论文标题
基于张量的基于张量的数值建模在许多粒子系统中的集体静电电势
Prospects of tensor-based numerical modeling of the collective electrostatic potential in many-particle systems
论文作者
论文摘要
最近，等级结构的张量方法表明，在许多粒子系统中对远程静电电位的数值处理以及相应的相互作用能量和力[39,40,2]。在本文中，我们概述了基于张量的数值建模的前景，用于晶格和多种类型的许多粒子系统中的集体静电电位。我们将最初引入的方法概括为基于等级结构网格的3D晶格[39]的集体电位计算，以$ l^{\ otimes d} $ lattices上的可变费用，并在fine $ n^{\ otimes d} $ dimimes d} $ cartes $ cartes $ cartes gridemension $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d.结果，相互作用势以$ O（d l n）$复杂性的参数低级别规范格式表示。然后以$ O（d l）$操作计算能量。大型生物分子中的静电是通过使用新型范围分离（RS）张量格式[2]建模的，该格式[2]维持在$ n \ times n \ times n \ times n \ times n \ times n $ grid上以$ O（n）$ o（n）$ - 复杂性为$ n \ times n \ timper n \ times n $ grid代表的多体系统的3D集体潜力的远程部分。我们表明，通过使用低级别RS格式的已经预先计算的电场可以轻松恢复力场。离散的DIRAC DELTA [45]的RS张量表示可以实现有效的能量维护正则化方案，以求解生物学中出现的强右侧右侧的3D椭圆PDES。我们得出的结论是，基于等级张量的近似技术为多体动态，蛋白质对接和分类问题以及数据科学中散射数据的低参数插值提供了有希望的数值工具。
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 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.
