We consider the scattering of a time-harmonic plane wave incident on a two-scale heterogeneous medium, which consists of scatterers that are much smaller than the wavelength and extended scatterers that are comparable to the wavelength. A generalized Foldy-Lax formulation is proposed to capture multiple scattering among point scatterers and extended scatterers. Our formulation is given as a coupled system, which combines the original Foldy-Lax formulation for the point scatterers and the regular boundary integral equation for the extended obstacle scatterers. An efficient physically motivated Gauss-Seidel iterative method is proposed to solve the coupled system, where only a linear system of algebraic equations for point scatterers or a boundary integral equation for a single extended obstacle scatterer is required to solve at each step of iteration. In contrast to the standard inverse obstacle scattering problem, the proposed inverse scattering problem is not only to determine the shape of the extended obstacle scatterer but also to locate the point scatterers. Based on the generalized Foldy-Lax formulation and the singular value decomposition of the response matrix constructed from the far-field pattern, an imaging function is developed to visualize the location of the point scatterers and the shape of the extended obstacle scatterer.

We prove that if for the isotropic Lamé system the coefficiem $\mu$ is a positive constant then both coefficents can be reconstructed from the partial Cauchy data.

In this talk, we shall consider the near-invisibility cloaking in acoustic scattering by non-singular transformation media. A general lossy layer is included into our construction. We are especially interested in the cloaking of active/radiating objects. Our results on the one hand show how to cloak active contents more efficiently, and on the other hand indicate how to choose the lossy layer optimally.

We consider boundary measurements for the wave equation on a bounded domain $M \subset \R^2$ or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in $M$. Computationally the method consists of solving a sequence of linear equations. We present some numerical results.