Range-Separated Tensor Format for Many-Particle Modeling

Abstract

We introduce and analyze the new range-separated (RS) canonical/Tucker tensor format which aims for numerical modeling of the 3D long-range interaction potentials in multiparticle systems. The main idea of the RS tensor format is the independent grid-based low-rank representation of the localized and global parts in the target tensor, which allows the efficient numerical approximation of $N$-particle interaction potentials. The single-particle reference potential, described by the radial basis function $p(\|x\|)$, $x \in \mathbb{R}^d$, say $p(\|x\|)=1/\|x\|$ for $d=3$, is split into a sum of localized and long-range low-rank canonical tensors represented on a fine 3D $n\times n\times n$ Cartesian grid. The smoothed long-range contribution to the total potential sum is represented on the 3D grid in $O(n)$ storage via the low-rank canonical/Tucker tensor. We prove that the Tucker rank parameters depend only logarithmically on the number of particles $N$ and the grid size $n$. Agglomeration of the short-rang...