The factorization method for reconstructing a penetrable obstacle with unknown buried objects (Q2840992)

The factorization method was introduced by \textit{A. Kirsch} [Inverse Probl. 14, No. 6, 1489--1512 (1998; Zbl 0919.35147)] to study scattering by impenetrable sound soft or sound hard obstacles. Since then, the factorization method has been developed for several problems in inverse scattering theory and impedance tomography. A striking feature of the factorization method is its sound theoretical foundation together with its simple and fast implementation. However, it should also be mentioned that the assumptions necessary for the derivation of the factorization method are very difficult to justify. Even up to now, it is still open for some basic models, e.g., scattering by scatterers with mixed boundary conditions, or by perfect conductors.NEWLINENEWLINEThe present paper is the first application of the factorization method to the acoustic scattering by penetrable obstacles. The factorization of the far field operator is motivated by [\textit{A. Charalambopoulos} et al., Inverse Probl. 23, No. 1, 27--51 (2007; Zbl 1111.74024)], where the factorization method was applied to recover a penetrable obstacle in inverse elastic transmission scattering. However, some new techniques have been introduced in this paper, e.g., the DtN map \(\Lambda_{n_0}\) with some chosen constant \(n_0 > 0\), which makes it possible to show the validity of the factorization method for more general transmission boundary conditions. It is thus a paper worth reading on a subject which is of interest to the inverse problems community.NEWLINENEWLINEInspired by the techniques provided in this paper, the factorization method has also been verified recently for anisotropic media [\textit{A. Kirsch} and \textit{X. Liu}, Math. Methods Appl. Sci. 37, No. 8, 1159--1170 (2014; Zbl 1290.35331)] and scatterers with conductivity boundary conditions [\textit{O. Bondarenko} and \textit{X. Liu}, Inverse Probl. 29, No. 9, Article ID 095021, 25 p. (2013; Zbl 1285.78004)].