Previous |  Up |  Next


hierarchical matrices; data-sparse approximations; formatted matrix operations; fast solvers
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.
[1] I. P. Gavrilyuk, W. Hackbusch, B. N. Khoromskij: $H$-matrix approximation for the operator exponential with applications. Numer. Math (to appear). MR 1917366
[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
[3] L. Grasedyck: Theorie und Anwendungen Hierarchischer Matrizen. Doctoral thesis, University Kiel, 2001.
[4] W. Hackbusch: A sparse matrix arithmetic based on $H$-Matrices. Part I: Introduction to $H$-Matrices. Computing 62 (1999), 89–108. DOI 10.1007/s006070050015 | MR 1694265
[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. DOI 10.1007/BF01396324 | MR 0972420
[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
[7] W. Hackbusch, B. N. Khoromskij: A sparse $H$-matrix arithmetic: general complexity estimates. J. Comput. Appl. Math. 125 (2000), 479–501. DOI 10.1016/S0377-0427(00)00486-6 | MR 1803209
[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
[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
[10] E. Tyrtyshnikov: Mosaic-skeleton approximation. Calcolo 33 (1996), 47–57. DOI 10.1007/BF02575706 | MR 1632459
Partner of
EuDML logo