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

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