Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays

In this paper, we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in $\mathbb{R}^d$, based on the idea of multigrid acceleration. The approach stands on successive reiterations of the orthogonal Tucker tensor approximation on a sequen...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 31; no. 4; pp. 3002 - 3026
Main Authors: Khoromskij, B. N., Khoromskaia, V.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
Subjects:
Accuracy
Algorithms
Applied mathematics
Approximation
Arrays
Cusps
Data structures
Decomposition
Electrons
Mathematical analysis
Mathematical models
Methods
Tensors
Three dimensional
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:In this paper, we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in $\mathbb{R}^d$, 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 us to transfer from the coarse-to-fine grids the important data structure information on the 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 three-dimensional (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 $10^{-6}$ in max-norm is achieved on large $n\times n\times n$ grids up to $n=1.6\cdot10^4$, with the time scale in several minutes.
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ISSN:1064-8275
1095-7197
DOI:10.1137/080730408