On Inverse Problems for Nonlocal Elliptic Operators -- Continuous vs Discrete Operators

Angkana Rüland, Heidelberg University, Germany

Inverse problem for prototypical nonlocal operators such as the fractional Laplacian display strikingly strong uniqueness, stability and single measurement results. These fundamentally rely on global variants of the unique continuation property for these nonlocal operators and dual flexibility properties in the form of Runge approximation results. In this talk, on the one hand, we discuss first  steps towards extensions of these rigidity and flexibility properties to more general classes of nonlocal operators enjoying only directional unique continuation properties. On the other hand, we compare and contrast the settings of continuous and discrete operators, showing quantitatively how discretization counteracts the rigidity of unique continuation.

This is based on a series of joint works with G. Covi, A. Fernandez-Bertolin, M. A. Garcia-Ferrero, L. Roncal, M. Salo, T. Ghosh, G. Uhlmann.