Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data

Abstract

We consider an inverse acoustic scattering problem in simultaneously recovering an embedded obstacle and its surrounding inhomogeneous medium by formally determined farfield data. It is shown that the knowledge of the scattering amplitude with a fixed incident direction and all observation angles along with frequencies from an open interval can be used to uniquely identify the embedded obstacle, sound-soft or sound-hard disregarding the surrounding medium. Furthermore, if the surrounding inhomogeneous medium is from an admissible class (still general), then the medium can be recovered as well. Our argument is based on deriving certain integral identities involving the unknowns and then inverting them by certain harmonic analysis techniques. Finally, based on our theoretical study, a fast and robust sampling method is proposed to reconstruct the shape and location of the buried targets and the support of the surrounding inhomogeneities.

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

“…The simultaneous recovery of an embedded obstacle and an unknown surrounding potential is also a longstanding problem in the literature and closely related to the so-called partial data Calderón problem [5,11]. The existing unique recovery results were established based on knowing the embedded obstacle to recover the unknown potential [11], or knowing the surrounding potential to recover the embedded obstacle [13,14,18,19,24], or using multiple spectral data to recover both of them [16].…”

Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement

“…Without loss of generality we suppose that U * := U 12 \ U 1 and U * is nonempty. We mention that the method is similar to the obstacle recovery result in [21]. We first consider the Neumann condition case, i.e., B[u] = ∂u ∂ν in (2.5).…”

“…∂B ∂u ∂ν +h , and k can be recovered. For recovery of the inner core, the numerical method can be applied by using the same scheme as in [21].…”

Recovery of an embedded obstacle and the surrounding medium for Maxwell's system

“…Finally, we briefly discuss the mathematical arguments to establish our unique recovery results. Our idea follows from a recent work [13] by two of the authors where the acoustic case was considered. First, we derive the integral representation of the solution to the scattering problem involving both the perfect conductor B and the medium (Ω\B, ǫ).…”