Some Recent Progress On Inverse Scattering Problems Within General Polyhedral Geometry

Unique identifiability by finitely many far-field measurements in the inverse scattering theory is a highly challenging fundamental mathematical topic. In this paper, we survey some recent progress on the inverse obstacle scattering problems and the inverse medium scattering problems associated with time-harmonic waves within a certain polyhedral geometry, where one can establish the unique identifiability results by finitely many measurements. Some unique identifiability issues on the inverse diffraction grating problems are also considered. Furthermore, the geometrical structures of Laplacian and transmission eigenfunctions are reviewed, which have important applications in the unique determination for inverse obstacle and medium scattering problems with finitely many measurements. We discuss the mathematical techniques and methods developed in the literature. Finally, we raise some intriguing open problems for the future investigation.