Numerische Mathematik

, Volume 130, Issue 1, pp 1–34 | Cite as

Wideband nested cross approximation for Helmholtz problems

  • M. BebendorfEmail author
  • C. Kuske
  • R. Venn


In this article, the construction of nested bases approximations to discretizations of integral operators with oscillatory kernels is presented. The new method has log-linear complexity and generalizes the adaptive cross approximation method to high-frequency problems. It allows for a continuous and numerically stable transition from low to high frequencies.

Mathematics Subject Classification

41A63 65D05 65D15 35J05 


  1. 1.
    Alpert, B.K., Beylkin, G., Coifman, R., Rokhlin, V.: Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput. 14, 159–184 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Amini, S., Profit, A.: Multi-level fast multipole solution of the scattering problem. Eng. Anal. Bound. Elements 27(5), 547–654 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, C.R.: An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Comput. 13(4), 923–947 (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Banjai, L., Hackbusch, W.: Hierarchical matrix techniques for low- and high-frequency Helmholtz problems. IMA J. Numer. Anal. 28(1), 46–79 (2008). doi: 10.1093/imanum/drm001 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Barnes, J., Hut, P.: A hierarchical \(\cal O({N}\log {N})\) force calculation algorithm. Nature 324, 446–449 (1986)CrossRefGoogle Scholar
  6. 6.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bebendorf, M.: Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. In: Lecture Notes in Computational Science and Engineering (LNCSE), vol. 63. Springer, Berlin (2008). ISBN 978-3-540-77146-3Google Scholar
  8. 8.
    Bebendorf, M., Venn, R.: Constructing nested bases approximations from the entries of non-local operators. Numer. Math. 121(4), 609–635 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Börm, S.: \({\cal H}^{2}\)-matrix arithmetics in linear complexity. Computing 77(1), 1–28 (2006)Google Scholar
  10. 10.
    Börm, S., Löhndorf, M., Melenk, J.M.: Approximation of integral operators by variable-order interpolation. Numer. Math. 99(4), 605–643 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Brakhage, H., Werner, P.: Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch. Math. 16, 325–329 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Brandt, A.: Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Comput. Phys. Commun. 65, 24–38 (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Buffa, A., Hiptmair, R.: A coercive combined field integral equation for electromagnetic scattering. SIAM J. Numer. Anal. 42(2), 621–640 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Burton, A.J., Miller, G.F.: The application of integral equation methods to the numerical solution of boundary value problems. Proc. R. Soc. Lond. A232, 201–210 (1971)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Candès, E., Demanet, L., Ying, L.: A fast butterfly algorithm for the computation of Fourier integral operators. Multiscale Model. Simul. 7(4), 1727–1750 (2009). doi: 10.1137/080734339 CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Spence, E.A.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numerica 21, 89–305 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Cheng, H., Crutchfield, W.Y., Gimbutas, Z., Greengard, L.F., Ethridge, J.F., Huang, J., Rokhlin, V., Yarvin, N., Zhao, J.: A wideband fast multipole method for the Helmholtz equation in three dimensions. J. Comput. Phys. 216(1), 300–325 (2006). doi: 10.1016/ CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Darve, E.: The fast multipole method: numerical implementation. J. Comput. Phys. 160(1), 195–240 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Engquist, B., Ying, L.: Fast directional multilevel algorithms for oscillatory kernels. SIAM J. Sci. Comput. 29(4), 1710–1737 (2007, electronic)Google Scholar
  20. 20.
    Engquist, B., Ying, L.: A fast directional algorithm for high frequency acoustic scattering in two dimensions. Commun. Math. Sci. 7(2), 327–345 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Francia, G.T.D.: Degrees of freedom of an image. J. Opt. Soc. Am. 59(7), 799–803 (1969)CrossRefGoogle Scholar
  22. 22.
    Giebermann, K.: Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im \(\mathbb{R}^3\). Ph.D. thesis, Universität Karlsruhe (1997)Google Scholar
  23. 23.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \({\cal H}\)-matrices. Computing 70, 295–334 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Greengard, L.: The Rapid Evaluation of Potential Fields in Particle Systems. MIT Press, Cambridge (1988)zbMATHGoogle Scholar
  25. 25.
    Greengard, L.F., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Greengard, L.F., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. In: Acta Numerica, vol. 6, pp. 229–269. Cambridge University Press, Cambridge (1997)Google Scholar
  27. 27.
    Hackbusch, W.: A sparse matrix arithmetic based on \(\cal H\)-matrices. Part I: introduction to \(\cal H\)-matrices. Computing 62(2), 89–108 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Hackbusch, W., Börm, S.: Data-sparse approximation by adaptive \({\cal H}^{2}\)-matrices. Computing 69(1), 1–35 (2002)Google Scholar
  29. 29.
    Hackbusch, W., Khoromskij, B.N.: A sparse \(\cal H\)-matrix arithmetic: general complexity estimates. J. Comput. Appl. Math. 125(1–2), 479–501 (2000). Numerical analysis 2000, vol. VI, Ordinary differential equations and integral equationsGoogle Scholar
  30. 30.
    Hackbusch, W., Khoromskij, B.N.: A sparse \(\cal H\)-matrix arithmetic. Part II: application to multi-dimensional problems. Computing 64(1), 21–47 (2000)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Hackbusch, W., Khoromskij, B.N., Sauter, S.A.: On \({\cal H}^{2}\)-matrices. In: Bungartz, H.J., Hoppe, R.H.W., Zenger, C. (eds.) Lectures on Applied Mathematics, pp. 9–29. Springer, Berlin (2000)Google Scholar
  32. 32.
    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4), 463–491 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Messner, M., Schanz, M., Darve, E.: Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation. J. Comput. Phys. 231, 1175–1196 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Michielssen, E., Boag, A.: A multilevel matrix decomposition for analyzing scattering from large structures. IEEE Trans. Antennas Propag. 44, 1086–1093 (1996)CrossRefGoogle Scholar
  35. 35.
    Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60(2), 187–207 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Rokhlin, V.: Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys. 86(2), 414–439 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Rokhlin, V.: Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harmon. Anal. 1(1), 82–93 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Sauer, T., Xu, Y.: On multivariate Lagrange interpolation. Math. Comput. 64(211), 1147–1170 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Tyrtyshnikov, E.E.: Mosaic-skeleton approximations. Calcolo 33(1–2), 47–57 (1998). Toeplitz matrices: structures, algorithms and applications (Cortona, 1996)Google Scholar
  40. 40.
    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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute for Numerical SimulationRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

Personalised recommendations