具有不同种类和权重的相互作用粒子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
摘要：
在这项工作中，我们考虑了我们以前的工作中提出的随机批处理方法的收敛性[Jin等人，J. Comput.物理、400（1），2020]的情况下，不同的物种和重量的相互作用的粒子。我们表明，强误差是$O(\sqrtτ)$，而弱误差是$O(τ)$，其中$τ$是两个随机划分的批次之间的时间步长。这两种类型的收敛在粒子数$N$中是一致的。强收敛的证明紧密遵循[Jin et al.，J. Comput.物理、400（1），2020]，但仍然有一些区别：由于现在没有交换，我们必须使用一定的加权平均的误差;一些改进的辅助引理必须证明与我们以前的工作相比。为了证明$N$中经验测度的弱收敛是一致的，对向后方程的导数进行了某些精确估计。弱收敛性分析也说明了随机批处理方法对$N$ -体Liouville方程的收敛性。
英文摘要：
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.
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