Tight two-level convergence of Linear Parareal and MGRIT: Extensions and implications in practice

TL;DR

This paper advances the theoretical understanding of linear Parareal and MGRIT methods by providing simplified convergence analysis and practical bounds, validated on standard hyperbolic PDEs.

Contribution

It offers a new simplified analysis of convergence for linear Parareal and MGRIT, extending previous theory with practical implications and validation on PDEs.

Findings

Convergence bounds accurately predict practical convergence.

Error propagation norm relates to minimum singular value of a block bidiagonal operator.

Validated on advection-diffusion and wave equations.

Abstract

Two of the most popular parallel-in-time methods are Parareal and multigrid-reduction-in-time (MGRIT). Recently, a general convergence theory was developed in Southworth (2019) for linear two-level MGRIT/Parareal that provides necessary and sufficient conditions for convergence, with tight bounds on worst-case convergence factors. This paper starts by providing a new and simplified analysis of linear error and residual propagation of Parareal, wherein the norm of error or residual propagation is given by one over the minimum singular value of a certain block bidiagonal operator. New discussion is then provided on the resulting necessary and sufficient conditions for convergence that arise by appealing to block Toeplitz theory as in Southworth (2019). Practical applications of the theory are discussed, and the convergence bounds demonstrated to predict convergence in practice to high accuracy on two standard linear hyperbolic PDEs: the advection(-diffusion) equation, and the wave equation in first-order form.