Invariant Formulations of Inverse Problems

Mathematical physics is based on the fundamental assumption that physical predictions must be the same, independently of the parameterization of the system. This principle even constitutes the very foundation of certain physical theories, of which the theory of relativity is perhaps the most notable. The importance of the principle is that it seeks to maintain objectivity: When two different analysts predict the evolution of the same physical system, but use different parameterizations (reference systems), their predictions must agree physically. Otherwise the theory would give results that depend on the individual analysts's preferences, and hence be subjective. Our question here is the following: What would happen if we impose the same constraints on modeling and data inversion? What if we required that our procedures for modeling and inversion should be designed such that no conflicts between analysts would appear? We will look into this problem by focusing on three different problem areas where possible inconsistencies may occur.