Convergence Analysis of a Mixed Precision Parareal Algorithm

We propose and analyze a mixed precision parareal algorithm that uses for the fine propagator \(\mathcal{F}\) and the coarse propagator \(\mathcal{G}\) a high precision \({\epsilon }_h\) and a low \({\epsilon }_l (\gg{\epsilon }_h)\) , 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 \(\mathcal{G}\) and \(\mathcal{F}\) . Low precision is also beneficial to reduce communication and memory costs, since we have to move and store fewer bits. Let \(\widehat{\rho }\) and \(\rho\) 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 \(N_t\) . Numerical results indicate that the converged solution of the mixed precision parareal algorithm attains the desired high precision \({\epsilon }_h\) , 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 \(\mathcal{G}\) and \(N_t\) is very large.