An introduction to hierarchical matrices

Wolfgang Hackbusch; Lars Grasedyck; Steffen Börm

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 2, page 229-241
  • ISSN: 0862-7959

Abstract

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We give a short introduction to a method for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods or as the inverses of partial differential operators. The result of the approximation will be the so-called hierarchical matrices (or short -matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity.

How to cite

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Hackbusch, Wolfgang, Grasedyck, Lars, and Börm, Steffen. "An introduction to hierarchical matrices." Mathematica Bohemica 127.2 (2002): 229-241. <http://eudml.org/doc/249049>.

@article{Hackbusch2002,
abstract = {We give a short introduction to a method for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods or as the inverses of partial differential operators. The result of the approximation will be the so-called hierarchical matrices (or short $\mathcal \{H\}$-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity.},
author = {Hackbusch, Wolfgang, Grasedyck, Lars, Börm, Steffen},
journal = {Mathematica Bohemica},
keywords = {hierarchical matrices; data-sparse approximations; formatted matrix operations; fast solvers; hierarchical matrices; data-sparse approximations; formatted matrix operations; fast solvers; matrix inversion; boundary integral methods; -matrices; matrix-matrix multiplication; optimal complexity},
language = {eng},
number = {2},
pages = {229-241},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An introduction to hierarchical matrices},
url = {http://eudml.org/doc/249049},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Hackbusch, Wolfgang
AU - Grasedyck, Lars
AU - Börm, Steffen
TI - An introduction to hierarchical matrices
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 2
SP - 229
EP - 241
AB - We give a short introduction to a method for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods or as the inverses of partial differential operators. The result of the approximation will be the so-called hierarchical matrices (or short $\mathcal {H}$-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity.
LA - eng
KW - hierarchical matrices; data-sparse approximations; formatted matrix operations; fast solvers; hierarchical matrices; data-sparse approximations; formatted matrix operations; fast solvers; matrix inversion; boundary integral methods; -matrices; matrix-matrix multiplication; optimal complexity
UR - http://eudml.org/doc/249049
ER -

References

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  2. H -matrix approximation for elliptic solution operators in cylindric domains, East-West J. Numer. Math. 9 (2001), 25–58. (2001) MR1839197
  3. Theorie und Anwendungen Hierarchischer Matrizen, Doctoral thesis, University Kiel, 2001. (2001) 
  4. A sparse matrix arithmetic based on H -Matrices. Part I: Introduction to H -Matrices, Computing 62 (1999), 89–108. (1999) MR1694265
  5. On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math. 54 (1989), 463–491. (1989) MR0972420
  6. A sparse H -matrix arithmetic. Part II: Application to multi-dimensional problems, Computing 64 (2000), 21–47. (2000) MR1755846
  7. A sparse H -matrix arithmetic: general complexity estimates, J. Comput. Appl. Math. 125 (2000), 479–501. (2000) MR1803209
  8. On H 2 -matrices, Lectures on applied mathematics, Hans-Joachim Bungartz, Ronald H. W. Hoppe, Christoph Zenger (eds.), Springer, Berlin, 2000, pp. 9–29. (2000) MR1767775
  9. H -matrix approximation on graded meshes, The Mathematics of Finite Elements and Applications X, MAFELAP 1999, John R. Whiteman (ed.), Elsevier, Amsterdam, 2000, pp. 307–316. (2000) MR1801984
  10. Mosaic-skeleton approximation, Calcolo 33 (1996), 47–57. (1996) MR1632459

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