L^∞ estimation of tensor truncations

Wolfgang Hackbusch

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Submission date: 19. Mar. 2012 (revised version: April 2012)
published in: Numerische Mathematik, 125 (2013) 3, p. 419-440 
DOI number (of the published article): 10.1007/s00211-013-0544-6
Bibtex
MSC-Numbers: 15A69, 15A18, 35J08, 46B70
Keywords and phrases: tensor calculus, tensor truncation, higher-order singular value decomposition (HOSVD), approximation, Gagliardo-Nirenberg inequality, Green function
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Abstract:
Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to ℓ² or L² norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require ℓ^∞ or L^∞ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of _∞ by ₂ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to _∞ can be derived from the Gagliardo-Nirenberg inequality because of the special nature of the SVD truncation.