论文标题
粒子系统事件链蒙特卡洛算法的PDMP表征
PDMP characterisation of event-chain Monte Carlo algorithms for particle systems
论文作者
论文摘要
当使用传统的Hasting-Metropolis可逆方案时，蒙特卡罗模拟了诸如硬球或软球等颗粒系统的系统，可以显示出较长的相转换时间。然后开发出称为事件链蒙特卡洛的有效算法以达到必要的加速度。它们基于非可逆连续时间马尔可夫流程。对于离散的时间方案而言，不能做到这类方案的不变性和实践性，并且缺乏理论框架，这阻碍了ECMC算法对更复杂的系统或过程的概括。在这项工作中，我们将ECMC中生成的Markov过程描述为分段确定性马尔可夫过程。它首先允许我们提出更多的一般方案，例如有关方向茶点。然后，我们证明了正确的固定分布的不变性。最后，我们显示了软性和硬球系统中过程的恐怖性，后者的密度条件。
Monte Carlo simulations of systems of particles such as hard spheres or soft spheres with singular kernels can display around a phase transition prohibitively long convergence times when using traditional Hasting-Metropolis reversible schemes. Efficient algorithms known as event-chain Monte Carlo were then developed to reach necessary accelerations. They are based on non-reversible continuous-time Markov processes. Proving invariance and ergodicity for such schemes cannot be done as for discrete-time schemes and a theoretical framework to do so was lacking, impeding the generalisation of ECMC algorithms to more sophisticated systems or processes. In this work, we characterize the Markov processes generated in ECMC as piecewise deterministic Markov processes. It first allows us to propose more general schemes, for instance regarding the direction refreshment. We then prove the invariance of the correct stationary distribution. Finally, we show the ergodicity of the processes in soft- and hard-sphere systems, with a density condition for the latter.
