Asynchronous parareal time discretization for partial differential equations

TL;DR

This paper introduces an asynchronous iteration method derived from the Parareal scheme for time-parallel PDE solutions, demonstrating comparable convergence properties and potential performance gains through analysis and experiments.

Contribution

It proposes a novel asynchronous time-parallel approach based on Parareal, with convergence analysis and evidence of performance improvements on high-performance computing platforms.

Findings

Asynchronous Parareal has similar convergence conditions to classical Parareal.

The method shows potential for significant speedup in PDE simulations.

Experimental results confirm the theoretical performance benefits.

Abstract

Asynchronous iterations are more and more investigated for both scaling and fault-resilience purpose on high performance computing platforms. While so far, they have been exclusively applied within space domain decomposition frameworks, this paper advocates a novel application direction targeting time-decomposed time-parallel approaches. Specifically, an asynchronous iterative model is derived from the Parareal scheme, for which convergence and speedup analysis are then conducted. It turned out that Parareal and async-Parareal feature very close convergence conditions, asymptotically equivalent, including the finite-time termination property. Based on a computational cost model aware of unsteady communication delays, our speedup analysis shows the potential performance gain from asynchronous iterations, which is confirmed by some experimental case of heat evolution on a homogeneous supercomputer. This primary work clearly suggests possible further benefits from asynchronous iterations.