Multigrid accelerated tensor approximation of function related multidimensional arrays. (English) Zbl 1197.65215

Khoromskij, B. N.; Khoromskaia, V.

SIAM J. Sci. Comput. 31, No. 4, 3002-3026 (2009).

The authors describe and analyse a novel tensor approximation method for discretized multidimensional functions and operators in \(\mathbb{R}^d\), based on the idea of multigrid acceleration. This approach allows to overcome the limitations of the single grid schemes in higher dimensions and for large grid size. The algorithm 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 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 the authors’ 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, one obtains 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.

Reviewer: Titus Petrila (Cluj-Napoca)

MSC:

65R10	Numerical methods for integral transforms
65F30	Other matrix algorithms (MSC2010)
65F50	Computational methods for sparse matrices
65F10	Iterative numerical methods for linear systems
15A69	Multilinear algebra, tensor calculus
44A35	Convolution as an integral transform
44A30	Multiple integral transforms
65Y20	Complexity and performance of numerical algorithms

Keywords:

orthogonal Tucker tensor decomposition; canonical model; discrete convolution; adaptive orthogonal tensor-product basis; multigrid acceleration; algorithm; linear complexity; electronic structure calculations; convolution product; Newton potential