Tensor-Product Approximation to Multi-Dimensional Integral Operators and Green's Functions

Wolfgang Hackbusch and Boris N. Khoromskij

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Submission date: 12. Apr. 2006
published in: SIAM journal on matrix analysis and applications, 30 (2008) 3, p. 1233-1253 
DOI number (of the published article): 10.1137/060657017
Bibtex
MSC-Numbers: 65F50, 65F30, 46B28, 47A80
Keywords and phrases: hierarchical matrices, kronecker tensor-product, Sinc approximation

Abstract:
The Kronecker tensor-product approximation combined with the -matrix techniques provides an efficient tool to represent integral operators as well as a discrete elliptic operator inverse in (the discrete Green's function) with a high spatial dimension d. In the present paper we give a survey on modern methods of the structured tensor-product approximation to multi-dimensional integral operators and Green's functions and present some new results on the existence of low tensor-rank decompositions to a class of function-related operators. The asymptotic complexity of the considered data-sparse representations is estimated by with q independent of d, where is the dimension of the discrete problem in one space direction. In particular, we apply the results to the Newton, Yukawa and Helmholtz kernels , and , respectively, with .