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

DOI10.1134/S0965542521050110zbMATH Open1469.82010arXiv2001.11393OpenAlexW3003894950MaRDI QIDQ2038503

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Publication date: 7 July 2021

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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 $L^{otimesd}$ lattices and discretized on fine $n^{otimesd}$ 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 $nimesnimesn$ 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 efficient energy preserving regularization scheme for solving the 3D elliptic PDEs with strongly singular right-hand side arising 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 and for low-parametric interpolation of scattered data in data science.

Full work available at URL: https://arxiv.org/abs/2001.11393

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