论文标题
用于多体系统的随机批量蒙特卡洛法
A random-batch Monte Carlo method for many-body systems with singular kernels
论文作者
论文摘要
我们提出了一种快速的潜在分裂马尔可夫链蒙特卡洛法，该方法的价格为$ o（1）$每一步，用于从平衡分布（Gibbs测量）中采样的时间，这些平衡分布（Gibbs测量）对应于具有单数相互作用内核的粒子系统。我们将相互作用的电势分解为两个部分，一个是远距离的，但很光滑，另一个是短距离，但可能是单数。为了置换粒子，我们首先使用随机批次的概念在平滑部分下使用随机微分方程（SDE）进化了一个粒子，这是随机梯度Langevin动力学中常用的。然后，我们使用短范围部分进行大都市的拒绝。与经典的langevin Dynamics不同，我们仅在短时间内运行随机批次的SDE动力学，因此第一步中的成本为$ O（p）$，其中$ p $是批处理大小。拒绝步骤的成本为$ o（1）$，因为所使用的交互作用短。我们证明了提出的随机批次蒙特卡洛方法是合理的，该方法在理论和数值实验中结合了随机批次和分裂策略。与经典的大都市杂货店算法相比，在典型的戴森·布朗尼运动和Lennard-Jones流体的典型示例中，我们的方法可以节省更多的时间。
We propose a fast potential splitting Markov Chain Monte Carlo method which costs $O(1)$ time each step for sampling from equilibrium distributions (Gibbs measures) corresponding to particle systems with singular interacting kernels. We decompose the interacting potential into two parts, one is of long range but is smooth, and the other one is of short range but may be singular. To displace a particle, we first evolve a selected particle using the stochastic differential equation (SDE) under the smooth part with the idea of random batches, as commonly used in stochastic gradient Langevin dynamics. Then, we use the short range part to do a Metropolis rejection. Different from the classical Langevin dynamics, we only run the SDE dynamics with random batch for a short duration of time so that the cost in the first step is $O(p)$, where $p$ is the batch size. The cost of the rejection step is $O(1)$ since the interaction used is of short range. We justify the proposed random-batch Monte Carlo method, which combines the random batch and splitting strategies, both in theory and with numerical experiments. While giving comparable results for typical examples of the Dyson Brownian motion and Lennard-Jones fluids, our method can save more time when compared to the classical Metropolis-Hastings algorithm.
