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

Xuanzhao Gao, Shidong Jiang, Jiuyang Liang, Zhenli Xu, Qi Zhou

TL;DR

This work addresses the computational bottleneck of long-range electrostatics in quasi-2D systems by introducing a fast spectral sum-of-Gaussians (SOG) decomposition of the Laplace kernel into near-field, mid-range, and long-range parts. The mid-range is solved via a Fourier spectral method, while the long-range employs a Fourier-Chebyshev approach in the free direction, enabled by range splitting and Gaussian separability that eliminates the need for upsampling or large zero-padding. The authors provide rigorous error analyses and parameter-selection guidelines, and demonstrate spectral accuracy with $O(N\log N)$ scaling across a range of geometries, including highly anisotropic boxes, through extensive numerical experiments. The method is kernel-agnostic (kernel-independent SOG) and holds promise for efficient MD/MC simulations in quasi-2D environments and beyond.

Abstract

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.

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

TL;DR

This work addresses the computational bottleneck of long-range electrostatics in quasi-2D systems by introducing a fast spectral sum-of-Gaussians (SOG) decomposition of the Laplace kernel into near-field, mid-range, and long-range parts. The mid-range is solved via a Fourier spectral method, while the long-range employs a Fourier-Chebyshev approach in the free direction, enabled by range splitting and Gaussian separability that eliminates the need for upsampling or large zero-padding. The authors provide rigorous error analyses and parameter-selection guidelines, and demonstrate spectral accuracy with scaling across a range of geometries, including highly anisotropic boxes, through extensive numerical experiments. The method is kernel-agnostic (kernel-independent SOG) and holds promise for efficient MD/MC simulations in quasi-2D environments and beyond.

Abstract

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 complexity with a small prefactor. The performance of the scheme is demonstrated via extensive numerical experiments.

Paper Structure

This paper contains 31 sections, 8 theorems, 133 equations, 6 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Sum-of-Gaussians decompositions of electrostatic interactions for quasi-2D systems
  3. Quasi-2D electrostatic interactions and kernel decomposition
  4. The SOG decomposition
  5. Decomposition error and parameter selection in quasi-2D systems
  6. A fast spectral sum-of-Gaussians method
  7. Range splitting of the far-field potential
  8. A Fourier spectral solver for the mid-range part
  9. Fourier-Chebyshev spectral solver for long-range part
  10. Grid discretization
  11. Complexity analysis
  12. Error analysis and parameter selection
  13. Approximation errors of the mid-range components
  14. Approximation errors of the long-ranged components
  15. Parameter selection
  16. ...and 16 more sections

Key Result

Proposition 2.1

The infinite summation in Eq. eq::Phi is absolutely convergent under the condition Eq. eq::chargeneutral.

Figures (6)

  • Figure 1: Relative error in (a) energy and (b) force due to the SOG decomposition, plotted against the number of truncated terms, $M$. The data are presented for various $b$, with other associated parameters detailed in Table \ref{['tabl:parameter']}. The dotted lines represent theoretical estimates as provided in Theorem \ref{['thm:SOG']}.
  • Figure 2: The relative error in evaluating $\Phi_{\text{SOG}}^{\rm mid}$ is plotted against (a) the window supports $\mathcal{P}$ and (b) the number of Fourier grids $I$ in each dimension. Data are shown for different type of window functions. The dashed lines in (a) and (b) represent the theoretical convergence rates provided by Theorems \ref{['thm::pad']} (associated with Eq. \ref{['eq::S']}) and \ref{['thm::trunFour']}, respectively. The range splitting factor is fixed as $\eta=0.294$, which corresponds to $m=8$.
  • Figure 3: The relative error in evaluating $\Phi_{\text{SOG}}^{\rm mid}$ as a function of (a) the range splitting factor $\eta$ and (b) the zero-padding factor $\lambda_z$. The dashed line represents the theoretical convergence rates presented in Theorem \ref{['thm::pad']}. In (b), the range splitting factor $\eta\approx0.39$, $0.68$, $1.18$, and $2.05$ corresponds to $\rm mid=10$, $14$, $18$, and $22$, respectively.
  • Figure 4: The relative error in evaluating $\Phi_{\text{SOG}}^{\rm long}$ as a function of (a) the number of Fourier modes $I^{\rm long}$ and (b) the number of Chebyshev basis $P$. Data are shown for different range splitting parameter $\eta$, where $\eta\approx0.11$, $0.15$, $0.19$, and $0.26$ correspond to $\rm mid=1$, $3$, $5$, and $7$, respectively. The dashed lines represent the theoretical convergence rate provided in Theorem \ref{['thm::Fcheb']}.
  • Figure 5: The relative error in computing $\Phi_{\text{SOG}}^{\rm long}$ against (a) window supports $\mathcal{P}$, (b) the number of Fourier grids $I^{\rm long}$ per dimension, (c) the number of Taylor expansion terms $Q$, and (d) the number of Chebyshev bases $P$. Data are shown for various aspect ratios $\gamma$. The dashed lines represent the theoretical convergence rate provided in this work.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • proof
  • Definition 3.2
  • ...and 19 more