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Abstract In this paper, we introduce a bilevel optimization framework for addressing inverse mean-field games, alongside an exploration of numerical methods tailored for this bilevel problem. The primary benefit of our bilevel formulation lies in maintaining the convexity of the objective function and the linearity of constraints in the forward problem. Our paper focuses on inverse mean-field games characterized by unknown obstacles and metrics. We show numerical stability for these two types of inverse problems. More importantly, we, for the first time, establish the identifiability of the inverse mean-field game with unknown obstacles via the solution of the resultant bilevel problem. The bilevel approach enables us to employ an alternating gradient-based optimization algorithm with a provable convergence guarantee. To validate the effectiveness of our methods in solving the inverse problems, we have designed comprehensive numerical experiments, providing empirical evidence of its efficacy. 

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.