New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-08-23 DOI:10.1137/23m1584502

Martin J. Gander, Liu-Di Lu

引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2048-2070, August 2024.
Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet–Neumann and Neumann–Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.

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抛物线最优控制问题的新时域分解方法 I. Dirichlet-Neumann 和 Neumann-Dirichlet 算法Dirichlet-Neumann 和 Neumann-Dirichlet 算法

SIAM 数值分析期刊》第 62 卷第 4 期第 2048-2070 页，2024 年 8 月。摘要。我们提出了新的 Dirichlet-Neumann 和 Neumann-Dirichlet 时域分解算法，应用于无约束抛物线最优控制问题。在空间半具体化之后，我们使用拉格朗日乘数方法推导出一个耦合的前向后向最优系统，然后可以使用时域分解来求解该系统。由于优化系统的前向-后向结构，可以为 Dirichlet-Neumann 算法和 Neumann-Dirichlet 算法找到三种变体。我们分析了它们的收敛行为，并确定了每种算法的最佳松弛参数。我们的分析表明，最自然的算法实际上只是很好的平滑器，还有更好的选择能带来高效的求解器。我们通过数值实验来说明我们的分析。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.