  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.
