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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 $d=3$.