In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds.
In particular we consider the case of a large human crowd trying to exit a room as fast as possible.
The motion of every pedestrian is determined by minimizing a cost functional,
which depends on his/her position, velocity, exit time and the overall density of people.
This microscopic setup leads in the mean-field limit to a parabolic optimal control problem.
We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate
to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers
and illustrate the behavior of the model with various numerical results.
The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied.
In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.
We propose a mixed finite element method for a class of
nonlinear diffusion equations, which is based on their
interpretation as gradient flows in optimal transportation
metrics. We introduce an appropriate linearization of the optimal
transport problem, which leads to a mixed symmetric formulation.
This formulation preserves the maximum principle in case of the
semi-discrete scheme as well as the fully discrete scheme for a
certain class of problems. In addition solutions of the mixed
formulation maintain exponential convergence in the relative
entropy towards the steady state in case of a nonlinear
Fokker-Planck equation with uniformly convex potential. We
demonstrate the behavior of the proposed scheme with 2D
simulations of the porous medium equations and blow-up questions
in the Patlak-Keller-Segel model.
In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll  to model knowledge growth in an economy.Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have.The existence of balanced growth path solutions implies exponential growth of the overall production in time. We prove existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfiesa Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange theknowledge level evolves by geometric Brownian motion.