Numerische Mathematik

, Volume 137, Issue 1, pp 1–34 | Cite as

Approximation of the high-frequency Helmholtz kernel by nested directional interpolation: error analysis

  • Steffen Börm
  • Jens M. Melenk


We present and analyze an approximation scheme for a class of highly oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as examples. The scheme is based on polynomial interpolation combined with suitable pre- and post-multiplication by plane waves. It is shown to converge exponentially in the polynomial degree and supports multilevel approximation techniques. Our convergence analysis may be employed to establish exponential convergence of certain classes of fast methods for discretizations of the Helmholtz integral operator that feature polylogarithmic-linear complexity.

Mathematics Subject Classification

35J05 65D05 65N38 41A10 65N12 


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards, U.S. Department of Commerce (1972)Google Scholar
  2. 2.
    Bebendorf, M., Kuske, C., Venn, R.: Wideband nested cross approximation for Helmholtz problems. Numer. Math. 130(1), 1–34 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Börm, S.: Directional \(\cal{H}^2\)-matrix compression for high-frequency problems.
  4. 4.
    Börm, S.: Approximation of integral operators by \({\cal{H}}^2\)-matrices with adaptive bases. Computing 74(3), 249–271 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Börm, S.: Efficient Numerical Methods for Non-local Operators: \({\cal{H}}^2\)-Matrix Compression, Algorithms and Analysis, volume 14 of EMS Tracts in Mathematics. EMS (2010)Google Scholar
  6. 6.
    Börm, S., Börst, C., Melenk, J.M.: An analysis of a butterfly algorithm. (2017)
  7. 7.
    Börm, S., Löhndorf, M., Melenk, J.M.: Approximation of integral operators by variable–order interpolation. Numer. Math. 99(4), 605–643 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brandt, A.: Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Comput. Phys. Commun. 65, 24–38 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Darve, E.: The fast multipole method: numerical implementation. J. Comput. Phys. 160(1), 195–240 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demanet, L., Ferrara, M., Maxwell, N., Poulson, J., Ying, L.: A butterfly algorithm for synthetic aperture radar imaging. SIAM J. Imaging Sci. 5(1), 203–243 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Engquist, B., Ying, L.: Fast directional multilevel algorithms for oscillatory kernels. SIAM J. Sci. Comput. 29(4), 1710–1737 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Greengard, L., Huang, J., Rokhlin, V., Wandzura, S.: Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Comput. Sci. Eng. 5(3), 32–38 (1998)CrossRefGoogle Scholar
  16. 16.
    Kunis, Stefan, Melzer, Ines: A stable and accurate butterfly sparse Fourier transform. SIAM J. Numer. Anal. 50(3), 1777–1800 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, Y., Yang, H., Ying, L.: Multidimensional butterfly factorization. Preprint available at (2015)
  18. 18.
    Messner, M., Schanz, M., Darve, E.: Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation. J. Comput. Phys. 231(4), 1175–1196 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Michielssen, E., Boag, A.: A multilevel matrix decomposition algorithm for analyzing scattering from large structures. IEEE Trans. Antennas Propag. 44, 1086–1093 (1996)CrossRefGoogle Scholar
  20. 20.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, Cambridge (1974)zbMATHGoogle Scholar
  21. 21.
    Rivlin, T.J.: The Chebyshev Polynomials. Wiley-Interscience, New York (1990)zbMATHGoogle Scholar
  22. 22.
    Rokhlin, V.: Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harm. Anal. 1, 82–93 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sauter, S.A.: Variable order panel clustering. Computing 64, 223–261 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institut für InformatikChristian-Albrechts-Universität zu Kiel, D-24118KielGermany
  2. 2.Institut für Analysis und Scientific ComputingTechnische Universität WienViennaAustria

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