Inverse obstacle scattering with non-over-determined data

t is proved that the scattering amplitude \(A(\beta, \alpha_0, k_0)\), known for all \(\beta\in S^2\), where \(S^2\) is the unit sphere in \(\mathbb{R}^3\), and fixed \(\alpha_0\in S^2\) and \(k_0>0\), determines uniquely the surface \(S\) of the obstacle \(D\) and the boundary condition on \(S\). The boundary condition on \(S\) is assumed to be the Dirichlet, or Neumann, or the impedance one. The uniqueness theorem for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades.A detailed proof of such a theorem is given in this paper for inverse scattering by obstacles for the first time. It follows from our results that the scattering solution vanishing on the boundary \(S\) of the obstacle cannot have closed surfaces of zeros in the exterior of the obstacle different from \(S\). To have a uniqueness theorem for inverse scattering problems with non-over-determined data is of principal interest because these are the minimal scattering data that allow one to uniquely recover the scatterer.

Windowed Green function method for the Helmholtz equation in the presence of multiply layered media

This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math. 76 , 1871–1898. ( doi:10.1137/15M1033782 )), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals.