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

TL;DR

GParareal introduces a Gaussian process-based time-parallel method for solving IVPs that improves convergence speed, handles challenging problems better, and leverages past solutions to accelerate computations.

Contribution

The paper presents GParareal, a novel Gaussian process emulation approach for time-parallel IVP solving, outperforming classic methods and utilizing legacy data for faster convergence.

Findings

GParareal converges in fewer iterations than parareal.

It can solve IVPs where parareal fails.

Utilizes legacy solutions to accelerate convergence.

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.