Adaptive Stepsize Algorithms For Langevin Dynamics

We discuss the design of an invariant measure-preserving transformation for the numerical treatment of Langevin dynamics based on a rescaling of time. The goal is to sample from an invariant measure. By using an appropriate monitor function that characterizes the numerical difficulty of the problem as a function of the system's state, this method allows for adaptive stepsize reduction only when necessary, thereby facilitating efficient recovery of long-time behavior. We study both overdamped and underdamped Langevin dynamics and investigate how to incorporate an appropriate correction term into a numerical splitting scheme to ensure preservation of the invariant measure. Finally, we demonstrate the technique on several model systems, including a Bayesian sampling problem with a steep prior.