Convergence Analysis of a Mixed Precision Parareal Algorithm

We propose and analyze a mixed precision parareal algorithm that uses for the fine propagator and the coarse propagator a high precision and a low , respectively. This paradigm potentially provides faster and more energy efficient coarse grid correction for parareal, compared to the original paradigm, which uses a uniform precision for both and . Low precision is also beneficial to reduce communication and memory costs, since we have to move and store fewer bits. Let and be, respectively, the decaying rate of the error of the parareal algorithm in the mixed and uniform precision modes. We perform a convergence analysis for the mixed precision parareal algorithm. The derived convergence rates are dependent on the precision of the coarse propagator and the number of coarse time steps . Numerical results indicate that the converged solution of the mixed precision parareal algorithm attains the desired high precision , while degeneration of the convergence rate is indeed observed in some worst case scenario, such as the situation that we use a half precision for and is very large.