On parareal algorithms for semilinear parabolic Stochastic PDEs

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a `parallel in real time' implementation. In this work, the fine integrator is given by the exponential Euler scheme. Two choices for the coarse integrator are considered: the linear implicit Euler scheme, and the exponential Euler scheme. The influence on the performance of the parareal algorithm, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations, is investigated, with theoretical analysis results and with extensive numerical experiments.