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MiS Preprint

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

Venera Khoromskaia and Boris N. Khoromskij


Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatic potentials in many-particle systems and the respective interaction energy and forces [39,40,2]. 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. We generalize the approach initially introduced for the rank-structured grid-based calculation of the collective potentials on 3D lattices [39] 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 result, the interaction potential is represented in a parametric low-rank canonical format in $O(d L n)$ complexity. The energy is then calculated in $O(d L)$ operations. Electrostatics in large biomolecules is modeled by using the novel range-separated (RS) tensor format [2], which maintains the long-range part of the 3D collective potential of the many-body system represented on $n\times n \times n$ grid in a parametric low-rank form in $O(n)$-complexity. We show that the 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 [45] enables the construction of the efficient energy preserving regularization scheme for solving the 3D elliptic partial differential equations with strongly singular right-hand side arising, in particular, in bio-sciences. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics, protein docking and classification problems, for low-parametric interpolation of scattered data in data science, as well as in machine learning in many dimensions.

Feb 5, 2020
Feb 8, 2020
MSC Codes:
65F30, 65F50, 65N35, 65F10
coulomb potential, range-separated tensor formats, low-rank tensor decomposition, summation of electrostatic potentials, energy and force calculations

Related publications

2021 Repository Open Access
Venera Khoromskaia and Boris N. Khoromskij

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

In: Computational mathematics and mathematical physics, 61 (2021) 5, pp. 864-886