Direct and inverse scattering for an isotropic medium with a second-order boundary condition

TL;DR

The paper addresses inverse and forward scattering for an isotropic penetrable medium with a delaminated boundary modeled by a second-order Robin condition. It develops a direct sampling method that yields a stable imaging functional to reconstruct the scatterer from boundary Cauchy data, and it connects this functional to Green's identities to localize D. The transmission eigenvalue problem is introduced and shown to have at most a discrete spectrum under favorable refractive-index conditions, with numerical demonstrations for disks and ellipses. Overall, the work provides 2D numerical validations for circular and non-circular scatterers, establishes the well-posedness of the direct problem, and outlines future directions including TE existence proofs and limited-aperture data scenarios.

Abstract

We consider the direct and inverse scattering problem for a penetrable, isotropic obstacle with a second-order Robin boundary condition, which asymptotically models the delamination of the boundary of the scatterer. We develop a direct sampling method to solve the inverse shape problem by numerically recovering the scatterer. Here, we assume that the corresponding Cauchy data is measured on the boundary of a region that fully contains the scatterer. Similar methods have been applied to other inverse shape problems, but they have not been studied for a penetrable, isotropic scatterer with a second-order Robin boundary condition. We also initiate the study of the corresponding transmission eigenvalue problem, which is derived from assuming zero Cauchy data is measured on the boundary of the region that fully contains the scatterer. We prove that the transmission eigenvalues for this problem are at most a discrete set. Numerical examples will be presented for the inverse shape problem in two dimensions for circular and non-circular scatterers. Further, transmission eigenvalues are computed numerically for various scatterers.

Figures (7)

• Figure 1: Reconstruction of a circular region with radius $R=0.5$. On the left: wavenumber $k = 3\pi/2$. On the right: wavenumber $k = 5\pi$.
• Figure 2: Reconstruction of a circular region with radius $R=0.5$. On the left: $5\%$ error was added to the Cauchy data. On the right: $20\%$ error was added to the Cauchy data.
• Figure 3: Reconstruction of small circular regions. On the left: two circular regions with centers at $(-0.5, -0.5)$ and $(0.5, 0.5)$. On the right: three circular regions with centers at $(0.35, 0.65)$, $(-0.6, 0.1)$, and $(-0.2, -0.7)$.
• Figure 4: Reconstructions where we approximate $\partial_{r}u^s \approx \text{i}k u^s$. Contour plots of $\widetilde{W}_{\text{nor}}(z)$ with $5\%$ error. On the left: data computed via separation of variables to recover $D = B(0,0.4)$ with $k=2\pi$. On the right: data computed via Lippmann-Schwinger equation to recover circular regions or radius 0.1 centered at $(-0.4 , -0.4)$ and $(0.4 , 0.4)$ with $k=2.5\pi$.
• Figure 5: Reconstruction of the circle shape where the Cauchy data was computed via BIEs. On the left: $5\%$ error was added to the data. On the right: $10 \%$ error was added to the data.

Theorems & Definitions (13)

• Lemma 2.1
• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Lemma 5.1
• proof
• Theorem 5.1