We work on different approaches to describe pedestrian dynamics based on mathematical techniques
from optimal control theory, mean field games and nonlinear partial differential equations. Individual
interactions are very complex and lead to interesting dynamical features such as lane formation or
segregation. We develop analytical and numerical methods to analyze these dynamics and give further
insights into the underlying microscopic interactions.
Kinetic models allow us to model interactions among individuals by 'collisions', using tools initially
developed in statistical mechanics to describe the behavior of thermodynamic systems. This approach has
been used successfully in many socio-economic applications such as opinion formation, price dynamics or
more recently knowledge growth in a society. The resulting Boltzmann type equations are often coupled to
other PDEs and require the development of sophisticated mathematical and numerical techniques.
A common characteristic of mean-field models is their gradient flow structure with respect to a certain
metric. Not all models have this underlying structure, but are only perturbed or asymptotic gradient flows.
While the analysis of gradient flows is fairly well understood, little is known in the latter cases. Based
on methods from gradient flow theory as well as optimal transportation problems we are working on existence
and stability results for these kind of PDEs as well as the development of structure preserving numerical
Electrostatic interaction and size constraints are among the main driving forces in the transport of charged
particles. The can be included in various ways on the microscopic level; in the mean-field limit the
resulting PDE systems are often highly nonlinear. We analyze these equations for different asymptotic limits,
develop numerical solvers to describe the formation of boundary layers close to highly charged walls
consistently and adress parameter identification problems for different applications of ion transport.