On The Mean Field Games System With the Lateral Cauchy Data via Carleman Estimates

We are concerned with the quantitative study of the second order Mean Field Game (MFG) system in a bounded domain with the lateral Cauchy data being prescribed. That is, both Dirichlet and Neumann boundary data of the MFG solutions are given. We derive a sharp H\"older stability estimate in quantifying the difference of the MFG solutions in terms of the corresponding difference of their lateral Cauchy data. This stability estimate implies uniqueness. That is, the lateral Cauchy data uniquely determine the MFG solutions. Some applications of practical interest are discussed. The main technical developments are two new Carleman estimates for the MFG system

Numerical Solution of the 3-D Travel Time Tomography Problem

The first numerical solution of the 3-D travel time tomography problem is presented. The globally convergent convexification numerical method is applied

Stability Estimates for Some Parabolic Inverse Problems With the Final Overdetermination via a New Carleman Estimate

This paper is about Holder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time {t=T}. In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem

Convexification Numerical Method for the Retrospective Problem of Mean Field Games

The convexification numerical method with the rigorously established global convergence property is constructed for a problem for the Mean Field Games System of the second order. This is the problem of the retrospective analysis of a game of infinitely many rational players. In addition to traditional initial and terminal conditions, one extra terminal condition is assumed to be known. Carleman estimates and a Carleman Weight Function play the key role. Numerical experiments demonstrate a good performance for complicated functions. Various versions of the convexification have been actively used by this research team for a number of years to numerically solve coefficient inverse problems

Convexification for a Coefficient Inverse Problem of Mean Field Games

The globally convergent convexification numerical method is constructed for a Coefficient Inverse Problem for the Mean Field Games System. A coefficient characterizing the global interaction term is recovered from the single measurement data. In particular, a new Carleman estimate for the Volterra integral operator is proven, and it stronger than the previously known one. Numerical results demonstrate accurate reconstructions from noisy data