Abstract. In this paper, we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in 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.T h em e t h o di s 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 × n × n grids up to n =1 .6 · 10 4 ,w i t ht h et i m es c a l ei ns e v e r a l minutes.

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Our recent method for low-rank tensor representation of sums of the arbitrarily positioned electrostatic potentials discretized on a 3D Cartesian grid reduces the 3D tensor summation to operations involving only 1D vectors however retaining the linear complexity scaling in the number of potentials. Here, we introduce and study a novel tensor approach for fast and accurate assembled summation of a large number of lattice-allocated potentials represented on 3D N × N × N grid with the computational requirements only weakly dependent on the number of summed potentials. It is based on the assembled low-rank canonical tensor representations of the collected potentials using pointwise sums of shifted canonical vectors representing the single generating function, say the Newton kernel. For a sum of electrostatic potentials over L × L × L lattice embedded in a box the required storage scales linearly in the 1D grid-size, O(N ), while the numerical cost is estimated by O(N L). For periodic boundary conditions, the storage demand remains proportional to the 1D grid-size of a unit cell, n = N/L, while the numerical cost reduces to O(N ), that outperforms the FFT-based Ewald-type summation algorithms of complexity O (N 3 log N ). The complexity in the grid parameter N can be reduced even to the logarithmic scale O(log N ) by using data-sparse representation of canonical N -vectors via the quantics tensor approximation. For justification, we prove an upper bound on the quantics ranks for the canonical vectors in the overall lattice sum. The presented approach is beneficial in applications which require further functional calculus with the lattice potential, say, scalar product with a function, integration or differentiation, which can be performed easily in tensor arithmetics on large 3D grids with 1D cost. Numerical tests illustrate the performance of the tensor summation method and confirm the estimated bounds on the tensor ranks. [48]. This paper is an essentially improved version of the preprint of the Max-Planck Institute for Mathematics in the Sciences