We discuss the design of an invariant measure-preserving transformed dynamics for the numerical treatment of Langevin dynamics based on rescaling of time, with the goal of sampling from an invariant measure. Given an appropriate monitor function which characterizes the numerical difficulty of the problem as a function of the state of the system, this method allows the stepsizes to be reduced only when necessary, facilitating efficient recovery of long-time behavior. We study both the overdamped and underdamped Langevin dynamics. We investigate how an appropriate correction term that ensures preservation of the invariant measure should be incorporated into a numerical splitting scheme. Finally, we demonstrate the use of the technique in several model systems, including a Bayesian sampling problem with a steep prior.

翻译：本文讨论了一种基于时间尺度重标度的不变测度保持变换动力学设计方法，用于Langevin动力学的数值处理，目标是从不变测度中采样。通过选取能够表征问题数值难度（作为系统状态函数）的适当监控函数，该方法仅在必要时减小步长，从而有效恢复长时间行为。我们研究了过阻尼和欠阻尼Langevin动力学，探讨如何在数值分裂方案中引入确保不变测度保持的适当修正项。最后，我们在多个模型系统中演示了该技术的应用，包括一个具有陡峭先验的贝叶斯采样问题。