The reciprocity gap functional method for the inverse scattering problem for cavities

In this paper, we concern with determining the shape of a perfectly conducting cavity from the Cauchy data on a curve inside the cavity. The near-field linear sampling method (LSM), i.e. the reciprocity gap (RG) functional method, is employed to reconstruct the shape of the cavity. The equivalence of the RG method and the linear sampling method with mere the scattered field is established. But from the examples, we can see that the reconstructions are as satisfactory as the exterior scattering problems. I think this behavior is due to our reconstruction method since this method is due to the Cauchy data, but the LSM with mere the scattered field is used. Numerical tests show that the methods can provide qualitative information on the cavity. The numerical influence of the proposed method with respect to the wave numbers, the curve for the Cauchy data on which are measured, and the curve which is used to construct the single-layer potential function, respectively, are also analyzed with some examples. In particular, we give the examples of determining the cavity from the Cauchy data measured on a portion of the curve inside the cavity.