Convergence Analysis of the Parareal Algorithm with Nonuniform Fine Time Grid

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-10-10 DOI:10.1137/23m1592481

Shu-Lin Wu, Tao Zhou

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2308-2330, October 2024.
Abstract. In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator as the fine propagator. Specifically, we consider two implementations of the coarse grid correction: the standard time-stepping mode and the parallel mode via the so-called diagonalization technique. For both cases, we prove that under certain conditions of the stability function of the fine propagator, the convergence factor of the parareal algorithm is not larger than that of the associated algorithm with a uniform fine time grid. Furthermore, we show that when such conditions are not satisfied, one can indeed observe degenerations of the convergence rate. The model that is used for performing the analysis is the Dahlquist test equation with nonnegative parameter, and the numerical results indicate that the theoretical results hold for nonlinear ODEs and linear ODEs where the coefficient matrix has complex eigenvalues.

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具有非均匀精细时间网格的 Parareal 算法的收敛性分析

SIAM 数值分析期刊》第 62 卷第 5 期第 2308-2330 页，2024 年 10 月。摘要本文研究了具有均匀粗时间网格和任意分布（非均匀）细时间网格（可在每次迭代中改变）的准噶尔算法的收敛特性。我们采用后向欧拉法作为粗传播器，采用一般的单步时间积分器作为细传播器。具体来说，我们考虑了粗网格修正的两种实现方式：标准时间步进模式和通过所谓对角化技术实现的并行模式。对于这两种情况，我们都证明了在精细传播器稳定函数的某些条件下，平行算法的收敛因子不会大于采用均匀精细时间网格的相关算法。此外，我们还证明，当不满足这些条件时，确实可以观察到收敛速率的退化。用于分析的模型是参数为非负的 Dahlquist 检验方程，数值结果表明，理论结果在非线性 ODE 和系数矩阵具有复特征值的线性 ODE 中均成立。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.