Quasi-2D Coulomb systems are of fundamental importance and have attracted much attention in many areas nowadays. Their reduced symmetry gives rise to interesting collective behaviors, but also brings great challenges for particle-based simulations. Here, we propose a novel algorithm framework to address the $\mathcal O(N^2)$ simulation complexity associated with the long-range nature of Coulomb interactions. First, we introduce an efficient Sum-of-Exponentials (SOE) approximation for the long-range kernel associated with Ewald splitting, achieving uniform convergence in terms of inter-particle distance, which reduces the complexity to $\mathcal{O}(N^{7/5})$. We then introduce a random batch sampling method in the periodic dimensions, the stochastic approximation is proven to be both unbiased and with reduced variance via a tailored importance sampling strategy, further reducing the computational cost to $\mathcal{O}(N)$. The performance of our algorithm is demonstrated via varies numerical examples. Notably, it achieves a speedup of $2\sim 3$ orders of magnitude comparing with Ewald2D method, enabling molecular dynamics (MD) simulations with up to $10^6$ particles on a single core. The present approach is therefore well-suited for large-scale particle-based simulations of Coulomb systems under confinement, making it possible to investigate the role of Coulomb interaction in many practical situations.

翻译：准二维库仑系统具有基础重要性，近年来在众多领域备受关注。其降维对称性虽能引发有趣的集体行为，但也给基于粒子的模拟带来巨大挑战。针对库仑相互作用长程特性导致的 $\mathcal O(N^2)$ 模拟复杂度，本文提出一种新型算法框架。首先，我们引入一种高效的指数和近似方法处理埃瓦尔德分裂中的长程核函数，该方法在粒子间距上实现均匀收敛，将复杂度降至 $\mathcal{O}(N^{7/5})$。随后，我们在周期性维度中引入随机批采样方法，通过定制的重要性采样策略，证明该随机近似既无偏且方差降低，进一步将计算成本压缩至 $\mathcal{O}(N)$。通过多个数值算例验证了算法性能，特别是与Ewald2D方法相比可实现 $2\sim 3$ 个数量级的加速，使得单核上可模拟高达 $10^6$ 粒子的分子动力学模拟。该方案特别适用于受限条件下库仑系统的大规模粒子模拟，为探究实际场景中库仑相互作用的作用机制提供了可能。