Structured Rank-(r₁,...,r_d) Decomposition of Function-related Tensors in ℝ^d

Boris N. Khoromskij

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Submission date: 17. Jan. 2006 (revised version: March 2006)
Pages: 35
published in: Computational methods in applied mathematics, 6 (2006) 2, p. 194-220 
DOI number (of the published article): 10.2478/cmam-2006-0010
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35
Keywords and phrases: low-rank matrices, $\mathcal{h}$-matrices, kronecker products, multi-dimensional operators
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Abstract:
The structured tensor-product approximation of multi-dimensional nonlocal operators by a two-level rank- 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 , , , and with .