Tensor decomposition in electronic structure calculations on 3D Cartesian grids
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
SIAM Journal on Matrix Analysis and Applications
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors
SIAM Journal on Scientific Computing
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Approximation of the electron density of Aluminium clusters in tensor-product format
Journal of Computational Physics
A tensor decomposition approach to data compression and approximation of ND systems
Multidimensional Systems and Signal Processing
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case
SIAM Journal on Scientific Computing
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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.