Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
23/2020

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

Venera Khoromskaia and Boris N. Khoromskij

Abstract

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.

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

Related publications

inJournal
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