A multigrid perspective on the parallel full approximation scheme in space and time

TL;DR

This paper establishes a rigorous mathematical foundation for PFASST by showing its equivalence to multigrid-in-time methods and analyzes its behavior using local Fourier analysis on diffusive and advective problems.

Contribution

It introduces a multigrid perspective on PFASST, providing the first rigorous analysis framework for this complex parallel-in-time method.

Findings

PFASST can be described as a multigrid-in-time method under certain assumptions.

Local Fourier analysis offers insights into PFASST's convergence for diffusive and advective problems.

The analysis helps understand PFASST's performance and guides future improvements.

Abstract

For the numerical solution of time-dependent partial differential equations, time-parallel methods have recently shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the "Parallel Full Approximation Scheme in Space and Time" (PFASST). PFASST already shows promising results for many use cases and many more is work in progress. However, a solid and reliable mathematical foundation is still missing. We show that under certain assumptions the PFASST algorithm can be conveniently and rigorously described as a multigrid-in-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using block-wise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.