Preprint 38/2006

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
MSC-Numbers: 65F50, 65F30, 46B28, 47A80
Keywords and phrases: hierarchical matrices, kronecker tensor-product, Sinc approximation

The Kronecker tensor-product approximation combined with the formula26-matrix techniques provides an efficient tool to represent integral operators as well as a discrete elliptic operator inverse formula28 in formula30 (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 formula34 with q independent of d, where formula40 is the dimension of the discrete problem in one space direction. In particular, we apply the results to the Newton, Yukawa and Helmholtz kernels formula42, formula44 and formula46, respectively, with formula48.

18.10.2019, 02:13