The first order interacting particle systems are ubiquitous. For example, they can be viewed as the overdamped Langevin equations. We first introduce a random algorithm, called Random Batch Method (RBM), for simulating first order systems.
The algorithms are motivated by the mini-batch idea in machine learning and statistics. Under some special conditions, we show the convergence of RBMs for the first marginal distribution under Wasserstein distance. Compared with traditional tree code and fast multipole expansion algorithms, RBM works for kernels that do not necessarily decay. We then apply RBM to Stein Variational Gradient Descent, a recent algorithm in statistics and machine learning, to obtain an efficient sampling method. This talk is based on joint works with Shi Jin (Shanghai Jiao Tong University), Jian-Guo Liu (Duke University), Jianfeng Lu (Duke University） and Zibu Liu (Duke University).