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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
139/2006

Tensor-Product Approximation to Operators and Functions in High Dimensions

Boris N. Khoromskij and Wolfgang Hackbusch

Abstract

In recent papers tensor-product structured Nyström and Galerkin type approximations of certain multi-dimensional integral operators have been introduced and analysed. In the present paper we focus on the analysis of the collocation type schemes with respect to the tensor-product basis in a high spatial dimension $d$. Approximations up to an accuracy $\mathcal{O} (N^{-\alpha/d})$ are proven to have the storage complexity $\mathcal{O} (dN^{1/d}\log^{q}N)$ with $q$ independent of $d$, where $N$ is the discrete problem size. In particular, we apply the theory to a collocation discretisation of the Newton potential with the kernel $\frac{1}{|x-y|}$, $x,y\in\mathbb{R}^{d}$, $d\geq3$. Numerical illustrations are given in the case of $d=3$.

Received:
Nov 28, 2006
Published:
Nov 28, 2006
MSC Codes:
65F50, 65F30, 46B28, 47A80
Keywords:
Tensor product approximation

Related publications

inJournal
2007 Repository Open Access
Wolfgang Hackbusch and Boris N. Khoromskij

Tensor-product approximation to operators and functions in high dimensions

In: Journal of complexity, 23 (2007) 4/6, pp. 697-714