Reconstruction of a crack with the incident waves and measurements inside a penetrable cavity

Abstract

The inverse scattering problem of determining a crack outside a known penetrable cavity from the measurements and point sources placed in a closed curve Λ inside the cavity is considered. The factorization method is used to reconstruct the crack on which the mixed-type boundary conditions are posed. To this end, we need to overcome two key difficulties. One of them is to find the relationship between the two data-to-data operators, which map the boundary data on the crack to the values of scattered fields on the closed curve Λ with outgoing radiation condition and incoming radiation condition, respectively. In addition, it is hard to decompose the near field operator in the usual way owing to the mixed-type boundary conditions. By the aid of an auxiliary operator, we can obtain the desired decomposition form in accordance with the theoretical framework of the factorization method.

A neural network method for the inverse scattering problem of impenetrable cavities

“…Recently, Zhang et al [33] proposed the idea of using a reference ball and the superimposition of two point sources as incident waves, and proved that the position, shape and boundary conditions of the cavity can be uniquely determined from phaseless near-field data. More studies on the inverse scattering problems of the cavities can also be found in [2,10,25].…”