Multigrid Accelerated Tensor Approximation in 3D Electronic Structure Calculations

We describe a novel tensor approximation method for discretised multi-dimensional 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 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 tensor product model, then our approximation method is proved to have linear scaling in the dimension parameter $d$, 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 $10^{-6}$ in max-norm is achieved on large $n\times n\times n$ grids up to $n=1.6\cdot 10^4$. The total computational time on such grids amounts to several minutes in Matlab implementation.