GParareal: A time-parallel ODE solver using Gaussian process emulation

Abstract

Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, "predicting" the solution serially using a cheap (coarse) solver and "correcting" these values using an expensive (fine) solver that runs in parallel on a number of temporal subintervals. In this work, we propose a time-parallel algorithm (GParareal) that solves IVPs by modelling the correction term, i.e. the difference between fine and coarse solutions, using a Gaussian process emulator. This approach compares favourably with the classic parareal algorithm and we demonstrate, on a number of IVPs, that GParareal can converge in fewer iterations than parareal, leading to an increase in parallel speed-up. GParareal also manages to locate solutions to certain IVPs where parareal fails and has the additional advantage of being able to use archives of legacy solutions, e.g. solutions from prior runs of the IVP for different initial conditions, to further accelerate convergence of the method -something that existing time-parallel methods do not do.

Stochastic Parareal: An Application of Probabilistic Methods to Time-Parallelization

“…Finally, we would aim to make use of the whole ensemble of fine propagated trajectories rather than using only one -avoiding the waste of valuable information about the solution at the coarse and fine resolutions. An approach in this direction has recently been proposed [33], making use of ideas from the field of probabilistic numerics to adopt a more Bayesian approach to this problem.…”

Error Bound Analysis of the Stochastic Parareal Algorithm

“…Others, aimed at reducing the number iterations in parareal, have also emerged. Approaches include learning the correction term in the PC using Gaussian process emulators (Pentland et al, 2022b) and building the coarse solver using a Krylov subspace of the set of PC solutions (Gander and Petcu, 2008).…”