一种用于准二维系统静电求和的快速谱高斯和方法
A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems
AI总结：
针对准二维周期静电系统，论文提出基于 SOG 核近似的快速谱求和方法，通过分区处理和免上采样、免零填充设计实现小常数 O(N log N) 复杂度。
摘要：
准二维静电系统的特征是在两个维度上具有周期性，而第三维自由，这类系统已在许多领域引起广泛关注。我们将高斯和（sum-of-Gaussians，SOG）近似应用于 Laplace 核，将相互作用划分为近场、中程和长程分量。近场分量具有奇异性，但在局部区域内紧支，因此直接计算。中程分量采用类似于三维非均匀快速 Fourier 变换的过程处理。长程分量包含大方差 Gaussian，在代理点上通过自由维的多项式插值/反插值以及另外两个维度的 Fourier 谱求解器处理。快速 Ewald 求和在高纵横比情形下需要大量零填充；与之不同，Gaussian 的可分离性使我们能够在自由方向上无需任何零填充即可处理这种情形。此外，尽管 NUFFT 通常依赖于每个维度上的一定上采样，截断核方法也会因核振荡引入额外的上采样因子，但由于 Gaussian 的光滑性，我们的方案消除了在任何方向上进行上采样的需要，从而显著降低了大规模问题的计算成本。最后，所有周期快速多极方法都需要将周期铺层划分为光滑的远场部分和包含最近邻单元的近场部分，而我们的方案直接在基本单元上运行，因此以更简单的实现获得更好的性能。我们给出了严格的误差分析，表明类 NUFFT 步骤不需要上采样，并以较小的前置因子实现 O(N log N) 复杂度。大量数值实验证明了该方案的性能。
英文摘要：
The quasi-2D electrostatic systems, characterized by periodicity in two dimensions with a free third dimension, have garnered significant interest in many fields. We apply the sum-of-Gaussians (SOG) approximation to the Laplace kernel, dividing the interactions into near-field, mid-range, and long-range components. The near-field component, singular but compactly supported in a local domain, is directly calculated. The mid-range component is managed using a procedure similar to nonuniform fast Fourier transforms in three dimensions. The long-range component, which includes Gaussians of large variance, is treated with polynomial interpolation/anterpolation in the free dimension and Fourier spectral solver in the other two dimensions on proxy points. Unlike the fast Ewald summation, which requires extensive zero padding in the case of high aspect ratios, the separability of Gaussians allows us to handle such case without any zero padding in the free direction. Furthermore, while NUFFTs typically rely on certain upsampling in each dimension, and the truncated kernel method introduces an additional factor of upsampling due to kernel oscillation, our scheme eliminates the need for upsampling in any direction due to the smoothness of Gaussians, significantly reducing computational cost for large-scale problems. Finally, whereas all periodic fast multipole methods require dividing the periodic tiling into a smooth far part and a near part containing its nearest neighboring cells, our scheme operates directly on the fundamental cell, resulting in better performance with simpler implementation. We provide a rigorous error analysis showing that upsampling is not required in NUFFT-like steps, achieving $O(N\log N)$ complexity with a small prefactor. The performance of the scheme is demonstrated via extensive numerical experiments.
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