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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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1609-9389
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Volume 6, Issue 2

Issues

Structured Rank-(r1, . . . , rd) Decomposition of Function-related Tensors in R_D

B.N. Khoromskij

Abstract

The structured tensor-product approximation of multidimensional nonlocal operators by a two-level rank-(r1, . . . , rd) decomposition of related higher-order tensors is proposed and analysed. In this approach, the 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 multilevel nested canonical decomposition is presented. The complexity analysis of the corresponding multilinear algebra shows an almost linear cost in the one-dimensional problem size. The existence of a low Kronecker rank two-level representation is proven for a class of function-related tensors.

Keywords: structured matrices; Kronecker products; higher-order tensors; multidimensional integral operators; Newton potential; matrix-valued functions

About the article

Received: 2006-03-12

Revised: 2006-06-05

Accepted: 2006-06-23

Published in Print:


Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., Volume 6, Issue 2, Pages 194–220, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2006-0010.

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© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 4.0

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[12]
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[13]
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[14]
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[15]
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Constructive Approximation, 2011, Volume 34, Number 2, Page 149
[16]
Boris N. Khoromskij
Constructive Approximation, 2011, Volume 34, Number 2, Page 257
[17]
M. V. Rakhuba and I. V. Oseledets
SIAM Journal on Scientific Computing, 2015, Volume 37, Number 2, Page A565
[18]
V. Khoromskaia and B.N. Khoromskij
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M. Bebendorf, A. Kühnemund, and S. Rjasanow
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[20]
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ESAIM: Mathematical Modelling and Numerical Analysis, 2010, Volume 44, Number 4, Page 759
[21]
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[22]
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SIAM Journal on Scientific Computing, 2011, Volume 33, Number 1, Page 364
[23]
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Constructive Approximation, 2009, Volume 30, Number 3, Page 599
[24]
B. N. Khoromskij and V. Khoromskaia
SIAM Journal on Scientific Computing, 2009, Volume 31, Number 4, Page 3002
[25]
Wolfgang Hackbusch and Boris N. Khoromskij
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[26]
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[27]
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