tba
Dienstag, 3.6.25, 14:15-15:15, Seminarraum 226, HH10
Higher gradient integrability for BD-minimizers
Dienstag, 17.6.25, 14:15-15:45, Seminarraum 226, HH10
Stabilized finite element approximation of Mean Field Game Partial Differential Inclusions
Dienstag, 8.7.25, 14:00-15:00, Seminarraum 226, HH10
In 2006, Lasry and Lions introduced Mean Field Games (MFG) as models for Nash equilibria of differential games of stochastic optimal control that involve a very large number of players. MFG equilibria are typically described by nonlinear forward-backward systems where a Hamilton--Jacobi--Bellman (HJB) equation, satisfied by the value function of a representative player, is coupled with a Kolmogorov--Fokker--Planck (KFP) equation for the player density. A range of applications of MFG have since appeared, such as in finance, economics, pedestrian dynamics, optimal transport, smart grid management, and traffic flow.
A typical assumption in the literature on MFG asserts the differentiability of the Hamiltonian in the HJB equation of the MFG system, which leads to an unambiguous advection in the KFP equation of the system that is based on the derivative of the Hamiltonian. However, it is known from applications of optimal control that optimal controls are not always unique, such as being bang-bang for example, thereby leading to HJB equations with convex but non-differentiable Hamiltonians.
In this talk I will introduce an extension of the MFG system to the case where the optimal controls for players are non-unique and the Hamiltonian of the MFG system is non-differentiable, where the KFP equation is generalized to a partial differential inclusion (PDI) based on selections of the subdifferential of the Hamiltonian. This results in a new class of models called Mean Field Game Partial Differential Inclusions (MFG PDI). I will introduce a monotone stabilized finite element discretisation of the weak formulation of time-dependent MFG PDI with Lipschitz, convex but (possibly) non-differentiable Hamiltonians. I will present theorems on the well-posedness of the discretisation and its strong convergence to weak solutions of the MFG PDI in the joint limit as the time-step and mesh-size vanish. The talk will be concluded with discussion of a numerical experiment for an MFG PDI with non-smooth solution. This talk is based on my doctoral thesis which was supervised by Dr. Iain Smears.
tba
Dienstag, 15.7.25, 14:15-15:45, Seminarraum 226, HH10
tba
Dienstag, 15.7.25, 14:15-15:45, Seminarraum 226, HH10