Stochastic parareal: an application of probabilistic methods to time-parallelisation

Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a solution in a fixed number of iterations using a predictor-corrector, stopping once a tolerance is met. This iterative process combines solutions located by inexpensive (coarse resolution) and expensive (fine resolution) numerical integrators. In this paper, we introduce a stochastic parareal algorithm aimed at accelerating the convergence of the deterministic parareal algorithm. Instead of providing the predictor-corrector with a deterministically located set of initial values, the stochastic algorithm samples initial values from dynamically varying probability distributions in each temporal sub-interval. All samples are then propagated in parallel using the expensive integrator. The set of sampled initial values yielding the most continuous (smoothest) trajectory across consecutive sub-intervals are fed into the predictor-corrector, converging in fewer iterations than the deterministic algorithm with a given probability. The performance of the stochastic algorithm, implemented using various probability distributions, is illustrated on low-dimensional systems of ordinary differential equations (ODEs). We provide numerical evidence that when the number of sampled initial values is large enough, stochastic parareal converges almost certainly in fewer iterations than the deterministic algorithm, maintaining solution accuracy. Given its stochastic nature, we also highlight that multiple simulations of stochastic parareal return a distribution of solutions that can represent a measure of uncertainty over the ODE solution.