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Optimizing Coarse Propagators in Parareal Algorithms

Bangti Jin, Qingle Lin, Zhi Zhou

TL;DR

This work tackles the critical bottleneck of parareal efficiency for parabolic problems by designing Optimized Coarse Propagators (OCP) through a stability- and accuracy-guided error-estimation framework. The authors formulate a parametric coarse-propagator stability function $R(\\lambda, \\boldsymbol{\\theta})$, enforce strict order-$q$ accuracy and strong stability, and solve a barrier-augmented optimization to minimize the per-mode convergence factor $\\kappa_c$. They prove that, for linear evolution, the parareal iterations converge when $\\kappa_c < 1$ and demonstrate near-constant, robust performance across a wide range of $J$ values. Numerical experiments on linear diffusion, Allen–Cahn, and Burgers models show substantial speedups and convergence improvements over conventional coarse propagators, with explicit OCP formulas provided. The approach offers a data-free, analytically grounded path to faster parallel-in-time solvers and suggests extensions to more challenging nonlinear, multiscale, and hyperbolic settings and to other parallel-in-time frameworks.

Abstract

The parareal algorithm represents an important class of parallel-in-time algorithms for solving evolution equations and has been widely applied in practice. To achieve effective speedup, the choice of the coarse propagator in the algorithm is vital. In this work, we investigate the use of {optimized} coarse propagators. Building upon the error estimation framework, we present a systematic procedure for constructing coarse propagators that enjoy desirable stability and consistent order. Additionally, we provide preliminary mathematical guarantees for the resulting parareal algorithm. Numerical experiments on a variety of settings, e.g., linear diffusion model, Allen-Cahn model, and viscous Burgers model, show that the optimizing procedure can significantly improve parallel efficiency when compared with the more ad hoc choice of some conventional and widely used coarse propagators.

Optimizing Coarse Propagators in Parareal Algorithms

TL;DR

This work tackles the critical bottleneck of parareal efficiency for parabolic problems by designing Optimized Coarse Propagators (OCP) through a stability- and accuracy-guided error-estimation framework. The authors formulate a parametric coarse-propagator stability function , enforce strict order- accuracy and strong stability, and solve a barrier-augmented optimization to minimize the per-mode convergence factor . They prove that, for linear evolution, the parareal iterations converge when and demonstrate near-constant, robust performance across a wide range of values. Numerical experiments on linear diffusion, Allen–Cahn, and Burgers models show substantial speedups and convergence improvements over conventional coarse propagators, with explicit OCP formulas provided. The approach offers a data-free, analytically grounded path to faster parallel-in-time solvers and suggests extensions to more challenging nonlinear, multiscale, and hyperbolic settings and to other parallel-in-time frameworks.

Abstract

The parareal algorithm represents an important class of parallel-in-time algorithms for solving evolution equations and has been widely applied in practice. To achieve effective speedup, the choice of the coarse propagator in the algorithm is vital. In this work, we investigate the use of {optimized} coarse propagators. Building upon the error estimation framework, we present a systematic procedure for constructing coarse propagators that enjoy desirable stability and consistent order. Additionally, we provide preliminary mathematical guarantees for the resulting parareal algorithm. Numerical experiments on a variety of settings, e.g., linear diffusion model, Allen-Cahn model, and viscous Burgers model, show that the optimizing procedure can significantly improve parallel efficiency when compared with the more ad hoc choice of some conventional and widely used coarse propagators.
Paper Structure (14 sections, 5 theorems, 61 equations, 11 figures, 3 tables, 2 algorithms)

This paper contains 14 sections, 5 theorems, 61 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Single step methods and parareal algorithm
  3. Single step solvers for parabolic equations
  4. Parareal algorithm
  5. Optimized coarse propagators
  6. Error estimation
  7. Optimizing coarse propagator
  8. Robustness with respect to the ratio $J$
  9. Numerical experiments and discussions
  10. Comparative study on convergence factor
  11. Illustration on model three problems
  12. Conclusion
  13. Additional plots
  14. Explicit formulas of OCPs

Key Result

Lemma 2.1

\newlabellem:conv-000 Let conditions (P1)-(P3) be fulfilled, $u(t)$ the solution to problem eqn:pde, and $u^n$ the solution to the scheme eqn:semi. Then there holds if $u_0\in {\mathcal{H}}$, $f^{(\ell)}\in C([0,T];\mathrm{Dom}({\mathcal{A}}^{q-\ell})$ with $0\le \ell\le q-1$ and $f^{(q)} \in L^1(0,T; {\mathcal{H}})$.

Figures (11)

  • Figure 1: Numerical bounds for the propagators, and convergence rate versus $J$.
  • Figure 1: The function $\phi (s)$ for three CPs and four FPs when $J\in\{2,4\}$.
  • Figure 1: The function $\phi (s)$ for three CPs and four FPs when $J\in\{16,32\}$.
  • Figure 1: The stability functions of the OCPs, for four FPs.
  • Figure 2: The convergence rate for Example \ref{['exam:diffusion']}(a) (smooth data). \newlabelfig:ex1 smooth0
  • ...and 6 more figures

Theorems & Definitions (17)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.2
  • Proof 1
  • Proposition 3.3
  • Proof 2
  • Remark 3.4
  • Theorem 3.5
  • ...and 7 more