A Fast Proximal Gradient Method and Convergence Analysis for Dynamic Mean Field Planning
A Fast Proximal Gradient Method and Convergence Analysis for Dynamic Mean Field Planning
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DOI：
10.1090/mcom/3879
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
Jiajia Yu;Rongjie Lai;Wuchen Li;S. Osher
中科院分区：
文献类型：
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作者：
Jiajia Yu;Rongjie Lai;Wuchen Li;S. Osher
本文提出了一种高效、灵活的基于加速近邻梯度法的动态平均场规划算法。在该算法中，除了易于实现的梯度下降步骤外，关键的投影步骤是求解椭圆型方程，该椭圆型方程的解可以用传统方法高效地获得。通过对算法中使用的迭代进行归纳，我们从理论上证明了当网格变得更细时，所提出的离散解收敛到基本的连续解。此外，我们将算法推广到平均场博弈问题，并使用多层和多网格策略进行加速。通过大量的数值实验验证了该算法的收敛分析，通过与现有算法的比较，验证了该算法的有效性和质量保持性，并说明了该算法处理各种平均场变分问题的灵活性。
In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid becomes finer. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrate its flexibility for handling various mean-field variational problems.