Uniqueness and stability in multi-dimensional inverse problems

Abstract

Recent results on uniqueness and stability of identification of coefficients and right sides of partial differential equations from overdetermined boundary data are described. Elliptic, hyperbolic, and parabolic equations and scattering theory are considered. Proofs are given or outlined whenever they contain a new and fruitful idea and are sufficiently short. This review is supposed to be quite comprehensive. In fact, we do not cover only inverse spectral theory. Some interesting numerical methods are mentioned, but numerics is also beyond the scope of this paper. A significant part is dedicated to so-called many boundary measurements (equations with given Dirichlet-to-Neumann map), but we also discuss results about single boundary measurements. An extensive bibliography contains basic papers in the field.