Hierarchical Kronecker tensor-product approximations

Wolfgang Hackbusch, Boris N. Khoromskij, and Eugene E. Tyrtyshnikov

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Submission date: 15. Apr. 2003 (revised version: November 2004)
Pages: 29
published in: Journal of numerical mathematics, 13 (2005) 2, p. 119-156 
DOI number (of the published article): 10.1163/1569395054012767
Bibtex
MSC-Numbers: 65F50, 65F30, 65N35, 65N38, 15A09
Keywords and phrases: integral equations, bem, low-rank matrices, hierarchical matrices, kronecker products, multi-dimensional matrices, tensors
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Abstract:
The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integro-differential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices. In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called ``profile-low-rank matrices'' or hierarchical matrices. We give a proof for the existence of such formats, present some observations and hypotheses, and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.