| Title:
|
An introduction to hierarchical matrices (English) |
| Author:
|
Hackbusch, Wolfgang |
| Author:
|
Grasedyck, Lars |
| Author:
|
Börm, Steffen |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
127 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
229-241 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
hierarchical matrices |
| Keyword:
|
data-sparse approximations |
| Keyword:
|
formatted matrix operations |
| Keyword:
|
fast solvers |
| MSC:
|
15A57 |
| MSC:
|
65F05 |
| MSC:
|
65F30 |
| MSC:
|
65F50 |
| MSC:
|
65N22 |
| MSC:
|
65N38 |
| MSC:
|
65N50 |
| MSC:
|
65Y20 |
| idZBL:
|
Zbl 1007.65032 |
| idMR:
|
MR1981528 |
| DOI:
|
10.21136/MB.2002.134156 |
| . |
| Date available:
|
2012-10-05T12:57:37Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134156 |
| . |
| Reference:
|
[1] I. P. Gavrilyuk, W. Hackbusch, B. N. Khoromskij: $H$-matrix approximation for the operator exponential with applications.Numer. Math (to appear). MR 1917366 |
| Reference:
|
[2] I. P. Gavrilyuk, W. Hackbusch, B. N. Khoromskij: $H$-matrix approximation for elliptic solution operators in cylindric domains.East-West J. Numer. Math. 9 (2001), 25–58. MR 1839197 |
| Reference:
|
[3] L. Grasedyck: Theorie und Anwendungen Hierarchischer Matrizen.Doctoral thesis, University Kiel, 2001. |
| Reference:
|
[4] W. Hackbusch: A sparse matrix arithmetic based on $H$-Matrices. Part I: Introduction to $H$-Matrices.Computing 62 (1999), 89–108. MR 1694265, 10.1007/s006070050015 |
| Reference:
|
[5] W. Hackbusch, Z. P. Nowak: On the fast matrix multiplication in the boundary element method by panel clustering.Numer. Math. 54 (1989), 463–491. MR 0972420, 10.1007/BF01396324 |
| Reference:
|
[6] W. Hackbusch, B. N. Khoromskij: A sparse $H$-matrix arithmetic. Part II: Application to multi-dimensional problems.Computing 64 (2000), 21–47. MR 1755846 |
| Reference:
|
[7] W. Hackbusch, B. N. Khoromskij: A sparse $H$-matrix arithmetic: general complexity estimates.J. Comput. Appl. Math. 125 (2000), 479–501. MR 1803209, 10.1016/S0377-0427(00)00486-6 |
| Reference:
|
[8] W. Hackbusch, B. N. Khoromskij, S. A. Sauter: On $H^{2}$-matrices.Lectures on applied mathematics, Hans-Joachim Bungartz, Ronald H. W. Hoppe, Christoph Zenger (eds.), Springer, Berlin, 2000, pp. 9–29. MR 1767775 |
| Reference:
|
[9] W. Hackbusch, B. N. Khoromskij: $H$-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. MR 1801984 |
| Reference:
|
[10] E. Tyrtyshnikov: Mosaic-skeleton approximation.Calcolo 33 (1996), 47–57. MR 1632459, 10.1007/BF02575706 |
| . |
