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

Venera Khoromskaia and Boris N. Khoromskij

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Submission date: 05. Feb. 2020
Pages: 32
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: coulomb potential, range-separated tensor formats, low-rank tensor decomposition, summation of electrostatic potentials, energy and force calculations
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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^⊗d lattices and discretized on fine n^⊗d Cartesian grids for arbitrary dimension d. As result, the interaction potential is represented in a parametric low-rank canonical format in O(dLn) complexity. The energy is then calculated in O(dL) 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 × n × 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.