A Fast Iterative PDE-Based Algorithm for Feedback Controls of Nonsmooth Mean-Field Control Problems

IF 3 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Scientific Computing Pub Date : 2024-08-20 DOI:10.1137/21m1441158

Christoph Reisinger, Wolfgang Stockinger, Yufei Zhang

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Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2737-A2773, August 2024.
Abstract. We propose a PDE-based accelerated gradient algorithm for optimal feedback controls of McKean–Vlasov dynamics that involve mean-field interactions both in the state and action. The method exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is approximated via a particle system, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. We present exhaustive numerical experiments for low- and high-dimensional mean-field control problems, including sparse stabilization of stochastic Cucker–Smale models, which reveal that our algorithm captures important structures of the optimal feedback control and achieves a robust performance with respect to parameter perturbation.

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基于 PDE 的非光滑平均场控制问题反馈控制快速迭代算法

SIAM 科学计算期刊》，第 46 卷第 4 期，第 A2737-A2773 页，2024 年 8 月。摘要我们提出了一种基于 PDE 的加速梯度算法，用于麦金-弗拉索夫（McKean-Vlasov）动力学的最优反馈控制。该方法利用前向-后向分裂方法，根据平滑代价的梯度、非平滑代价的近似图和动态更新的动量参数迭代改进近似控制。每一步都通过粒子系统对状态动态进行近似，并通过非局部线性 PDE 耦合系统评估所需梯度。后者根据状态维度，通过有限差分近似或基于神经网络的残差近似来求解。我们针对低维和高维均场控制问题（包括随机 Cucker-Smale 模型的稀疏稳定）进行了详尽的数值实验，结果表明我们的算法捕捉到了最优反馈控制的重要结构，并在参数扰动方面实现了稳健的性能。

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 Scientific Computing 数学-应用数学

CiteScore

5.50

自引率

3.20%

发文量

209

审稿时长

1 months

期刊介绍： The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems. SISC papers are classified into three categories: 1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms. 2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist. 3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.