Recovering complex elastic scatterers by a single far-field pattern

We consider the inverse scattering problem of reconstructing multiple impenetrable bodies embedded in an unbounded, homogeneous and isotropic elastic medium. The inverse problem is nonlinear and ill-posed. Our study is conducted in an extremely general and practical setting: the number of scatterers is unknown in advance; and each scatterer could be either a rigid body or a cavity which is not required to be known in advance; and moreover there might be components of multiscale sizes presented simultaneously. We develop several locating schemes by making use of only a single far-field pattern, which is widely known to be challenging in the literature. The inverse scattering schemes are of a totally "direct" nature without any inversion involved. For the recovery of multiple small scatterers, the nonlinear inverse problem is linearized and to that end, we derive sharp asymptotic expansion of the elastic far-field pattern in terms of the relative size of the cavities. The asymptotic expansion is based on the boundary-layer-potential technique and the result obtained is of significant mathematical interest for its own sake. The recovery of regular-size/extended scatterers is based on projecting the measured far-field pattern into an admissible solution space. With a local tuning technique, we can further recover multiple multiscale elastic scatterers.