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MiS Preprint

Tensor-Product Approximation to Operators and Functions in High Dimensions

Boris N. Khoromskij and Wolfgang Hackbusch


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 $d=3$.

MSC Codes:
65F50, 65F30, 46B28, 47A80
Tensor product approximation

Related publications

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