Coefficient inverse problem for the wave equation: a story of well-posedness and reconstruction.

Let us start by assuming that we are interested in the study of a dynamic phenomenon that is modeled by a partial differential equation (PDE). The first topic of interest for a mathematician then would be the direct problem, where, knowing all the physical coefficients, source terms and initial conditions, one wants to prove the existence of a solution and study its behavior and properties. In science, an inverse problem is the process of studying from a set of observations, the causal factors that produced them. Thus, for instance, one could be interested in an inverse problem related to PDEs, such as the determination of some unknown physical coefficient from a local measurement of the behavior of the solution. Inverse problems are very relevant in mathematics because they focus on the study of parameters that we cannot directly observe.