Convergence Analysis of a Mixed Precision Parareal Algorithm
Xiaoqiang Yue, Zhiyong Wang, Shu‐Lin Wu · SIAM Journal on Scientific Computing · 2023
Abstract. We propose and analyze a mixed precision parareal algorithm that uses for the fine propagator [Formula: see text] and the coarse propagator [Formula: see text] a high precision [Formula: see text] and a low [Formula: see text], 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 [Formula: see text] and [Formula: see text]. Low precision is also beneficial to reduce communication and memory costs, since we have to move and store fewer bits. Let [Formula: see text] and [Formula: see text] 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 [Formula: see text]. Numerical results indicate that the converged solution of the mixed precision parareal algorithm attains the desired high precision [Formula: see text], 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 [Formula: see text] and [Formula: see text] is very large.