  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 underlying
interactions among the population based on a limited set of noisy partial
observations of the population dynamics under the limited aperture. Due to its
severe ill-posedness, obtaining a good quality reconstruction is very
difficult. Nonetheless, it is vital to recover the model parameters stably and
efficiently in order to uncover the underlying causes for population dynamics
for practical needs.
  Our work focuses on the simultaneous recovery of running cost and interaction
energy in the MFG equations from a \emph{finite number of boundary
measurements} of population profile and boundary movement. To achieve this
goal, we formalize the inverse problem as a constrained optimization problem of
a least squares residual functional under suitable norms. We then develop a
fast and robust operator splitting algorithm to solve the optimization using
techniques including harmonic extensions, three-operator splitting scheme, and
primal-dual hybrid gradient method. Numerical experiments illustrate the
effectiveness and robustness of the algorithm.
