Introduction to Hierarchical Matrices with Applications

Steffen Börm, Lars Grasedyck, and Wolfgang Hackbusch

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Submission date: 20. Feb. 2002
Pages: 28
published in: Engineering analysis with boundary elements, 27 (2003) 5, p. 405-422 
DOI number (of the published article): 10.1016/S0955-7997(02)00152-2
Bibtex
MSC-Numbers: 65F05, 65F30, 65F50, 65N50
Keywords and phrases: hierarchical matrices, formatted matrix operations, lyapunov equation, riccati equation, fast solvers
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Abstract:
We give a short introduction to methods for the data-sparse approximationof matrices resulting from the discretisation of non-local operators occurringin boundary integral methods, as the inverses of partial differentialoperators or as solutions of control problems.

The result of the approximation will be so-called hierarchicalmatrices (or short H-matrices). These matrices form a subset ofthe set of all matrices and have a data-sparse representation. The essentialoperations for these matrices (matrix-vector and matrix-matrix multiplication,addition and inversion) can be performed in, up to logarithmic factors, optimalcomplexity.

We give a review of specialised variants of H-matrices, especially ofH²-matrices, and finally consider applications of the different methodsto problems from integral equations, partial differential equationsand control theory.