Inverse Problems and Imaging: Past, Present and Future

Inverse problems focus on the problem of determining parameters and data inherent in the mathematical model of a physical or biological phenomenon from measurements of the observable data. Such problems are almost invariably ill-posed in the sense that in general existence, uniqueness or continuous dependence on the data is no longer true. In this chapter we have chosen three “canonical” examples of inverse problems, have described their physical origin and then present mathematical methods which in each case address the basic issues of existence, uniqueness and stability for what are fundamentally ill-posed problems. These examples are the backwards heat equation, computerized tomography and the inverse scattering problem for acoustic waves. To aid the reader in our discussion we have also presented a brief introduction to the theory of Hilbert spaces where we have only supposed a pre-requisite of elementary linear algebra and calculus.