Prospects of tensor-based numerical modeling of the collective electrostatic potential 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 (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 RS format. The 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.