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MiS Preprint

Approximation of boundary element operators by adaptive ${\mathcal{H}}^{2}$-matrices

Steffen Börm and Wolfgang Hackbusch


The discretization of integral operators corresponding to non-local kernel functions typically gives rise to densely populated matrices. In order to be able to treat these matrices in an efficient manner, they have to be compressed, e.g., by panel clustering algorithms, multipole expansions or wavelet techniques.

By choosing the correct panel clustering approach, the resulting approximation of the matrix can be written in the form of a so-called $\mathcal{H}^2$-matrix. The $\mathcal{H}^2$-matrix representation can be computed for fairly general kernel functions by a black box algorithm that requires only pointwise evaluations of the kernel function.

Although this technique leads to good results, the expansion system tends to contain a certain level of redundancy that leads to an unnecessarily high complexity for the memory requirements and the matrix-vector multiplication. We present two variants of the original method that can compress the matrix even further. Both methods work on the fly, i.e., it is not necessary to keep the original $\mathcal{H}^2$-matrix in memory, and both methods perform an algebraic compression, so that the black box character of the algorithm is preserved.

MSC Codes:
65F05, 65F30, 65F50, 65N38, 68P05, 45B05, 35C20
hierarchical matrices, nested bases, bem

Related publications

2004 Repository Open Access
Steffen Börm and Wolfgang Hackbusch

Approximation of boundary element operators by adaptive \(\mathscr {H}^2\)-matrices

In: Foundations of computational mathematics : Minneapolis 2002 (FoCM 2002) ; selected papers based on the plenary talks presented at FoCM 2002, Minneapolis, MN, USA, August 5-14, 2002 / Felipe Cucker... (eds.)
Cambridge : Cambridge University Press, 2004. - pp. 58-75
(London Mathematical Society lecture note series ; 312)