Abstract

Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatics in many-particle systems and the respective interaction energy and forces. 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. Our approach, initially introduced for the rank-structured grid-based calculation of the interaction potentials on 3D lattices is generalized here 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 a result, the interaction potential is represented in a parametric low-rank canonical format in \(O(dLn)\) complexity. The total interaction energy can be then calculated in \(O(dL)\) operations. Electrostatics in large bio-molecular systems is discretized on a fine \({{n}^{{ \otimes 3}}}\) grid by using the novel range-separated (RS) tensor format, which maintains the long-range part of the 3D collective potential of a many-body system in a parametric low-rank form in \(O(n)\)-complexity. We show how the energy and 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 (conservative) regularization scheme for solving the 3D elliptic partial differential equations with strongly singular right-hand side arising in scientific computing. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics in bio-sciences, protein docking and classification problems, for low-parametric interpolation of scattered data in data science, as well as in machine learning in many dimensions.

中文翻译：

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基于张量的多粒子系统集体静电数值建模的前景

摘要

最近，秩结构张量方法表明在多粒子系统中长程静电的数值处理以及各自的相互作用能量和力方面取得了进展。在本文中，我们概述了基于张量的对晶格上和一般类型多粒子系统中的集体静电势进行数值建模的前景。我们的方法最初是为 3D 晶格上的基于秩结构网格的相互作用势计算而引入的，这里将其推广到具有可变电荷的多粒子系统的情况\({{L}^{{ \otimes d} }}\)格子并在精细\({{n}^{{ \otimes d}}}\)笛卡尔网格上离散化，用于任意维度\(d\). 因此，相互作用势以\(O(dLn)\)复杂度的参数低秩规范格式表示。然后可以在\(O(dL)\)操作中计算总相互作用能。大型生物分子系统中的静电通过使用新颖的距离分隔 (RS) 张量格式在精细的\({{n}^{{ \otimes 3}}}\)网格上离散化，该格式保留了远程部分\(O(n)\) 中参数低秩形式的多体系统的 3D 集体潜力-复杂。我们展示了如何通过使用低秩 RS 格式中已经预先计算的电场来轻松恢复能量和力场。离散化狄拉克 delta 的 RS 张量表示能够构建有效的能量保持（保守）正则化方案，用于求解科学计算中出现的具有强奇异右手侧的 3D 椭圆偏微分方程。我们得出的结论是，基于秩结构张量的近似技术为生物科学中的多体动力学、蛋白质对接和分类问题、数据科学中分散数据的低参数插值以及在许多维度的机器学习中。