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| Zeit: | Freitag, 15.10.1999, 11:45 Uhr |
| Ort: | Reichenhainer Straße 70, B202 |
| Autor: | B. Khoromskij (Leipzig) |
| Thema: | H-Matrix Approximation in FE and BE Methods |
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A class of hierarchical matrices (H-matrices) has been introduced in
[1] which are data-sparse and allow an approximate matrix arithmetic
of almost linear complexity. Several types of H-matrices were
analysed in [2,3,4,5,6], which are able to approximate nonlocal
operators in FEM and BEM applications in the case of quasi-uniform
unstructured as well as graded tensor product meshes.
The consistency error and general complexity estimates for H-matrix approximations to integral operators defined on domains/manifolds in Rd, d=2,3, will be discussed. The basic idea behind local kernel expansions of the optimal order is based on a reduction to the surface of clusters, which build an admissible block partitioning. The resulting H-matrices retain the approximation power of the exact Galerkin scheme, on the one hand, and provide linear (resp. linear-logarithmic) complexity for matrix-vector multiplications in FEM (resp. BEM) applications, on the other. The matrix-matrix product and matrix-inversion in hierarchical formats are shown to have a linear-logarithmic complexity as well. Emphasis will be placed upon the optimization of the H-matrix arithmetic, in particular, using an H2-matrix concept. Modifications for the case of nonuniform meshes are also considered. We present numerical results illustrating the data sparsity of hierarchical approximations for the integral and inverse to the elliptic differential operators. [1] W. Hackbusch: A Sparse Matrix Arithmetic based on H-Matrices. Part I: Introduction to H-Matrices. Computing 62 (1999), 89-108. [2] W. Hackbusch and B. N. Khoromskij: A Sparse H-Matrix Arithmetic. Part II. Application to Multi-Dimensional Problems. Preprint No. 22, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 1999. Computing (to appear). [3] W. Hackbusch and B. N. Khoromskij: A Sparse H-Matrix Arithmetic: General Complexity Estimates. MPI Leipzig 1999, submitted. [4] W. Hackbusch, B. N. Khoromskij and S. Sauter: On H2-matrices. Preprint No. 50, MPI, Leipzig, 1999. Submitted. [5] W. Hackbusch and B. N. Khoromskij: H-Matrix Approximation on Graded Meshes. Preprint No. 54, MPI, Leipzig, 1999. Proceedings of the conference MAFELAP 1999 (J.R. Whiteman, ed.). [6] W. Hackbusch and B. N. Khoromskij: Towards H-Matrix Approximation of the Linear Complexity. A contribution to the proc. of the conference TMP11 1998 in Chemnitz. | |
| Das Seminar wird von Prof. Schneider geleitet. Interessenten sind herzlich eingeladen. | |