Structured Rank-(r1, …, rd) Decomposition of Function-related Tensors in ℝ

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 ensor 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. In particular, we apply the results to d-th order tensors generated by the multivariate functions [formula omitted]. © 2006, Institute of Mathematics, NAS of Belarus. All rights reserved.