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MiS Preprint

$L^{\infty}$ estimation of tensor truncations

Wolfgang Hackbusch


Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to $\ell^{2}$ or $L^{2}$ 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 $\ell^{\infty}$ or $L^{\infty}$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $\left\Vert \cdot\right\Vert _{\infty}$ by $\left\Vert \cdot\right\Vert _{2}$ 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 $\left\Vert \cdot\right\Vert _{\infty}$ can be derived from the Gagliardo-Nirenberg inequality because of the special nature of the SVD\ truncation.

MSC Codes:
15A69, 15A18, 35J08, 46B70
tensor calculus, tensor truncation, higher-order singular value decomposition (HOSVD), approximation, Gagliardo-Nirenberg inequality, Green function

Related publications

2013 Repository Open Access
Wolfgang Hackbusch

\(L^{\infty} \) estimation of tensor truncations

In: Numerische Mathematik, 125 (2013) 3, pp. 419-440