Tensor-Product Approximation to Operators and Functions in High Dimensions

revised version: March 2007
Boris N. Khoromskij and Wolfgang Hackbusch
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Submission date: 28. Nov. 2006
Pages: 17
published in: Journal of complexity, 23 (2007) 4/6, p. 697-714 
DOI number (of the published article): 10.1016/j.jco.2007.03.007
MSC-Numbers: 65F50, 65F30, 46B28, 47A80
Keywords and phrases: Tensor product approximation
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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 are proven to have the storage complexity 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 , , . Numerical illustrations are given in the case of d=3.