We deal with the Data Completion Poisson problem, also called the Cauchy’s problem. The special point is that: on a portion of the boundary, Neumann and Dirichlet conditions are given, while the complementary part, no conditions, no data are available. We start by explaining how to obtain an appropriate variational formulation of the problem. Its main properties are state and in particular its severe ill posedeness is pointed out. A Lavrentiev regularizing process is proposed and analyzed. Next, we show how to use a domain extension to improve the computational performance of the Lavrentiev method. We therefore switch to the finite element approximations, announce the main estimates and provide some comments. This is the most recent part of the research on the subject (20021-2023). To our knowledge this is the first complete convergence analysis of the full discrete-regularized solution of the data completion problem. The proofs are technical and tedious and are not given in the talk.

An acoustic medium occupying a compact domain with non-constant sound speed, is probed by an impulsive plane wave, and the far-field response is measured in all directions for all frequencies. A longstanding open problem, called the fixed angle scattering inverse problem, is the recovery of the sound speed from this far-field response. In some situations, the acoustic properties of the medium are modeled by a Lorentzian metric and then the goal is the recovery of this metric from the far field measurements corresponding to a finite number of incoming plane waves. We consider a time domain, near field version of this problem and show that natural fixed angle measurements distinguish between a constant velocity (the Minkowski metric) medium and a non-constant velocity (a general Lorentzian metric) medium. The talk is based on a joint work with Rakesh (Delaware) and Mikko Salo (Jyväskylä).