Numerical Algorithms

, Volume 64, Issue 3, pp 567–592 | Cite as

Low-rank approximation of integral operators by using the Green formula and quadrature

Original Paper

Abstract

Approximating integral operators by a standard Galerkin discretisation typically leads to dense matrices. To avoid the quadratic complexity it takes to compute and store a dense matrix, several approaches have been introduced including \(\mathcal {H}\)-matrices. The kernel function is approximated by a separable function, this leads to a low rank matrix. Interpolation is a robust and popular scheme, but requires us to interpolate in each spatial dimension, which leads to a complexity of \(m^d\) for \(m\)-th order. Instead of interpolation we propose using quadrature on the kernel function represented with Green’s formula. Due to the fact that we are integrating only over the boundary, we save one spatial dimension compared to the interpolation method and get a complexity of \(m^{d-1}\).

Keywords

Integral equations Data-sparse approximation Quadrature Green’s formula Hierarchical matrices 

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References

  1. 1.
    Anderson, C.R.: An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Comput. 13, 923–947 (1992)CrossRefMATHGoogle Scholar
  2. 2.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bebendorf, M., Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing 70(1), 1–24 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Börm, S.: Efficient Numerical Methods for Non-local Operators: \({\mathcal H}^2\)-Matrix Compression, Algorithms and Analysis. EMS Tracts in Mathematics, vol. 14. EMS (2010)Google Scholar
  5. 5.
    Börm, S., Grasedyck, L.: Low-rank approximation of integral operators by interpolation. Computing 72, 325–332 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101, 221–249 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Börm, S., Grasedyck, L., Hackbusch,W.: Introduction to hierarchical matrices with applications. Eng. Anal. Bound. Elem. 27, 405–422 (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Cheng, H., Gimbutas, Z., Martinsson, P.-G., Rokhlin, V.: On the compression of low rank matrices. SIAM J. Sci. Comput. 26(4), 1389–1404 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dahmen,W., Schneider, R.:Wavelets on manifolds I: construction and domain decomposition. SIAM J. Math. Anal. 31, 184–230 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261, 1–22 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goreinov, S.A., Zamarashkin, N.L., Tyrtyshnikov, E.E.: Pseudo-skeleton approximations by matrices of maximal volume. Math. Notes 62, 515–519 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Green, G.: An essay on the application of mathematical analysis to the theories of electricity and magnetism. Nottingham (1828)Google Scholar
  13. 13.
    Greengard, L., Gueyffier, D., Martinsson, P.-G., Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numer. 18, 243–275 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the laplace in three dimensions. In: Acta Numerica 1997, pp. 229–269. Cambridge University Press, Cambridge, MA (1997)Google Scholar
  16. 16.
    Hackbusch, W.: Elliptic Differential Equations. Theory and Numerical Treatment. Springer-Verlag, Berlin (1992)MATHGoogle Scholar
  17. 17.
    Hackbusch, W.: A sparse matrix arithmetic based on \(\mathcal {H}\)-matrices. Part I: Introduction to \(\mathcal {H}\)-matrices. Computing 62, 89–108 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hackbusch, W.: Hierarchische Matrizen—Algorithmen und Analysis. Springer, New York (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Hackbusch, W., Khoromskij, B.N.: A sparse matrix arithmetic based on \(\mathcal {H}\)-matrices. Part II: Application to multi-dimensional problems. Computing 64, 21–47 (2000)MathSciNetMATHGoogle Scholar
  20. 20.
    Hackbusch, W., Khoromskij, B.N., Sauter, S.A.: On \(\mathcal {H}^2\)-matrices. In: Bungartz, H., Hoppe, R., Zenger, C. (eds.) Lectures on Applied Mathematics, pp. 9–29. Springer-Verlag, Berlin (2000)Google Scholar
  21. 21.
    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54, 463–491 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Makino, J.: Yet another fast multipole method without multipoles—pseudoparticle multipole method. J. Comput. Phys. 151, 910–920 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Martinsson, P.-G., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205, 1–23 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic-skeleton method. Computing 64, 367–380 (2000)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Christian-Albrechts-Universität zu KielKielGermany

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