Determining a stationary mean field game system from full/partial boundary measurement

TL;DR

This paper develops a method to uniquely identify parameters in a stationary mean field game system using boundary measurements, addressing challenges like nonlinear PDE coupling, partial data, and multiple parameter recovery.

Contribution

It introduces an enhanced higher-order linearization technique for inverse problems in MFG systems, capable of handling partial boundary data and multiple unknown parameters.

Findings

Successfully identifies multiple parameters from boundary data

Effective for both full and partial boundary measurements

Addresses nonlinear PDE coupling and probability constraints

Abstract

In this paper, we propose and study the utilization of the Dirichlet-to-Neumann (DN) map to uniquely identify the discount functions $r, k$ and cost function $F$ in a stationary mean field game (MFG) system. This study features several technical novelties that make it highly intriguing and challenging. Firstly, it involves a coupling of two nonlinear elliptic partial differential equations. Secondly, the simultaneous recovery of multiple parameters poses a significant implementation challenge. Thirdly, there is the probability measure constraint of the coupled equations to consider. Finally, the limited information available from partial boundary measurements adds another layer of complexity to the problem. Considering these challenges and problems, we present an enhanced higher-order linearization method to tackle the inverse problem related to the MFG system. Our proposed approach involves linearizing around a pair of zero solutions and fulfilling the probability measurement constraint by adjusting the positive input at the boundary. It is worth emphasizing that this technique is not only applicable for uniquely identifying multiple parameters using full-boundary measurements but also highly effective for utilizing partial-boundary measurements.