PhD research topic - Elisabetta Carlini
Numerical Analysis of Partial Differential Equations
Mean Field Games, Fokker-Planck-Kolmogorov equations
Recently macroscopic models for large populations of interacting rational agents, known as Mean Filed Games model (MFG), has been proposed.
These games are typically expressed as a system of partial differential equations: a Hamilton–Jacobi equation coupled with a Fokker–Planck (FP) equation (see  for an extensive introduction). Analytical solutions of this problem are not known, so numerical techniques are necessary to approximate them.
Semi-Lagrangian (SL) schemes have been extensively applied to the context of hyperbolic and optimal control problems (see  and the references therein). The main advantage is that they are explicit and, at the same time, allow large time steps. They have been also shown to be adaptable to approximate degenerate elliptic problems.
In the case of first order and degenerate second order Mean Field Games problem, SL schemes have been proposed in [3, 4].
We are interested to construct efficient numerical approximation for Mean Filed games problem and Fokker Plank equations based on SL scheme; to develop a convergence analysis and error estimates for such new schemes and finally to validate them by numerical simulations.
In particular, some objectives of the research are
-to extend SL schemes for treating MFG models defined on bounded domains with different types of boundary conditions.
-to construct efficient algorithms for the approximation of the fully discrete SL scheme (manly based on Newton or quasi Newton type schemes)
-to develop high order numerical SL scheme. In particular, we seek to develop higher order SL scheme for Fokker Plank equations
-to develop a Semi-Lagrangian scheme for FP equations with jumps, and derive a fully discrete scheme for MFG with controlled jumps which have interesting application for instance in illiquid interbank market model.