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

TL;DR

This paper introduces a numerical algorithm to stably recover parameters in mean-field game models from limited, noisy boundary data, addressing the severe ill-posedness of the inverse problem.

Contribution

It proposes a novel constrained optimization approach combined with an operator splitting algorithm to efficiently solve the inverse problem in mean-field games from partial boundary measurements.

Findings

Algorithm effectively recovers model parameters from noisy data.

Numerical experiments demonstrate robustness and efficiency.

Method addresses ill-posedness in inverse mean-field game problems.

Abstract

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.