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：我们讨论了基于时间重新标度的数值处理朗之万动力学时，用于保持不变测度的变换动力学设计，其目标是从不变测度中进行采样。给定一个适当的监控函数（该函数根据系统状态表征问题的数值难度），此方法允许仅在必要时减小步长，从而促进长时间行为的有效恢复。我们研究了过阻尼和欠阻尼朗之万动力学。我们探讨了如何将确保不变测度保持的适当修正项纳入数值分裂格式中。最后，我们展示了该技术在多个模型系统中的应用，包括一个具有陡峭先验的贝叶斯采样问题。