Inverse Obstacle Scattering with Modified or Reduced Data

Roughly speaking, one can distinguish between two different approaches for the approximate solution of the inverse obstacle scattering problem for time-harmonic waves. In a first group of methods, the inverse obstacle problem is separated into a linear ill-posed part for the reconstruction of the scattered wave from its far field pattern and a nonlinear well-posed part for finding the location of the boundary of the scatterer from the boundary condition for the total field. In a second group of methods, the inverse obstacle problem is either considered as an ill-posed nonlinear operator equation or reformulated as a nonlinear optimization problem.

We will present and review some mathematical foundations for methods of the second group such as regularized Newton iterations. This will be done both for the basic problem to recover the scatterer from the far field pattern for one incident direction and all observation directions and for modifications with reduced data, i.e., reconstructions from limited-aperture observations, reconstructions from the amplitude of the far field, or reconstructions from backscattering data.