An inverse problem of tomographic type in population dynamics

Abstract

In this paper we address an inverse problem on populations described by probability distributions. From a theoretical point of view, this problem can be seen as a natural extension to the classical observability problem. We consider a population that is described by a classical linear finite-dimensional system in which the initial state is a random vector subject to a non-parametric probability distribution. The problem is to reconstruct this initial state distribution from the time-evolution of the probability distribution of the output. We reveal as a novel viewpoint, that, at its core, this problem is a tomography problem which is a well-known subject in the field of inverse problems. Furthermore we show how this tomography problem is inherently linked with the observability properties of the finite-dimensional system thereby establishing a beautiful link between a control theoretic question and tomography problems.