A modified linear sampling method valid for all frequencies

The linear sampling method was recently introduced by Colton and Kirsch (1996) for solving inverse scattering problems in the resonance region. This method reduces the inverse problem to the solution of a linear integral equation of the first kind. A major advantage of the method is that it requires no a priori information on the physical properties and boundary of the scatterer and takes into account multiple scattering.;However, it is known that the linear sampling formulation may fail at some interior eigenvalues due to the lack of completeness of the far field data. In other words, there can exist values of the wave number for which the sampling operator is no longer injective with dense range and the linear sampling method is not applicable. These eigenvalues depend on the scatterer, and therefore are not known in advance, which may be a serious drawback of the method.;In this thesis, we consider the inverse obstacle problem and the inverse inhomogeneous medium problem for acoustic waves, and propose a modified version of the linear sampling method that remains valid for all positive wave numbers. This modification is based on ideas of Jones and Ursell in the context of direct acoustic scattering problems. It consists of adding a special series to the kernel of the sampling integral operator so that we "recover" its injectivity and range density for all positive wave numbers. The validity of the modified sampling method is illustrated with several numerical examples. For wave numbers close to an interior eigenvalue, our numerical experiments also indicate that the standard linear sampling reconstructions may be completely disrupted.