Multiscale modeling and simulation of crowded transport in the life and social sciences

Research Topics

Pedestrian dynamics

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.
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Kinetic and mean-field models in socio-economic applications

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.
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Gradient flows and optimal transportation problems

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 schemes.
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Ion transport in narrow pores

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.
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