A unified analysis framework for iterative parallel-in-time algorithms

TL;DR

This paper introduces a unified framework to analyze and compare various iterative parallel-in-time algorithms, providing new insights into their convergence properties and enabling easier development of future methods.

Contribution

It presents a common notation and convergence analysis for Parareal, PFASST, MGRIT, and STMG, allowing direct comparison and revealing their super-linear convergence.

Findings

All four methods converge super-linearly.

The generating function framework enables detailed comparison.

Numerical experiments support theoretical convergence results.

Abstract

Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations, and the convergence estimates that are available are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly, and also compare them numerically. The generating function framework provides further opportunities to explore and analyze existing and new methods.