The Inverse Scattering Problem for a Penetrable Cavity with Internal Measurements

Abstract

We consider the inverse scattering problem for a cavity that is bounded by a penetrable inhomogeneous medium of compact support and seek to determine the shape of the cavity from internal measurements on a curve or surface inside the cavity. We prove uniqueness and establish a linear sampling method for determining the shape of the cavity. A central role in our analysis is played by an unusual non-selfadjoint eigenvalue problem which we call the exterior transmission eigenvalue problem.

The reciprocity gap method for a cavity in an inhomogeneous medium

“…The total field u = u s + u i , where u s is the scattered field. In a similar way as Theorem 5.1 in [3], it is not difficult to show that u i and u s satisfy the reciprocity relations…”

“…In [12], Jakubik and Potthast used the solutions of the Cauchy problem by potential methods and the range test to study the integrity of the boundary of some cavity by acoustics. Later, the linear sampling method [21,22,25,11,3,23], the nonlinear integral equation method [20], the decomposition method [26], the factorization method [15,18], and the near-field imaging method [14] were applied.…”

“…As in [3], the physical properties of the medium in R 2 \D are described by the 2 × 2 symmetric matrix valued function A with L ∞ (R 2 \D) entries such that ξ • Re(A)ξ ≥ α ξ 2 and ξ • Im(A)ξ ≤ 0 for all ξ ∈ C and some α > 0, and the bounded function…”

An inverse spectral uniqueness in exterior transmission problem

“…The inverse problem is to find the index of refraction by the measurements in the far-fields. We refer the scattering and inverse scattering theory of this problem to [1,4,13,14,15,22]. To ensure the well-posedness of the scattered wave fields, we impose the Sommerfeld radiation conditions to (1.2).…”

The factorization method for a partially coated cavity in inverse scattering

“…The typical inverse scattering problems are exterior problems where the measurements are taken outside of the objects (see [5,6]). but in recent years there has been extensive interest in the interior scattering problem for determining the scattering objects and the structures [7,13,21,23,25,26,29,30]. The scattering problem with both point sources (incident waves) and measurements (scattered waves) inside a cavity is called the interior scattering problem.…”