We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing N-independent time steps and also capture, in the N → ∞ limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems. 

金石, 现为上海交通大学自然科学研究院院长。先后获北京大学学士学位, 美国亚利桑那大学博士学位, 历任美国纽约大学库朗数学研究所博士后, 美国佐治亚理工学院助理教授, 副教授, 美国威斯康星大学(麦迪逊)正教授, 数学系系主任, Vilas杰出成就教授, 上海交通大学数学系讲席教授, 系主任。他曾获得冯康科学计算奖, 国家自然科学基金杰出青年基金(海外), 教育部长江讲座教授(清华大学), 国际华人数学家大会晨兴数学银奖。他是美国数学会(AMS)首批会士, 工业与应用数学学会(SIAM)会士, 及2018年国际数学家大会邀请报告人。他在计算流体力学, 动理学方程, 双曲型守恒律方程, 高频波计算, 计算物理和多尺度问题的计算方法等领域均做出了重要贡献。金石教授是International Press杂志CMS的创刊及现任主编, 并担任过SISC, SIAP等国际著名计算和应用数学杂志的编委。