The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems. In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is the same as the kinetic Monte Carlo (KMC) method for the pairwise interacting particle system of size $N$. In the RBM-r algorithm, one randomly picks a small batch of size $p \ll N$, and only the particles in the picked batch interact among each other within the batch for a short time, where the weak interaction (of strength $\frac{1}{N-1}$) in the original system is replaced by a strong interaction (of strength $\frac{1}{p-1}$). Then one repeats this pick-interact process. This KMC algorithm dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step, and provides an unbiased approximation of the original force/velocity field of the interacting particle system. In this paper, We give a rigorous proof of this approximation with an explicit convergence rate. An improved rate is also obtained when there is no diffusion in the system. Notably, the techniques in our analysis can potentially be applied to study KMC for other systems, including the stochastic Ising spin system.

翻译：[Jin et al. J Comput Phys, 2020] 提出的随机批处理方法（RBM）是一种用于模拟相互作用粒子系统的高效算法。本文研究带放回的随机批处理方法（RBM-r），该方法对于规模为 $N$ 的成对相互作用粒子系统而言，等同于动力学蒙特卡洛（KMC）方法。在 RBM-r 算法中，每一时间步随机选取一个规模为 $p \ll N$ 的小批量，仅该批次内的粒子在短时间内发生批次内相互作用，其中原系统中的弱相互作用（强度为 $\frac{1}{N-1}$）被替换为强相互作用（强度为 $\frac{1}{p-1}$）。随后重复此选取-相互作用过程。该 KMC 算法将每个时间步的计算成本从 $O(N^2)$ 显著降低至 $O(pN)$，并为原始相互作用粒子系统的力场/速度场提供了无偏近似。本文通过显式收敛率对该近似给出了严格证明。当系统中不存在扩散项时，我们进一步得到了更优的收敛率。值得注意的是，本文分析所采用的技术可推广至研究其他系统的 KMC 方法，包括随机伊辛自旋系统。