The Generalized Linear Sampling Method for Limited Aperture Measurements

Abstract

We extend the so-called generalized linear sampling method (GLSM) to the case of limited aperture data at a fixed frequency. In this case the factorization of the sampling operator does not obey the symmetry required in the justification of the GLSM introduced by Audibert and Haddar [Inverse Probl., 30 (2004), p. 035011]. We propose a new formulation by adding an extra penalty term that asymptotically corrects the nonsymmetry of the GLSM original penalty term. The analysis of the new formulation is first presented in an abstract framework. We then show how to apply our setting to the scalar problem with far field measurements or near field measurements on a limited aperture. We finally validate the method through some numerical tests in two dimensions and for far field measurements.