The Random Batch Method proposed in our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] is not only a numerical method for interacting particle systems and its mean-field limit, but also can be viewed as a model of particle system in which particles interact, at discrete time, with randomly selected mini-batch of particles. In this paper we investigate the mean-field limit of this model as the number of particles $N \to \infty$. Unlike the classical mean field limit for interacting particle systems where the law of large numbers plays the role and the chaos is propagated to later times, the mean field limit now does not rely on the law of large numbers and chaos is imposed at every discrete time. Despite this, we will not only justify this mean-field limit (discrete in time) but will also show that the limit, as the discrete time interval $τ\to 0$, approaches to the solution of a nonlinear Fokker-Planck equation arising as the mean-field limit of the original interacting particle system in Wasserstein distance.
Random-batch method for multi-species stochastic interacting particle systems
提出了一种多种群相互作用粒子系统的随机批处理方法，推广了S。金湖，澳-地李和J- G. Liu [J. Comput. 400（2020），108877]。算法的思想是在每个时间步将粒子群随机分成小批，然后在批内演化每个粒子的相互作用，直到下一个时间步。这将计算成本降低了一个数量级，同时保持一定的精度。证明了误差过程的$L^2$误差表现为时间步长的平方根，在时间上是一致的，从而提供了格式的收敛性。数值效率进行了测试的一些例子，和数值模拟的意见动力学在一个分层的公司，由工人，经理和CEO，提出。
Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration
