论文标题
将粒子与不同物种相互作用的随机批处理方法的收敛性
Convergence of Random Batch Method for interacting particles with disparate species and weights
论文作者
论文摘要
我们在这项工作中考虑了我们先前工作中提出的随机批处理方法的收敛[Jin等，J。Comput。 Phys。，400（1），2020]，用于与不同物种和重量的情况相互作用。我们表明，强误差为$ O（\sqrtτ）$，而弱错误为$ o（τ）$，其中$τ$是两个随机批处理之间的时间步长。两种类型的收敛性在$ n $（粒子数量）中都是均匀的。强烈收敛的证据紧随[Jin等，J。Comput。物理学，400（1），2020年]对于难以区分的粒子，但仍然存在一些差异：由于现在没有交换性，我们必须使用一定的加权误差平均值；与我们以前的工作相比，必须证明一些精致的辅助引理。为了表明经验度量的弱收敛性在$ n $中是统一的，已经使用了向后方程的衍生物的某些尖锐估计。弱收敛分析也说明了$ n $ boby liouville方程的随机批次方法的收敛。
We consider in this work the convergence of Random Batch Method proposed in our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] for interacting particles to the case of disparate species and weights. We show that the strong error is of $O(\sqrtτ)$ while the weak error is of $O(τ)$ where $τ$ 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(1), 2020] 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 Random Batch Method for $N$-body Liouville equations.
