ON PARAREAL ALGORITHMS FOR SEMILINEAR PARABOLIC STOCHASTIC PDEs

摘要

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations to achieve a "parallel-in-real-time" implementation. These algorithms proceed as two-level integrators, with the fine integrator being given by the exponential Euler scheme in this work. Two choices for the coarse integrator are considered: the linear implicit Euler scheme and the exponential Euler scheme. It is proved that as the number of iterations increases, the order of convergence is limited by the regularity of the noise, whereas for the exponential Euler case, the order of convergence always increases. The influences on the performance of the parareal algorithms, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations are also illustrated by extensive numerical experiments.