\({\mathcal{H}}^{2}\)-Matrix Compression

  • Steffen BörmEmail author
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Representing a matrix in a hierarchical data structure instead of the standard two-dimensional array can offer significant advantages: submatrices can be compressed efficiently, different resolutions of a matrix can be handled easily, and even matrix operations like multiplication, factorization or inversion can be performed in the compressed representation, thus saving computation time and storage. \({\mathcal{H}}^{2}\)-matrices use a subdivision of the matrix into a hierarchy of submatrices in combination with a hierarchical basis, similar to a wavelet basis, to handle \(n \times n\) matrices in \(\mathcal{O}(nk)\) units of storage, where k is a parameter controlling the compression error. This chapters gives a short introduction into the basic concepts of the \({\mathcal{H}}^{2}\)-matrix method, particularly concerning the compression of arbitrary matrices.



Most of the research results described in this chapter are the result of several years of work as a researcher at the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, and I particularly owe a debt of gratitude to Wolfgang Hackbusch and Lars Grasedyck for many insightful discussions.


  1. 1.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)Google Scholar
  2. 2.
    Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)Google Scholar
  3. 3.
    Bebendorf, M., Hackbusch, W.: Existence of \(\mathcal{H}\)-matrix approximants to the inverse FE-matrix of elliptic operators with L -coefficients. Numer. Math. 95, 1–28 (2003)Google Scholar
  4. 4.
    Bebendorf, M., Grzhibovskis, R.: Accelerating Galerkin BEM for linear elasticity using adaptive cross approximation. Math. Meth. Appl. Sci. 29, 1721–1747 (2006)Google Scholar
  5. 5.
    Bendoraityte, J., Börm, S.: Distributed \({\mathcal{H}}^{2}\)-matrices for non-local operators. Comput. Vis. Sci. 11, 237–249 (2008)Google Scholar
  6. 6.
    Beylkin, G., Coifman, R., Rokhlin, V.: The fast wavelet transform and numerical algorithms. Commun. Pure Appl. Math. 44, 141–183 (1991)Google Scholar
  7. 7.
    Börm, S.: \({\mathcal{H}}^{2}\)-matrix arithmetics in linear complexity. Computing 77(1), 1–28 (2006)Google Scholar
  8. 8.
    Börm, S.: Adaptive variable-rank approximation of dense matrices. SIAM J. Sci. Comput. 30(1), 148–168 (2007)Google Scholar
  9. 9.
    Börm, S.: Data-sparse approximation of non-local operators by \({\mathcal{H}}^{2}\)-matrices. Linear Algebra Appl. 422, 380–403 (2007)Google Scholar
  10. 10.
    Börm, S.: Construction of data-sparse \({\mathcal{H}}^{2}\)-matrices by hierarchical compression. SIAM J. Sci. Comput. 31(3), 1820–1839 (2009)Google Scholar
  11. 11.
    Börm, S.: Approximation of solution operators of elliptic partial differential equations by \(\mathcal{H}\)- and \({\mathcal{H}}^{2}\)-matrices. Numer. Math. 115(2), 165–193 (2010)Google Scholar
  12. 12.
    Börm, S.: Efficient Numerical Methods for Non-local Operators: \({\mathcal{H}}^{2}\)-Matrix Compression, Algorithms and Analysis. EMS Tracts in Mathematics, vol. 14. EMS, Zürich (2010)Google Scholar
  13. 13.
    Börm, S., Hackbusch, W.: Data-sparse approximation by adaptive \({\mathcal{H}}^{2}\)-matrices. Computing 69, 1–35 (2002)Google Scholar
  14. 14.
    Börm, S., Grasedyck, L.: Low-rank approximation of integral operators by interpolation. Computing 72, 325–332 (2004)Google Scholar
  15. 15.
    Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101, 221–249 (2005)Google Scholar
  16. 16.
    Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Eng. Anal. Bound. Elem. 27, 405–422 (2003)Google Scholar
  17. 17.
    Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)Google Scholar
  18. 18.
    Ewald, P.P.: Die Berechnung optischer und elektrostatischer Gitterpotentiale. Annalen der Physik 369(3), 253–287 (1920)Google Scholar
  19. 19.
    Grasedyck, L.: Theorie und Anwendungen Hierarchischer Matrizen. Doctoral thesis, Universität Kiel (2001)Google Scholar
  20. 20.
    Grasedyck, L.: Adaptive recompression of \(\mathcal{H}\)-matrices for BEM. Computing 74(3), 205–223 (2004)Google Scholar
  21. 21.
    Grasedyck, L.: Existence of a low-rank or \(\mathcal{H}\)-matrix approximant to the solution of a Sylvester equation. Numer. Linear Algebra Appl. 11, 371–389 (2004)Google Scholar
  22. 22.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \(\mathcal{H}\)-matrices. Computing 70, 295–334 (2003)Google Scholar
  23. 23.
    Grasedyck, L., LeBorne, S.: \(\mathcal{H}\)-matrix preconditioners in convection-dominated problems. SIAM J. Math. Anal. 27(4), 1172–1183 (2006)Google Scholar
  24. 24.
    Grasedyck, L., Hackbusch, W., Khoromskij, B.N.: Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Computing 70, 121–165 (2003)Google Scholar
  25. 25.
    Grasedyck, L., Hackbusch, W., LeBorne, S.: Adaptive geometrically balanced clustering of \(\mathcal{H}\)-matrices. Computing 73, 1–23 (2004)Google Scholar
  26. 26.
    Grasedyck, L., Kriemann, R., LeBorne, S.: Domain decomposition based \(\mathcal{H}\)-LU preconditioning. Numer. Math. 112(4), 565–600 (2009)Google Scholar
  27. 27.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)Google Scholar
  28. 28.
    Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. In: Acta Numerica 1997, pp. 229–269. Cambridge University Press, Cambridge/New York (1997)Google Scholar
  29. 29.
    Hackbusch, W.: A sparse matrix arithmetic based on \(\mathcal{H}\)-matrices. Part I: introduction to \(\mathcal{H}\)-matrices. Computing 62, 89–108 (1999)Google Scholar
  30. 30.
    Hackbusch, W.: Hierarchische Matrizen: Algorithmen und Analysis. Springer, Berlin/ Heidelberg (2009)Google Scholar
  31. 31.
    Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54, 463–491 (1989)Google Scholar
  32. 32.
    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)Google Scholar
  33. 33.
    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, Berlin (2000)Google Scholar
  34. 34.
    Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60, 187–207 (1985)Google Scholar
  35. 35.
    Sauter, S.A.: Variable order panel clustering. Computing 64, 223–261 (2000)Google Scholar
  36. 36.
    Tausch, J., White, J.: Multiscale bases for the sparse representation of boundary integral operators on complex geometries. SIAM J. Sci. Comput. 24(5), 1610–1629 (2003)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Computer ScienceChristian-Albrechts-Universität zu KielKielGermany

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