Finite-Time Convergence Guarantees for Time-Parallel Methods
This work addresses the problem of analyzing and ensuring convergence for time-parallel algorithms in chaotic systems, which is incremental as it builds on existing methods but introduces new theoretical criteria.
The paper tackled the limited convergence analysis of time-parallel methods like Parareal for nonlinear, chaotic systems by introducing a theoretical framework that provides finite-time convergence guarantees, showing the convergence factor scales as O(h^2) and enabling convergence to statistical attractors without pointwise accuracy in chaotic regimes.
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 $β$ 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.