Computing

, Volume 77, Issue 1, pp 1–28 | Cite as

Open image in new window-Matrix Arithmetics in Linear Complexity

Article

Abstract

For hierarchical matrices, approximations of the matrix-matrix sum and product can be computed in almost linear complexity, and using these matrix operations it is possible to construct the matrix inverse, efficient preconditioners based on approximate factorizations or solutions of certain matrix equations.

Open image in new window-matrices are a variant of hierarchical matrices which allow us to perform certain operations, like the matrix-vector product, in ``true'' linear complexity, but until now it was not clear whether matrix arithmetic operations could also reach this, in some sense optimal, complexity.

We present algorithms that compute the best-approximation of the sum and product of two Open image in new window-matrices in a prescribed Open image in new window-matrix format, and we prove that these computations can be accomplished in linear complexity. Numerical experiments demonstrate that the new algorithms are more efficient than the well-known methods for hierarchical matrices.

MSC Subject Classification

65F30 

Keywords

Hierarchical matrices formatted matrix operations 

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References

  1. Bebendorf, M., Hackbusch, W. 2003Existence of Open image in new window-matrix approximants to the inverse FE-matrix of elliptic operators with L-coefficientsNumer. Math.95128CrossRefMathSciNetGoogle Scholar
  2. Börm, S. 2004Open image in new window-matrices – multilevel methods for the approximation of integral operatorsComput. Visual. Sci.7173181MATHGoogle Scholar
  3. Börm, S. 2005Approximation of integral operators by Open image in new window-matrices with adaptive basesComputing74249271MATHMathSciNetGoogle Scholar
  4. Börm, S.: Data-sparse approximation of non-local operators by Open image in new window-matrices. Preprint 44/2005, Max-Planck Institute for Mathematics in the Sciences, 2005.Google Scholar
  5. Börm, S., Grasedyck, L. 2005Hybrid cross approximation of integral operatorsNumer. Math.101221249Google Scholar
  6. Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices. Lecture Note 21, Max-Planck Institute for Mathematics in the Sciences, 2003.Google Scholar
  7. Börm, S., Hackbusch, W. 2002Data-sparse approximation by adaptive Open image in new window-matricesComputing69135MathSciNetGoogle Scholar
  8. Börm, S., Hackbusch, W. 2002Open image in new window-matrix approximation of integral operators by interpolationAppl. Numer. Math.43129143MathSciNetGoogle Scholar
  9. Börm, S., Löhndorf, M., Melenk, J. M. 2005Approximation of integral operators by variable-order interpolationNumer. Math.99605643MathSciNetGoogle Scholar
  10. Börm, S., Sauter, S. A. 2005BEM with linear complexity for the classical boundary integral operatorsMath. Comput.7411391177Google Scholar
  11. Dahmen, W., Schneider, R. 1999Wavelets on manifolds I: construction and domain decompositionSIAM J. Math. Anal.31184230CrossRefMathSciNetGoogle Scholar
  12. Grasedyck, L.: Theorie und Anwendungen hierarchischer Matrizen. PhD thesis, Universität Kiel, 2001.Google Scholar
  13. Grasedyck, L. 2004Adaptive recompression of Open image in new window-matrices for BEMComputing74205223MathSciNetGoogle Scholar
  14. Grasedyck, L., Hackbusch, W. 2003Construction and arithmetics of Open image in new window-matricesComputing70295334MathSciNetGoogle Scholar
  15. Greengard, L., Rokhlin, V. 1987A fast algorithm for particle simulationsJ. Comput. Phys.73325348CrossRefMathSciNetGoogle Scholar
  16. Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace in three dimensions. Acta Numerica 1997, Cambridge University Press, pp. 229–269.Google Scholar
  17. Hackbusch, W. 1999A sparse matrix arithmetic based on Open image in new window-matrices. Part I : Introduction to Open image in new window-matricesComputing6289108CrossRefMATHMathSciNetGoogle Scholar
  18. Hackbusch, W.: Hierarchische Matrizen – Algorithmen und Analysis. Available online at http://www.mis.mpg.de/scicomp/fulltext/hmvorlesung.ps, 2004.Google Scholar
  19. Hackbusch, W., Khoromskij, B. 2000A sparse matrix arithmetic based on Open image in new window-matrices. Part II: Application to multi-dimensional problemsComputing642147MathSciNetGoogle Scholar
  20. Hackbusch, W., Khoromskij, B., Sauter, S. 2000

    On Open image in new window-matrices

    Bungartz, H.Hoppe, R.Zenger, C. eds. Lectures on Applied MathematicsSpringerBerlin929
    Google Scholar
  21. Hackbusch, W., Nowak, Z. P. 1989On the fast matrix multiplication in the boundary element method by panel clusteringNumer. Math.54463491CrossRefMathSciNetGoogle Scholar
  22. Rokhlin, V. 1985Rapid solution of integral equations of classical potential theoryJ. Comput. Phys.60187207CrossRefMATHMathSciNetGoogle Scholar
  23. Sauter, S.: Variable order panel clustering (extended version). Preprint 52/1999, Max-Planck-Institut für Mathematik, Leipzig, Germany, 1999.Google Scholar
  24. Sauter, S. 2000Variable order panel clusteringComputing64223261CrossRefMATHMathSciNetGoogle Scholar
  25. Tausch, J., White, J. 2003Multiscale bases for the sparse representation of boundary integral operators on complex geometriesSIAM J. Sci. Comput.2416101629CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2005

Authors and Affiliations

  1. 1.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

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