Prospects of tensor-based numerical modeling of the collective electrostatics in many-particle systems

Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatics in many-particle systems and the respective interaction energy and forces. In this paper, we outline the prospects for tensor-based numerical modeling of the collective electrostatic potential on lattices and in many-particle systems of general type. Our approach, initially introduced for the rank-structured grid-based calculation of the interaction potentials on 3D lattices is generalized here to the case of many-particle systems with variable charges placed on $L^{\otimes d}$ lattices and discretized on fine $n^{\otimes d}$ Cartesian grids for arbitrary dimension $d$. As a result, the interaction potential is represented in a parametric low-rank canonical format in $O(dLn)$ complexity. The total interaction energy can be then calculated in $O(dL)$ operations. Electrostatics in large bio-molecular systems is discretized on a fine $n^{\otimes 3}$ grid by using the novel range-separated $(\mathrm{RS})$ tensor format, which maintains the long-range part of the 3D collective potential of a many-body system in a parametric low-rank form in $O(n)$-complexity. We show how the energy and force field can be easily recovered by using the already precomputed electric field in the low-rank $\mathrm{RS}$ format. The $\mathrm{RS}$ tensor representation of the discretized Dirac delta enables the construction of the efficient energy preserving (conservative) regularization scheme for solving the 3D elliptic partial differential equations with strongly singular right-hand side arising in scientific computing. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics in bio-sciences, protein docking and classification problems, for low-parametric interpolation of scattered data in data science, as well as in machine learning in many dimensions.