On the use of reduced basis methods to accelerate and stabilize the Parareal method

We propose a modified parallel-in-time - Parareal - multi-level time integration method which, in contrast to previous methods, uses a coarse solver based on the information from the fine solver at each iteration through the construction of a reduced model. This approach is demonstrated to offer two substantial advantages: it accelerates convergence of the original parareal method and we furthermore demonstrate that the reduced basis stabilizes the parareal method for purely advective problems where instabilities are known to arise. When combined with empirical interpolation techniques (EIM), we discuss the use of this approach to solve both linear and nonlinear problems and highlight the minimal changes required to utilize this algorithm to accelerate existing implementations. We illustrate advantages through the algorithmic design, through analysis of stability, convergence, and computational complexity, and through several numerical examples.