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In this work, we consider a novel inverse problem in mean-field games (MFG).We aim to recover the MFG model parameters that govern the underlyinginteractions among the population based on a limited set of noisy partialobservations of the population dynamics under the limited aperture. Due to itssevere ill-posedness, obtaining a good quality reconstruction is verydifficult. Nonetheless, it is vital to recover the model parameters stably andefficiently in order to uncover the underlying causes for population dynamicsfor practical needs.  Our work focuses on the simultaneous recovery of running cost and interactionenergy in the MFG equations from a \emph{finite number of boundarymeasurements} of population profile and boundary movement. To achieve thisgoal, we formalize the inverse problem as a constrained optimization problem ofa least squares residual functional under suitable norms. We then develop afast and robust operator splitting algorithm to solve the optimization usingtechniques including harmonic extensions, three-operator splitting scheme, andprimal-dual hybrid gradient method. Numerical experiments illustrate theeffectiveness and robustness of the algorithm.

A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem