Skip to main content
SpringerLink
Log in

Fast computation of adaptive wavelet expansions

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we describe and analyze an algorithm for the fast computation of sparse wavelet coefficient arrays typically arising in adaptive wavelet solvers. The scheme improves on an earlier version from Dahmen et al. (Numer. Math. 86, 49–101, 2000) in several respects motivated by recent developments of adaptive wavelet schemes. The new structure of the scheme is shown to enhance its performance while a completely different approach to the error analysis accommodates the needs put forward by the above mentioned context of adaptive solvers. The results are illustrated by numerical experiments for one and two dimensional examples.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barinka A., Barsch T., Dahlke S. and Konik M. (2001). Some remarks on quadrature formulas for refinable functions and wavelets. ZAMM 81(12): 839–855

    Article  MathSciNet  MATH  Google Scholar 

  2. Barinka, A., Dahmen, W., Schneider, R.: Fast computation of adaptive wavelet expansions (extended version). IGPM report #244, RWTH, Aachen (2004)

  3. Canuto A., Tabacco A. and Urban K. (1999). The wavelet element method, part I: construction and analysis. Appl. Comp. Harm. Anal. 6: 1–52

    Article  MathSciNet  MATH  Google Scholar 

  4. Canuto A., Tabacco A. and Urban K. (2000). The wavelet element method, part II: realization and additional features. Appl. Comp. Harm. Anal. 8: 123–165

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen A., Dahmen W. and DeVore R. (2001). Adaptive wavelet methods for elliptic operator equations—Convergence rates. Math. Comp. 70: 27–75

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen A., Dahmen W. and DeVore R. (2002). Adaptive wavelet methods II— Beyond the elliptic case. Found. Comput. Math. 2: 203–245

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal. 41, 1785–1823 (2003)

    Google Scholar 

  8. Cohen A., Dahmen W. and DeVore R. (2003). Sparse evaluation of nonlinear functionals of multiscale expansions. SIAM J. Math. Anal. 35: 279–303

    Article  MathSciNet  MATH  Google Scholar 

  9. Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet techniques in Numerical Simulation. In: Encyclopedia of Computational Mechanics, R. De Borste, T. Hughes, E. Stein (eds) (2004), 157–197

  10. Cohen A. and Masson R. (1997). Wavelet adaptive methods for second order elliptic problems, boundary conditions and domain decomposition. Numer. Math. 8: 21–47

    Google Scholar 

  11. Dahlke S., Dahmen W. and Urban K. (2002). Adaptive wavelet methods for saddle point problems—Convergence rates. SIAM J. Numer. Anal. 40(4): 1230–1262

    Article  MathSciNet  MATH  Google Scholar 

  12. Dahmen, W.: Multiscale and wavelet methods for operator equations, C.I.M.E. Lecture Notes, vol. 1825, Springer, Berlin Heidelberg New York, 2003, 31–96

  13. Dahmen W. and Micchelli C.A. (1993). Using the refinement equation for evaluating integrals of wavelets. SIAM J. Numer. Anal. 30: 507–537

    Article  MathSciNet  MATH  Google Scholar 

  14. Dahmen W. and Schneider R. (1999). Composite wavelet bases for operator equations. Math. Comp. 68: 1533–1567

    Article  MathSciNet  MATH  Google Scholar 

  15. Dahmen W. and Schneider R. (1999). Wavelets on manifolds I: construction and domain decomposition. SIAM J. Math. Anal. 31: 184–230

    Article  MathSciNet  MATH  Google Scholar 

  16. Dahmen W., Schneider R. and Xu Y. (2000). Nonlinear functions of wavelet expansions—adaptive reconstruction and fast evaluation. Numer. Math. 86: 49–101

    Article  MathSciNet  MATH  Google Scholar 

  17. DeVore R. (1998). Nonlinear approximation.. Acta Numer. 7: 51–150

    Article  MathSciNet  Google Scholar 

  18. Xu Y. and Zou Q. (2003). Adaptive wavelet methods for elliptic operator equations with nonlinear terms. Adv. Comput. Math. 19(1): 99–146

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056, Aachen, Germany

    W. Dahmen

  2. ITERGO Informationstechnologie GmbH, Viktoriaplatz 2, 40477, Düsseldorf, Germany

    A. Barinka

  3. Scientific Computing, CAU Kiel, Christian-Albrechts-Platz 4, 24118, Kiel, Germany

    R. Schneider

Authors
  1. A. Barinka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Dahmen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. Dahmen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barinka, A., Dahmen, W. & Schneider, R. Fast computation of adaptive wavelet expansions. Numer. Math. 105, 549–589 (2007). https://doi.org/10.1007/s00211-006-0050-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-006-0050-1

Keywords

Search

Navigation