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MiS Preprint
6/2006
Structured Rank-$(r_1,...,r_d)$ Decomposition of Function-related Tensors in $\mathbb{R}^{d} $
Boris N. Khoromskij
Abstract
The structured tensor-product approximation of multi-dimensional nonlocal operators by a two-level rank-$(r_1,...,r_d)$ decomposition of related higher-order tensors is proposed and analysed. In this approach, a construction of the desired approximant to a target tensor is a reminiscence of the Tucker-type model, where the canonical components are represented in a fixed (uniform) basis, while the core tensor is given in the canonical format. As an alternative, the multi-level nested canonical decomposition is presented. The complexity analysis of the corresponding multi-linear algebra indicates almost linear cost in one-dimensional problem size.
The existence of a low Kronecker rank two-level representation is proven for a class of function-related tensors. In particular, we apply the results to $d$-th order tensors generated by the multi-variate functions $\frac{1}{|x|^2} $, $\frac{1}{|x-y|} $, $e^{-\alpha |x-y|} $, $\frac{e^{-|x-y| }}{|x-y|} $ and $|x|^\lambda sinc(|x|\,|y|)$ with $x,y \in \mathbb{R}^{d} $.
