Abstract

This Ph.D. thesis presents a threefold revisitation and reformulation of the linear sampling method (LSM) for the qualitative solution of inverse scattering problems (in the resonance region and in time-harmonic regime): 1) from the viewpoint of its implementation (in a 3D setting), the LSM is recast in appropriate Hilbert spaces, whereby the set of algebraic systems arising from an angular discretization of the far-field equation (written for each sampling point of the numerical grid covering the investigation domain and for each sampling polarization) is replaced by a single functional equation. As a consequence, this 'no-sampling' LSM requires a single regularization procedure, thus resulting in an extremely fast algorithm: complex 3D objects are visualized in around one minute without loss of quality if compared to the traditional implementation; 2) from the viewpoint of its application (in a 2D setting), the LSM is coupled with the reciprocity gap functional in such a way that the influence of scatterers outside the array of receiving antennas is excluded and an inhomogeneous background inside them can be allowed for: then, the resulting 'no-sampling' algorithm proves able to detect tumoural masses inside numerical (but rather realistic) phantoms of the female breast by inverting the data of an appropriate microwave scattering experiment; 3) from the viewpoint of its theoretical foundation, the LSM is physically interpreted as a consequence of the principle of energy conservation (in a lossless background). More precisely, it is shown that the far-field equation at the basis of the LSM (which does not follow from physical laws) can be regarded as a constraint on the power flux of the scattered wave in the far-field region: if the flow lines of the Poynting vector carrying this flux verify some regularity properties (as suggested by numerical simulations), the information contained in the far-field constraint is back-propagated to each point of the background up to the near-field region, and the (approximate) fulfilment of such constraint forces the L^2-norm of any (approximate) solution of the far-field equation to behave as a good indicator function for the unknown scatterer, i.e., to be 'small' inside the scatterer itself and 'large' outside.