SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 746-768, January 2021.
We consider in this work the convergence of the random batch method proposed in our previous work [Jin et al., J. Comput. Phys., 400(2020), 108877] for interacting particles to the case of disparate species and weights. We show that the strong error is of $O(\sqrt{\tau})$ while the weak error is of $O(\tau)$ where $\tau$ is the time step between two random divisions of batches. Both types of convergence are uniform in $N$, the number of particles. The proof of strong convergence follows closely the proof in [Jin et al., J. Comput. Phys., 400(2020), 108877] for indistinguishable particles, but there are still some differences: Since there is no exchangeability now, we have to use a certain weighted average of the errors; some refined auxiliary lemmas have to be proved compared with our previous work. To show that the weak convergence of empirical measure is uniform in $N$, certain sharp estimates for the derivatives of the backward equations have been used. The weak convergence analysis is also illustrating for the convergence of the Random Batch Method for $N$-body Liouville equations.

中文翻译：

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具有不同种类和重量的粒子相互作用的随机批处理方法的收敛性

SIAM数值分析学报，第59卷，第2期，第746-768页，2021年1月。
我们在这项工作中考虑了先前工作中提出的随机批处理方法的收敛性[Jin等人，J。Comput。Phys。，400（2020），108877]，以使粒子相互作用以实现不同种类和不同重量的情况。我们显示，强误差为$ O（\ sqrt {\ tau}）$，而弱误差为$ O（\ tau）$，其中$ \ tau $是批次的两个随机划分之间的时间步长。两种类型的收敛在粒子数量$ N $中是一致的。强收敛的证明紧随[Jin et al。，J. Comput。Phys。，400（2020），108877]对于不可区分的粒子，但是仍然存在一些差异：由于现在没有可交换性，我们必须使用一定的误差加权平均值；与我们以前的工作相比，必须证明一些改进的辅助引理。为了表明经验测度的弱收敛在$ N $中是均匀的，对反向方程的导数使用了某些清晰的估计。弱收敛分析还说明了$ N $体Liouville方程的随机批次方法的收敛性。