The final version of “Error bound analysis of the stochastic parareal algorithm” by Kamran Pentland, Massimiliano Tamborrino, and myself has just appeared online in the SIAM Journal on Scientific Computing (SISC).
K. Pentland, M. Tamborrino, and T. J. Sullivan. “Error bound analysis of the stochastic parareal algorithm.” SIAM Journal on Scientific Computing 45(5):A2657–A2678, 2023.
Abstract. Stochastic Parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as Parareal. Similarly to Parareal, it combines fine- and coarse-grained solutions to an ODE using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.