On the ensemble observability problem for nonlinear systems

We consider the ensemble observability problem for nonlinear systems and illuminate it from both theoretical and practical perspectives. In the ensemble observability problem we would like to reconstruct a density of initial states from the evolution of the density of outputs under a nonlinear system. For the theoretical question of ensemble observability, i.e. the uniqueness of the reconstruction of the initial state density, we discuss possible approaches. One approach is based on the natural connection between the ensemble observability problem with nonlinear tomography problems. Through this formulation, the problem also becomes amenable to computational solutions. We present an implementation of Algebraic Reconstruction Techniques from computerized tomography which is well-suited for reconstructing the initial state density. We illustrate this method on a simulation example.