The linear sampling method and energy conservation

In this paper we explain the linear sampling method and its performances in various scattering conditions by means of an analysis of the far-field equation based on the principle of energy conservation. Specifically, we consider the conservation of energy along the flow strips of the Poynting vector associated with the scattered field whose far-field pattern is one of the two terms in the far-field equation. The behavior of these flow lines is numerically investigated and theoretically described. Appropriate assumptions on the flow lines, based on the numerical results, allow characterizing a set of approximate solutions of the far-field equation which can be used to visualize the boundary of the scatterer in the framework of the linear sampling method. In particular, under the same assumptions, we can show that Tikhonov regularized solutions belong to this set of approximate solutions for appropriate choices of the regularization parameter.