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 ordinary differential equation (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.

翻译：随机帕拉雷尔（SParareal）是流行的并行时间算法帕拉雷尔的一种概率变体。与帕拉雷尔类似，它通过预测校正（PC）格式结合常微分方程（ODE）的粗粒度和细粒度解。关键区别在于，PC中加入了精心选择的随机扰动，以加速ODE随机解的定位。本文针对不同类型的扰动，推导了应用于非线性ODE系统的SParareal的超线性与线性均方误差界。我们在线性ODE系统与标量非线性ODE上通过数值方法验证了这些误差界，结果表明理论与数值计算具有良好的一致性。