Skip to main content
SpringerLink
Log in
Book cover

Problems and Methods in Mathematical Physics pp 148–164Cite as

Wavelet Approximation Methods for Integral and Pseudodifferential Equations

  • Chapter

  • 71 Accesses

Part of the book series: TEUBNER-TEXTE zur Mathematik ((TTZM))

Abstract

While initially typical applications of wavelets were concerned with signal and image analysis there have been recent attempts of applying wavelets to the solution of integral and differential equations (see e.g. [2], [34], [19], [21], [6], [33], [15]–[18], [28], [29]).

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

eBook
USD   24.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   34.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, D.N., Wendland, W.L.: The convergence of spline collocation for strongly elliptic equations on curves. Numer. Math. 47 (1985), 317–431.

    Article  MathSciNet  MATH  Google Scholar 

  2. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Comm. Pure and Appl. Math., Vol. XLIV (1991), 141–183.

    Article  MathSciNet  Google Scholar 

  3. de Boor, C., DeVore, R., Ron, A.: On the construction of multivariate (pre)wavelets. Technical Summary Report No. 92-09, Center for the Mathematical Sciences, University of Wisconsin-Madison 1992.

    Google Scholar 

  4. Brandt, A., Lubrecht, A.A.: Multilevel matrix multiplication and fast solution of integral equations. J. Comp. Phys. 90 (1991), 348–370.

    Article  MathSciNet  Google Scholar 

  5. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision. Memoirs of the American Math. Soc, Vol. 93, No. 453, 1991.

    Google Scholar 

  6. Chui, C.K. (ed.): Wavelets: A Tutorial in Theory and Applications. Academic Press, Boston 1992.

    MATH  Google Scholar 

  7. Chui, C.K., Stöckler, J., Ward, J.D.: Compactly supported box spline wavelets. Technical report, Preprint 1990.

    Google Scholar 

  8. Cohen, A., Daubechies, I.: Non-separable bidimensional wavelet bases. Preprint AT & T Bell Laboratories 1992.

    Google Scholar 

  9. Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure and Appl. Math. 45 (1992), 485–560.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cohen, A., Schlenker, J.M.: Compactly supported bidimensional wavelet bases with hexagonal symmetry. Preprint AT & T Bell Laboratories 1992, to appear in Constructive Approximation.

    Google Scholar 

  11. Costabel, M.: Boundary integral operators on Lipschitz-domains: Elementary results. SIAM J. Math. Anal. 19(3) (1988), 613–626.

    Article  MathSciNet  MATH  Google Scholar 

  12. Costabel, M., McLean, W.: Spline collocation for strongly elliptic equations on the torus. Numer. Math. 62 (1992), 511–538.

    Article  MathSciNet  MATH  Google Scholar 

  13. Costabel, M., Wendland, W.: Strong ellipticity of boundary integral operators. J. Reine Angew. Math. 372 (1986), 39–63.

    MathSciNet  Google Scholar 

  14. Dahmen, W.: Locally finite decompositions of nested spaces and applications to operator equations. In: Algorithms for Approximation, M.G. Cox and J.C. Mason (eds), to appear.

    Google Scholar 

  15. Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations I: Stability and convergence. Preprint No. 7, Institut für Angewandte Analysis und Stochastik, Berlin 1992; Math. Zeitschrift, to appear.

    Google Scholar 

  16. Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution. Advances in Computational Mathematics, 2nd Issue, 1 (1993), 259–335.

    Article  MathSciNet  MATH  Google Scholar 

  17. Dahmen, W., Prössdorf, S., Schneider, R.: Multiscale methods for pseudodifferential equations. In: Recent Advances in Wavelet Analysis (eds. L.L. Schumaker and G. Webb), Academic Press 1993, 191-235.

    Google Scholar 

  18. Dahmen, W., Kleemann, B., Prössdorf, S., Schneider, R.: A multiscale method for the double layer potential equation on a polyhedron. Preprint No. 76, Institut für Angewandte Analysis und Stochastik, Berlin 1993; Advances in Computational Mathematics (eds. H.P. Dikshit and C.A. Micchelli) 1994, to appear.

