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MiS Preprint

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

Wolfgang Hackbusch and Boris N. Khoromskij


The Kronecker tensor-product approximation combined with the $\mathcal{H} $-matrix techniques provides an efficient tool to represent integral operators as well as a discrete elliptic operator inverse $A^{-1}\in\mathbb{R}^{N\times N}$ in $\mathbb{R}^{d}$ (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 $\mathcal{O}(d n\log^{q}n)$ with $q$ independent of $d$, where $n=N^{1/d}$ is the dimension of the discrete problem in one space direction. In particular, we apply the results to the Newton, Yukawa and Helmholtz kernels $\frac{1}{|x-y|} $, $\frac{e^{-\lambda|x-y| }}{|x-y|} $ and $\frac{\cos(\lambda|x-y| )}{|x-y|}$, respectively, with $x,y \in\mathbb{R}^{d}$.

Apr 12, 2006
Apr 12, 2006
MSC Codes:
65F50, 65F30, 46B28, 47A80
hierarchical matrices, kronecker tensor-product, Sinc approximation

Related publications

2008 Repository Open Access
Boris N. Khoromskij and Wolfgang Hackbusch

Tensor-product approximation to multidimensional integral operators and Green's functions

In: SIAM journal on matrix analysis and applications, 30 (2008) 3, pp. 1233-1253