We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
38/2006
Tensor-Product Approximation to Multi-Dimensional Integral Operators and Green's Functions
Wolfgang Hackbusch and Boris N. Khoromskij
Abstract
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 $x,y \in\mathbb{R}^{d}$.
