Identification of nonlinearities in transport-diffusion models of crowded motion

The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied. In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.