Numerische Mathematik

, Volume 133, Issue 3, pp 409–442 | Cite as

Approximation of integral operators by Green quadrature and nested cross approximation

  • Steffen BörmEmail author
  • Sven Christophersen


We present a fast algorithm that constructs a data-sparse approximation of matrices arising in the context of integral equation methods for elliptic partial differential equations. The new algorithm uses Green’s representation formula in combination with quadrature to obtain a first approximation of the kernel function, and then applies nested cross approximation to obtain a more efficient representation. The resulting \({\mathcal H}^2\)-matrix representation requires \({\mathcal O}(n k)\) units of storage for an \(n\times n\) matrix, where k depends on the prescribed accuracy.

Mathematics Subject Classification

65N38 65N80 65D30 45B05 


  1. 1.
    Anderson, C.R.: An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Comput. 13, 923–947 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bebendorf, M., Grzhibovskis, R.: Accelerating Galerkin BEM for linear elasticity using adaptive cross approximation. Math. Meth. Appl. Sci. 29, 1721–1747 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bebendorf, M., Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing 70(1), 1–24 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bebendorf, M., Venn, R.: Constructing nested bases approximations from the entries of non-local operators. Numer. Math. 121(4), 609–635 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Börm, S.: Efficient numerical methods for non-local operators: \({\cal {H}}^2\)-matrix compression, algorithms and analysis. EMS Tracts Math. 14, (2010). doi: 10.4171/091
  7. 7.
    Börm, S., Gördes, J.: Low-rank approximation of integral operators by using the Green formula and quadrature. Numer. Algorithms 64(3), 567–592 (2013). doi: 10.1007/s11075-012-9679-2 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101, 221–249 (2005). doi: 10.1007/s00211-005-0618-1 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical Matrices. Lecture Note 21 of the Max Planck Institute for Mathematics in the Sciences (2003)Google Scholar
  10. 10.
    Börm, S., Hackbusch, W.: Data-sparse approximation by adaptive \({\cal {H}}^2\)-matrices. Computing 69, 1–35 (2002). doi: 10.1007/s00607-002-1450-4
  11. 11.
    Börm, S., Hackbusch, W.: \({\cal {H}}^2\)-matrix approximation of integral operators by interpolation. Appl. Numer. Math. 43, 129–143 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brandt, A.: Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Comput. Phys. Commun. 65(1–3), 24–38 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brandt, A., Lubrecht, A.A.: Multilevel matrix multiplication and fast solution of integral equations. J. Comput. Phys. 90, 348–370 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations I: stability and convergence. Math. Z. 215, 583–620 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dahmen, W., Schneider, R.: Wavelets on manifolds I: construction and domain decomposition. SIAM J. Math. Anal. 31, 184–230 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Giebermann, K.: Multilevel approximation of boundary integral operators. Computing 67, 183–207 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261, 1–22 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \({\cal {H}}\)-matrices. Computing 70, 295–334 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hackbusch, W.: Elliptic Differential Equations. Theory and Numerical Treatment. Springer-Verlag, Berlin (1992)zbMATHGoogle Scholar
  21. 21.
    Hackbusch, W.: A sparse matrix arithmetic based on \({\cal {H}}\)-matrices. Part I: introduction to \({\cal {H}}\)-matrices. Computing 62, 89–108 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hackbusch, W.: Hierarchische Matrizen: Algorithmen und Analysis. Springer, New York (2009)Google Scholar
  23. 23.
    Hackbusch, W., Khoromskij, B.N.: A sparse matrix arithmetic based on \({\cal {H}}\)-matrices. Part II: application to multi-dimensional problems. Computing 64, 21–47 (2000)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54, 463–491 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Harbrecht, H., Schneider, R.: Wavelet Galerkin schemes for boundary integral equations: implementation and quadrature. SIAM J. Sci. Comput. 27, 1347–1370 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. No. 164 in Appl. Math. Sci. Springer, New York (2008)CrossRefGoogle Scholar
  27. 27.
    Maaskant, R., Mittra, R., Tijhuis, A.: Fast analysis of large antenna arrays using the characteristic basis function method and the adaptive cross approximation algorithm. IEEE Trans. Antennas Propag. 56(11), 3440–3451 (2008)CrossRefGoogle Scholar
  28. 28.
    Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60, 187–207 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sauter, S.A.: Cubature techniques for 3-d Galerkin BEM. In: Hackbusch, W., Wittum, G. (eds.) Boundary Elements: Implementation and Analysis of Advanced Algorithms, pp. 29–44. Vieweg-Verlag, Berlin (1996)CrossRefGoogle Scholar
  30. 30.
    Sauter, S.A., Schwab, C.: Randelementmethoden. Teubner, Berlin (2004)CrossRefGoogle Scholar
  31. 31.
    Schöberl, J.: NETGEN: an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1(1), 41–52 (1997)CrossRefzbMATHGoogle Scholar
  32. 32.
    Tamayo, J.M., Heldring, A., Rius, J.M.: Multilevel adaptive cross approximation. IEEE Trans. Antennas Propag. 59(12), 4600–4608 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tyrtyshnikov, E.E.: Mosaic-skeleton approximation. Calcolo 33, 47–57 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic-skeleton method. Computing 64, 367–380 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196(2), 591–626 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of KielKielGermany

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