From interacting particle systems to kinetic equations

n this seminar we concentrate on collocation and stochastic Galerkin methods for the uncertainty quantification of Vlasov--Fokker--Planck (VFP) equations with nonlocal flux. In particular, we develop methods that preserve their structural properties and that are spectrally accurate in the random space. In contrast to a direct application of classical uncertainty quantification methods, which are typically lead to the loss of positivity and of entropy properties, the proposed schemes are capable to achieve high accuracy in the random space without losing non-negativity of the solution, which dissipates the entropy under suitable assumptions. Applications of the developed methods will be presented in the context of social and traffic dynamics.