhängt. Unsere Analyse beruht auf Eigenschaften von Wavelet-Paketen, die formuliert und bewiesen werden. Numerische Schätzungen der Konvergenzgeschwindigkeit erläutern die theoretischen Ergebnisse.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Bank, R. E., Dupont, T.F., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math.52, 427–458 (1988).

  2. [2]

    Beylkin, G.: On the representation of operators in bases of compactly supported wavelets. SIAM J. Numer. Anal.6, 1716–1740 (1992).

  3. [3]

    Bramble, J. H., Pasciak, J. E., Wang, J., Xu, J.: Convergence estimate for multigrid algorithms without regularity assumptions. Math. Comp.,57, 427–458 (1991).

  4. [4]

    Ciarlet, P. G.: The finite element methods for elliptic problems. New York: North-Holland 1987.

  5. [5]

    Coifman, R. R., Meyer, Y., Wickerhauser, M. V.: Size properties of wavelet packets. In: Ruskai (ed.) Wavelets and their applications, pp. 453–470. Boston: Jones and Bartlett 1992.

  6. [6]

    Daubechies, I.: Orthonormal bases of compactly supporte wavelets. Comm. Pure Appl. Math.41, 906–966 (1988).

  7. [7]

    Daubechies, I.: Ten lectures on wavelets. CBMS-NSF Series in Applied Mathematics. Philadelphia: SIAM Publications 1992.

  8. [8]

    Davis, P. J.: Circulant matrices. New York: John Wiley 1979.

  9. [9]

    Eirola, T.: Sobolev characterization of solutions of dilation equations. SIAM J. Math. Anal.23, 1015–1030 (1992).

  10. [10]

    Hackbusch, W.: Multi-grid methods and applications. Springer Series in Computational Mathematics. New York: Springer 1985.

  11. [11]

    Hackbusch, W.: The frequency decomposition multi-grid method, part I: Application to anisotropic equations. Numer. Math.56, 229–245 (1989).

  12. [12]

    Hackbusch, W.: The frequency decomposition multi-grid method, part II: Convergence analysis based on the additive Schwarz method. Numer. Math.63, 433–453 (1992).

  13. [13]

    Latto, A., Resnikoff, H. L., Tenenbaum, E.: The evaluation of connection coefficients of compactly supported wavelets. In: Proceedings of the USA-French Workshop on Wavelets and Turbulence. Princeton University 1991.

  14. [14]

    Mallat, S.: Multiresolution approximation and wavelet orthonormal bases ofL 2(R). Trans. Amer. Math. Soc.315, 69–87 (1989).

  15. [15]

    Meyer, Y.: Wavelets: algorithms and applications. Philadelphia: SIAM Publications 1993.

  16. [16]

    Rieder, A., Wells, R. O., Jr., Zhou, X.: A wavelet approach to robust multilevel solvers for anisotropic elliptic problems. Technical Report 93-07, Rice University, 1993. Computational Mathematics Laboratory. Accepted for publication in Appl. Comput. Harmonic Anal. (ACHA)

  17. [17]

    Wang, J.: Convergence analysis without regularity assumptions for multigrid algorithms based on SOR smoothing. SIAM J. Numer. Anal.29, 987–1001 (1992).

  18. [18]

    Wells, R. O., Jr., Zhou, X.: Wavelet interpolation and approximate solutions of elliptic partial differential equations. In: Wilson, R., Tanner, E. A. (eds.) Noncompact lie groups. Dordrecht: Kluwer, 1994. To appear. Proceedings of NATO Advanced Research Workshop.

Download references

Author information

Additional information

Supported partially by AFOSR under grant number 90-0334 which was funded by DARPA.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rieder, A., Zhou, X. On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method. Computing 53, 155–171 (1994). https://doi.org/10.1007/BF02252987

Download citation

AMS Subject Classifications

  • 65F10
  • 65N30

Key words

  • Wavelets
  • wavelet packets
  • robust multilevel methods
  • V-cycle