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 size increases. 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 illustrates its flexibility for handling various mean-field variational problems.

中文翻译：

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动态均值场规划的快速近邻梯度法和收敛性分析

在本文中，我们提出了一种有效且灵活的算法来解决基于加速近端梯度法的动态平均场规划问题。除了该算法中易于实现的梯度下降步骤外，关键的投影步骤也变成了求解椭圆方程的步骤，该椭圆方程的求解可以通过常规方法有效地获得。通过对算法中使用的迭代进行归纳，我们从理论上表明，随着网格大小的增加，所提出的离散解收敛到底层的连续解。此外，我们将算法推广到均值博弈问题，并使用多级和多网格策略对其进行加速。我们进行了全面的数值实验，以确认所提出算法的收敛性分析，