Simulation and optimization of interacting particle systems with application in pedestrian dynamics

We propose an anisotropic interaction model which allows for collision avoidance in pairwise interactions by a rotation of forces, including the agents’ body size. The influence of the body size on the self-organization of the agents in the channel and crossing scenarios, as well as the fundamental diagram, is studied. Since the model is stated as a coupled system of ordinary differential equations, we are able to give a rigorous well-posedness analysis. Then we state a parameter calibration problem that involves data from real experiments. We prove the existence of a minimizer and derive the corresponding first-order optimality conditions. With the help of these conditions, we propose a gradient descent algorithm based on mini-batches of the data set. We employ the proposed algorithm to fit the parameter of the collision avoidance and the strength parameters of the interaction forces from real experiments. The results underpin the feasibility of the method. We extend an anisotropic interaction model with the rigorous derivation of the mean-field limit. Then we analyze the convergence of the empirical probability measure corresponding to the particle system, showing that it is a weak solution to the mean-field equation. Next, we derive a hydrodynamic description for interaction dynamics with the help of a moment approximation of the kinetic equation. Finally, we determine the conditions for stationary states in uni- and bi-directional flows and characterize time-invariant states of the dynamics using the hydrodynamic description.