Numerische Mathematik

, Volume 99, Issue 4, pp 605–643 | Cite as

Approximation of Integral Operators by Variable-Order Interpolation

  • Steffen Börm
  • Maike Löhndorf
  • Jens M. MelenkEmail author


We employ a data-sparse, recursive matrix representation, so-called Open image in new window -matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lower-order ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence Open image in new window (h) is retained for the classical double layer potential discretized with piecewise constant functions.


Double Layer Tensor Product Kernel Function Integral Operator Matrix Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amini, S., Profit, A.T.J.: Multi-level fast multipole solution of the scattering problem. Engineering Analysis with Boundary Elements. 27, 547–564 (2003)Google Scholar
  2. 2.
    Börm, S., Grasedyck L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Engineering Analysis with Boundary Elements. 27, 405–422 (2003)Google Scholar
  3. 3.
    Börm, S., Hackbusch, W.: Open image in new window-matrix approximation of integral operators by interpolation. Applied Numerical Mathematics. 43, 129–143 (2002)Google Scholar
  4. 4.
    Börm, S., Löhndorf, M., Melenk, J.M.: Approximation of integral operators by variable-order interpolation. Technical Report 82, Max Planck Institute for Mathematics in the Sciences, 2002. Extended preprint.Google Scholar
  5. 5.
    Brandt A., Lubrecht, A.A.: Multilevel matrix multiplication and fast solution of integral equations. J. Comput. Phys. 90, 348–370 1990Google Scholar
  6. 6.
    Dahmen, W., Faermann, B., Graham, I.G., Hackbusch, W., Sauter, S.A.: Inverse inequalities on non-quasiuniform meshes and applications to the mortar element method. Math. Comp. 73, 1107–1138 (2004)Google Scholar
  7. 7.
    Dahmen, W., Schneider, R.: Wavelets on manifolds I: Construction and domain decomposition. SIAM Journal of Mathematical Analysis. 31, 184–230 (1999)Google Scholar
  8. 8.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer-Verlag, 1993Google Scholar
  9. 9.
    Giebermann, K.: Multilevel approximation of boundary integral operators. Computing. 67, 183–207 (2001)Google Scholar
  10. 10.
    Grasedyck, L.: Theorie und Anwendungen Hierarchischer Matrizen. PhD thesis, Universität Kiel, 2001Google Scholar
  11. 11.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of Open image in new window-matrices. Computing. 70(4), 295–334 (2003)Google Scholar
  12. 12.
    Greengard L., Rokhlin V.: A fast algorithm for particle simulations. Journal of Computational Physics. 73, 325–348 (1987)Google Scholar
  13. 13.
    Hackbusch, W.: A sparse matrix arithmetic based on Open image in new window-matrices. Part I: Introduction to Open image in new window-matrices. Computing. 62, 89–108 (1999)Google Scholar
  14. 14.
    Hackbusch, W., Khoromskij, B.: A sparse matrix arithmetic based on Open image in new window-matrices. Part II: Application to multi-dimensional problems. Computing. 64, 21–47 (2000)Google Scholar
  15. 15.
    Hackbusch, W., Khoromskij, B., Sauter, S.: On Open image in new window-matrices. In H. Bungartz, R. Hoppe, and C. Zenger, editors, Lectures on Applied Mathematics. Springer-Verlag, Berlin, 2000 pp. 9–29Google Scholar
  16. 16.
    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numerische Mathematik. 54, 463–491 (1989)Google Scholar
  17. 17.
    Krezebek, N., Sauter, S.: Fast cluster techniques for BEM. Technical Report 04, Institut für Mathematik, University of Zurich, 2002Google Scholar
  18. 18.
    Lage, C., Schmidlin, G., Schwab, C.: Rapid solution of first kind boundary integral equations in ℝ3. Engineering Analysis with Boundary Elements. 27, 469–490 (2003)Google Scholar
  19. 19.
    Löhndorf, M.: Effiziente Behandlung von Integraloperatoren mit Open image in new window-Matrizen variabler Ordnung. PhD thesis, Universität Leipzig, 2003Google Scholar
  20. 20.
    Löhndorf, M., Melenk, J.M.: Approximation of integral operators by variable-order approximation. part II: Non-smooth domains. In preparation.Google Scholar
  21. 21.
    Rivlin, T.J.: The Chebyshev Polynomials. Wiley-Interscience, New York, 1984Google Scholar
  22. 22.
    Sauter, S.: Variable order panel clustering (extended version). Technical Report 52, Max-Planck-Institut für Mathematik, Leipzig, Germany, 1999Google Scholar
  23. 23.
    Sauter, S.: Variable order panel clustering. Computing. 64, 223–261 (2000)Google Scholar
  24. 24.
    Schmidlin, G.: Fast Solution Algorithm for Integral Equations in ℝ3. PhD thesis, ETH Zürich, 2003Google Scholar
  25. 25.
    Tausch, J.: A variable order wavelet method for the sparse representation of layer potentials in the non-standard form. Journal of Numerical Mathematics. 12 (3), 233–254 (2004)Google Scholar
  26. 26.
    Tyrtyshnikov, E.: Mosaic-skeleton approximation. Calcolo. 33, 47–57 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Steffen Börm
    • 1
  • Maike Löhndorf
    • 1
  • Jens M. Melenk
    • 2
    Email author
  1. 1.Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22–26LeipzigGermany
  2. 2.Department of MathematicsThe University of ReadingUnited Kingdom

Personalised recommendations