An Adaptive Parareal Algorithm: Application to the Simulation of Molecular Dynamics Trajectories

The aim of this article is to design parareal algorithms in the context of thermostated molecular dynamics. In its original setup, the fine and coarse propagators used in the parareal algorithm solve the same dynamics with different time-steps, with the goal of achieving accuracy in the limit of small time-step of the integrators involved. This is typically not useful in molecular dynamics, where one is interested in extremely long trajectories and where the time-step of the fine propagator is in practice chosen as large as possible, that is, close to the limit of stability of the numerical scheme. In this article, we consider a version of the parareal algorithm which is better suited to molecular dynamics simulations, and wherein the propagators involved use the same time-step while employing different potential energy landscapes to drive the dynamics. Although the parareal algorithm always converges, it suffers from various limitations in this context: intermediate blow-up of the trajectory (which makes it impossible to postprocess) may be observed; in certain cases the trajectory encounters undefined values before converging (the way the algorithm handles them might depend on the computer architecture); the algorithm does not provide any computational gain in terms of wall-clock time (compared to a standard sequential integration) in the limit of increasingly long time horizons. We highlight these issues with numerical experiments and provide some elements of theoretical analysis. We then present a modified version of the parareal algorithm wherein the algorithm adaptively divides the entire time horizon into smaller time-slabs where the aforementioned issues are circumvented. Using numerical experiments on toy examples, we show that the adaptive algorithm overcomes the various limitations of the standard parareal algorithm, thereby allowing for significantly improved gains.