Microwave imaging defines an inverse problem that is severely ill-posed but with interesting potential applications. In this talk, we will first talk about the direct problem, involving Maxwell's equations, and then we will iterate a lot of things for the inverse problem.
In particular, we will talk about the question of data completion, with two goals: first, provide a uniqueness result, and then, provide a tool to help solving the full problem. After that, we will differentiate the solution of Maxwell's equations many times in order to regularize a cost function that can then be minimized, more or less easily (but more less than more).
We will end by a lot of numerical experiments and some questions that remain open.
Jérémy Heleine, David Lafontaine
