Summary: The inverse problem under consideration is to reconstruct the shape information of obstacles or inclusions embedded in the (inhomogeneous) background medium from boundary measurements of propagating waves. This article is a survey of enclosure-type methods implementing exponential complex geometrical optics waves as boundary illumination. The equations for acoustic waves, electromagnetic waves and elastic waves are considered for a medium with impenetrable obstacles and penetrable inclusions (characterized by a jump discontinuity in the parameters). We also outlined some open problems along this direction of research.
For the entire collection see [Zbl 1277.65002].