Finite-Time Convergence Guarantees for Time-Parallel Methods

TL;DR

This paper develops a finite-time convergence analysis for time-parallel algorithms like Parareal, especially for nonlinear and chaotic systems, providing explicit estimates and a chaos-aware convergence criterion.

Contribution

Introduces a contraction mapping framework for finite-time convergence analysis of time-parallel methods, with explicit estimates and a new chaos-aware convergence criterion.

Findings

Finite-time convergence guarantees derived for nonlinear problems.

Explicit convergence factor estimates scale as $ ext{O}(h^2)$ with grid refinement.

Chaos-aware criterion enables convergence to statistical attractors despite chaos.

Abstract

Time-parallel algorithms, such as Parareal, are well-understood for linear problems, but their convergence analysis for nonlinear, chaotic systems remains limited. This paper introduces a new theoretical framework for analysing time-decomposition methods as contraction mappings that converge in a finite number of iterations. We derive a finite-time guarantee linking the initial error, convergence rate, and iteration count, defined via a geometric outer--inner-ball condition. We apply this framework to Parareal, deriving explicit estimates for the convergence factor $\beta$ on nonlinear problems and showing it scales as $\mathcal{O}(h^2)$ when the macroscopic time grid is uniformly refined. Further, we address the failure of standard convergence criteria in chaotic regimes by introducing a proximity function. This chaos-aware criterion weighs solution discontinuities by the system's Lyapunov exponent (or the solver's Lipschitz constant), allowing the algorithm to converge to the correct statistical attractor without enforcing futile pointwise accuracy on divergent trajectories. Numerical experiments on the Logistic, Lorenz, and Lorenz-96 systems demonstrate that this approach decouples the iteration count from the total simulation time. By isolating the intrinsic mathematical bounds from hardware-dependent overheads, we establish that the method is strictly algorithmically scalable.