Why linear sampling works

The linear sampling method is a method to reconstruct the shape of an obstacle in time-harmonic inverse scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. Although it has been proven numerically to be a fast and reliable method in many situations, no mathematical argument has yet been found to prove why this is so. Using results obtained by Kirsch in deriving the related factorization method, we show in this paper that for a large class of scattering problems, linear sampling can be interpreted rigorously as a numerical method to reconstruct the shape of an obstacle; or in other words that linear sampling must work for problems in this class.