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
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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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  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 , 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  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.