Elastic scatterer reconstruction via the adjoint sampling method

An inverse problem dealing with the reconstruction of voids in a uniform semi-infinite solid from near-field elastodynamic waveforms is investigated via the linear sampling method. To cater to active imaging applications that are characterized by a limited density of illuminating sources, existing formulation of the linear sampling method is advanced in terms of its adjoint statement that features integration over the receiver surface rather than its source counterpart. To deal with an ill-posedness of the integral equation that is used to reconstruct the obstacle, the problem is solved by alternative means of Tikhonov regularization and a preconditioned conjugate gradient method. Through a set of numerical examples, it is shown (i) that the adjoint statement elevates the performance of the linear sampling method when dealing with scarce illuminating sources, and (ii) that a combined use of the existing formulation together with its adjoint counterpart represents an effective tool for exposing an undersampling of the experimental input, e.g., in terms of the density of source points used to illuminate the obstacle.