Mean field games with nonlinear mobilities in pedestrian dynamics
Martin Burger Marco Di Francesco Peter A. Markowich Marie-Therese Wolfram
Discrete & Continuous Dynamical Systems - B 2014, 19(5): 1311-1333 doi: 10.3934/dcdsb.2014.19.1311
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.
keywords: mean field limit optimal control numerical simulations. Pedestrian dynamics calculus of variations
Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth
Martin Burger Alexander Lorz Marie-Therese Wolfram
Kinetic & Related Models 2017, 10(1): 117-140 doi: 10.3934/krm.2017005

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [15] 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.

keywords: Mean-field games Boltzmann-type equations Hamilton-Jacobi equations travelling wave solutions
Kinetic description of collision avoidance in pedestrian crowds by sidestepping
Adriano Festa Andrea Tosin Marie-Therese Wolfram
Kinetic & Related Models 2018, 11(3): 491-520 doi: 10.3934/krm.2018022

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

keywords: Crowd dynamics collision avoidance Boltzmann-type kinetic model mean-field approximation
Identification of nonlinearities in transport-diffusion models of crowded motion
Martin Burger Jan-Frederik Pietschmann Marie-Therese Wolfram
Inverse Problems & Imaging 2013, 7(4): 1157-1182 doi: 10.3934/ipi.2013.7.1157
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.
keywords: nonlinear drift-diffusion equations. identifiability Inverse problems
A mixed finite element method for nonlinear diffusion equations
Martin Burger José A. Carrillo Marie-Therese Wolfram
Kinetic & Related Models 2010, 3(1): 59-83 doi: 10.3934/krm.2010.3.59
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.
keywords: porous medium equation Patlak-Keller-Segel model. mixed finite element method optimal transportation problem Nonlinear diffusion problems

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