Although convergence of the Parareal and multigrid‐reduction‐in‐time (MGRIT) parallel‐in‐time algorithms is well studied, results on their optimality is limited. Appealing to recently derived tight bounds of two‐level Parareal and MGRIT convergence, this article proves (or disproves) h_x‐ and h_t‐independent convergence of two‐level Parareal and MGRIT, for linear problems of the form ${\mathbf{u}}^{\prime }(t)+ℒ\mathbf{u}(t)=f(t)$, where $ℒ$ is symmetric positive definite and Runge‐Kutta time integration is used. The theory presented in this article also encompasses analysis of some modified Parareal algorithms, such as the θ‐Parareal method, and shows that not all Runge‐Kutta schemes are equal from the perspective of parallel‐in‐time. Some schemes, particularly L‐stable methods, offer significantly better convergence than others as they are guaranteed to converge rapidly at both limits of small and large h_tξ, where ξ denotes an eigenvalue of $ℒ$ and h_t time‐step size. On the other hand, some schemes do not obtain h‐optimal convergence, and two‐level convergence is restricted to certain regimes. In certain cases, an $𝒪(1)$ factor change in time step h_t or coarsening factor k can be the difference between convergence factors ρ≈0.02 and divergence! The analysis is extended to skew‐symmetric operators as well, which cannot obtain h‐independent convergence and, in fact, will generally not converge for a sufficiently large number of time steps. Numerical results confirm the analysis in practice and emphasize the importance of a priori analysis in choosing an effective coarse‐grid scheme and coarsening factor. A Mathematica notebook to perform a priori two‐grid analysis is available at https://github.com/XBraid/xbraid‐convergence‐est.

中文翻译：

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使用Runge-Kutta时间积分的“最佳”与h无关的Parareal和多重网格实时收敛

尽管对Parareal和Multigrid-in-time-time（MGRIT）并行时间算法的收敛性进行了很好的研究，但其最优性结果有限。呼吁最近推导的两层Parareal和MGRIT收敛的紧边界，本文证明（或证明）两层Parareal和MGRIT的h _x和h _t独立收敛，用于形式的线性问题${\mathbf{ü}}^{\prime }（Ť）+ℒ\mathbf{ü}（Ť）=F（Ť）$， 在哪里 $ℒ$是对称正定的，并使用了Runge-Kutta时间积分。本文介绍的理论还包括对某些修改过的Parareal算法（例如θ- Paraareal方法）的分析，并表明从并行时间的角度来看，并非所有Runge-Kutta方案都是相等的。一些方案中，特别是L-稳定方法，提供了比其他显著更好的收敛，因为它们迅速地在小的和大两个界限保证收敛ħ_吨ξ，其中ξ表示的特征值$ℒ$和ħ_吨时间步长。另一方面，某些方案没有获得h最优收敛，并且两级收敛仅限于某些方案。在某些情况下，$𝒪（\mathrm{1个}）$在时间步长因子变化ħ_吨或粗大化因子ķ可以是收敛因子之间的差ρ ≈0.02和发散！该分析也扩展到了偏对称算子，该算子无法获得独立于h的收敛，实际上，通常不会在足够多的时间步长上收敛。数值结果证实了该分析在实践中的重要性，并强调了先验分析在选择有效的粗网格方案和粗化因子中的重要性。https://github.com/XBraid/xbraid-convergence-est上提供了Mathematica笔记本，用于执行先验两网格分析。