标题： 一种用于准二维系统静电求和的快速谱高斯函数求和方法 显示英文标题

标题： A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems

摘要： 准二维静电系统，具有在两个维度上的周期性以及一个自由的第三维度，已在许多领域引起了广泛关注。 我们将高斯函数之和（SOG）近似应用于拉普拉斯核，将相互作用分为近场、中程和远场成分。 近场成分在局部区域内具有奇异性但紧支撑，直接进行计算。 中程成分使用类似于三维非均匀快速傅里叶变换的处理方法。 远场成分包含方差较大的高斯函数，通过在自由维度上的多项式插值/反插值以及在其他两个维度上代理点的傅里叶谱求解器进行处理。 与需要在高纵横比情况下进行大量零填充的快速Ewald求和不同，高斯函数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此类情况。 此外，虽然NUFFT通常依赖于每个维度的某种上采样，而截断核方法由于核的振荡引入了额外的上采样因子，但由于高斯函数的平滑性，我们的方案消除了任何方向上的上采样需求，显著降低了大规模问题的计算成本。 最后，尽管所有周期性快速多极方法都需要将周期性铺砌划分为一个光滑的远场部分和一个包含其最近邻单元的近场部分，我们的方案直接在基本单元上运行，从而实现了更好的性能和更简单的实现。 我们提供了严格的误差分析，表明在类似NUFFT的步骤中不需要上采样，实现了$O(N\log N)$的复杂度，且具有较小的预因子。 该方案的性能通过大量的数值实验得到了验证。 显示英文摘要

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.