Preprint 40/2008

Multigrid Accelerated Tensor Approximation of Function Related Multi-dimensional Arrays

Boris N. Khoromskij and Venera Khoromskaia

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Submission date: 28. Apr. 2008 (revised version: November 2009)
Pages: 27
published in: SIAM journal on scientific computing, 31 (2009) 4, p. 3002-3026 
DOI number (of the published article): 10.1137/080730408
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35
Keywords and phrases: Orthogonal Tucker tensor decomposition, discrete convolution, multigrid acceleration
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Abstract:
In this paper, we describe and analyse a novel tensor approximation method for discretized multi-dimensional functions and operators in formula9, based on the idea of multigrid acceleration. The approach stands on successive reiterations of the orthogonal Tucker tensor approximation on a sequence of nested refined grids. On the one hand, it provides a good initial guess for the nonlinear iterations to find the approximating subspaces on finer grids, on the other hand, it allows to transfer from the coarse-to-fine grids the important data structure information on location of the so-called most important fibers in directional unfolding matrices. The method indicates linear complexity with respect to the size of data representing the input tensor. In particular, if the target tensor is given by using the rank-R canonical model, then our approximation method is proved to have linear scaling in the univariate grid size n, and in the input rank R.

The method is tested by 3D electronic structure calculations. For the multigrid accelerated low Tucker-rank approximation of the all electron densities having strong nuclear cusps, we obtain high resolution of their 3D convolution product with the Newton potential. The accuracy of order formula17 in max-norm is achieved on large formula19 grids up to formula21, with the time scale in several minutes.

03.07.2017, 01:41