  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.
