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In recent years, the description of controllable (or active) particle systems using methods of kinetic gas theory has been achieved and allows now to tackle a wide range of applications as for example traffic flow. In this research project, the inherent hierarchy exploited extensively in kinetic theory for theoretical and numerical considerations will be investigated in order to develop novel analytical and numerical methods for control problems posed on multiple scales as well as under aspects of non-smoothness in the control. The work program includes the analysis of consistent optimality conditions within the model hierarchies, numerical analysis for control aspects relevant in particular on the highest level of the model hierarchy, as well as the development of numerical methods for time-dynamic non-smooth optimization problems on all levels. In addition to the sensitivity of non-smooth kinetic equations, the multi-scale nature of the equations raises questions for boundary control problems of nonlocal hyperbolic equations and switching systems.
Kinetic theory may help to analyze novel financial marekt models where investors are described as heterogeneous interacting agents. Such novel models help to gain insights in the creation of financial crashes. Furthermore, we focus on the connection between the design of the microscopic financial agents and macroscopic behaviour of the kinetic model. We hope to investigate new answers regarding the relationship between the psychological behaviour of investors and empirical observations on financial markets, known as stylized facts.
The goal of this project is to use kinetic and mean-field theory to gain novel insights into data science applications. Examples include high dimensional global optimization, large data clustering or inverse problems. In deep learning we are especially interested to understand the good performance of neural networks in many data science applications.