Adaptive Recompression of -Matrices for BEM

Lars Grasedyck

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Submission date: 13. Apr. 2004 (revised version: July 2004)
Pages: 22
published in: Computing, 74 (2005) 3, p. 205-223 
DOI number (of the published article): 10.1007/s00607-004-0103-1
Bibtex
Keywords and phrases: hierarchical matrices, preconditioning, boundary elements
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Abstract:
The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use a hierarchical low-rank approximation, the so-called -matrix, that approximates the dense stiffness matrix in admissible blocks (corresponding to domains where the underlying kernel function is smooth) by low rank matrices. The low rank matrices are assembled by the ACA+ algorithm, an improved variant of the well-known ACA method. We present an algorithm that can determine a coarser block structure that minimises the storage requirements (enhanced compression) and speeds up the arithmetic (e.g., inversion) in the -matrix format. This coarse approximation is done adaptively and on-the-fly to a given accuracy such that the matrix is assembled with minimal storage requirements while keeping the desired approximation quality. The benefits of this new recompression technique are demonstrated by numerical examples.