Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation
Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation
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DOI：
10.1016/j.cpc.2014.08.015
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发表时间：
2014-05
期刊：
影响因子：
--
通讯作者：
V. Khoromskaia;B. Khoromskij
中科院分区：
文献类型：
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作者：
V. Khoromskaia;B. Khoromskij
我们最近在 3D 笛卡尔网格上离散化的任意位置静电势之和的低秩张量表示方法将 3D 张量求和减少为仅涉及 1D 向量的运算，但保留了势数量的线性复杂度缩放。在这里，我们介绍并研究了一种新颖的张量方法，用于快速准确地对 3D N×N×N 网格上表示的大量晶格分配势进行组装求和，计算要求仅微弱地依赖于求和势的数量。它基于使用代表单个生成函数（例如牛顿核）的移位规范向量的逐点和来组装所收集势的低秩规范张量表示。对于嵌入盒子中的 L×L×L 晶格上的静电势总和，所需的存储在一维网格大小 O (N) 中线性缩放，而数值成本则由 O (N L) 估计。对于周期性边界条件，存储需求仍然与单元格的一维网格大小成正比，n= N/L，而数值成本降低到 O (N)，优于复杂度为 O (N 3 log N) 的基于 FFT 的 Ewald 型求和算法。通过量子张量近似使用规范 N 向量的数据稀疏表示，网格参数 N 的复杂性甚至可以降低到对数尺度 O (log N)。为了证明合理性，我们证明了整个晶格和中规范向量的量子等级的上限。所提出的方法在需要进一步具有晶格势的函数微积分的应用中是有益的，例如，具有函数、积分或微分的标量积，这可以在具有 1D 成本的大型 3D 网格上的张量算术中轻松执行。数值测试说明了张量求和方法的性能并确认了张量秩的估计界限。
Our recent method for low-rank tensor representation of sums of the arbitrarily positioned electrostatic potentials discretized on a 3D Cartesian grid reduces the 3D tensor summation to operations involving only 1D vectors however retaining the linear complexity scaling in the number of potentials. Here, we introduce and study a novel tensor approach for fast and accurate assembled summation of a large number of lattice-allocated potentials represented on 3D N× N× N grid with the computational requirements only weakly dependent on the number of summed potentials. It is based on the assembled low-rank canonical tensor representations of the collected potentials using pointwise sums of shifted canonical vectors representing the single generating function, say the Newton kernel. For a sum of electrostatic potentials over L× L× L lattice embedded in a box the required storage scales linearly in the 1D grid-size, O (N), while the numerical cost is estimated by O (N L). For periodic boundary conditions, the storage demand remains proportional to the 1D grid-size of a unit cell, n= N/L, while the numerical cost reduces to O (N), that outperforms the FFT-based Ewald-type summation algorithms of complexity O (N 3 log N). The complexity in the grid parameter N can be reduced even to the logarithmic scale O (log N) by using data-sparse representation of canonical N-vectors via the quantics tensor approximation. For justification, we prove an upper bound on the quantics ranks for the canonical vectors in the overall lattice sum. The presented approach is beneficial in applications which require further functional calculus with the lattice potential, say, scalar product with a function, integration or differentiation, which can be performed easily in tensor arithmetics on large 3D grids with 1D cost. Numerical tests illustrate the performance of the tensor summation method and confirm the estimated bounds on the tensor ranks.