SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1486-A1509, January 2020.
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 a 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 and is often chosen to be $O(1)$. 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.

中文翻译：

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具有奇异核的多体系统的随机批量蒙特卡罗方法

SIAM科学计算杂志，第42卷，第3期，第A1486-A1509页，2020年1月。
我们提出了一种快速的潜在分裂马尔可夫链蒙特卡洛方法，该方法从与具有奇异相互作用核的粒子系统相对应的平衡分布（Gibbs度量）中进行采样，每步花费$ O（1）$的时间。我们将相互作用的潜力分解为两个部分：一个是远距离但平滑，而另一个是短距离但可能是奇异的。为了置换粒子，我们首先使用随机微分方程（在随机梯度Langevin动力学中通常使用）在平滑部分下使用随机微分方程（SDE）生成选定粒子。然后，我们使用短程部分进行大都会拒绝。与经典的Langevin动力学不同，我们仅在短时间内以随机批次运行SDE动力学，因此第一步的成本为$ O（p）$，其中$ p是批处理大小，通常选择为$ O（1）$。拒绝步骤的成本为$ O（1）$，因为使用的交互作用范围很短。我们证明了提出的随机批次蒙特卡洛方法的合理性，该方法在理论上和数值实验上都结合了随机批次和分裂策略。在为戴森·布朗运动和Lennard-Jones流体的典型示例提供可比的结果时，与经典的Metropolis-Hastings算法相比，我们的方法可以节省更多时间。