    Google Scholar 

  19. Glowinski, R.R., Lawton, W.M., Ravachol, M., Tenenbaum, E.: Wavelet solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Preprint, Aware Inc., Cambridge, Mass. 1989.

    Google Scholar 

  20. Harten, A., Yad-Shalom, I.: Fast multiresolution algorithms for matrixvector multiplication. ICASE Report No. 92-55, October 1992.

    Google Scholar 

  21. Jaffard, S.: Wavelet methods for fast resolution of elliptic problems. SIAM J. Numer. Anal. 29 (1992), 965–986.

    Article  MathSciNet  MATH  Google Scholar 

  22. Jia, R.Q., Micchelli, C.A.: Using the refinement equation for the construction of pre-wavelets. In: Curves and Surfaces, P. Laurent, A. Le Méhauté, and L.L. Schumaker (eds.), Academic Press, New York 1991, 209–246.

    Google Scholar 

  23. Kohn, J., Nirenberg, L.: On the algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18 (1965), 269–305.

    Article  MathSciNet  MATH  Google Scholar 

  24. Mallat, S.: Multiresolution approximation and wavelet orthogonal bases of L 2(ℝn). Trans. Amer. Math. Soc. 315 (1989), 67–87.

    MathSciNet  Google Scholar 

  25. McLean, W.: Local and global description of periodic pseudodifferential operators. Math. Nachr. 150 (1991), 151–161.

    Article  MathSciNet  MATH  Google Scholar 

  26. McLean, W.: Periodic pseudodifferential operators and periodic function spaces. Technical Report, University of New South Wales, Australia 1989.

    Google Scholar 

  27. Meyer, Y.: Wavelets and Operators. Proc. Special Year in Modern Analysis, Urbana 1986/87.

    Google Scholar 

  28. Meyer, Y.: Ondelettes et Opérateurs 1: Ondelettes. Hermann, Paris 1990.

    Google Scholar 

  29. Meyer, Y.: Ondelettes et Opérateurs 2: Opérateur de Caldéron-Zygmund. Hermann, Paris 1990.

    Google Scholar 

  30. Prössdorf, S.: Ein Lokalisierungsprinzip in der Theorie der Spline-Approximationen und einige Anwendungen. Math. Nachr. 119 (1984), 239–255.

    Article  MathSciNet  MATH  Google Scholar 

  31. Riemenschneider, S., Shen, Z.: Wavelets and pre-wavelets in low dimensions. Technical Report 1991.

    Google Scholar 

  32. Schatz, A., Thomée, V., Wendland, W.: Mathematical Theory of Finite and Boundary Element Methods. DMV Seminar, Birkhäuser Verlag, Basel, Boston, Berlin 1990.

    Book  MATH  Google Scholar 

  33. Schumaker, L.L., Webb, G. (eds.): Recent Advances in Wavelet Analysis. Academic Press, Boston 1993.

    Google Scholar 

  34. Wavelet analysis and the numerical solutions of partial differential equations. Progress Report: June 1990, Aware Inc., Cambride, Mass.

    Google Scholar 

  35. Wendland, W.L.: On some mathematical aspects of boundary element methods for elliptic problems. In: Mathematics of Finite Elements and Applications V, J. Whiteman (ed.), Academic Press, London 1985, 193–227.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D - 10117, Berlin, Germany

    Siegfried Prössdorf

Authors
  1. Siegfried Prössdorf
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Technical University, Chemnitz-Zwickau, Deutschland

    Lothar Jentsch  & Fredi Tröltzsch  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Fachmedien Wiesbaden

About this chapter

Cite this chapter

Prössdorf, S. (1994). Wavelet Approximation Methods for Integral and Pseudodifferential Equations. In: Jentsch, L., Tröltzsch, F. (eds) Problems and Methods in Mathematical Physics. TEUBNER-TEXTE zur Mathematik. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-85161-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-322-85161-1_14

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-322-85162-8

  • Online ISBN: 978-3-322-85161-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation