Uniqueness of an inverse electromagnetic coefficient problem with partial boundary data and its numerical resolution through an iterated sensitivity equation

Abstract

In this paper we study an inverse boundary value problem for Maxwell's equations. The goal is to reconstruct perturbations in the refractive index of the medium inside an object from the knowledge of the tangential trace of an electric field on a part of the boundary of the domain. We first provide a uniqueness result for this inverse problem. Then, we propose a complete procedure to reconstruct numerically the perturbations, based on the minimization of a cost functional involving an iterated sensitivity equation.

Figures (15)

• Figure 1: Example of possible configuration for the domain $\Omega$. The accessible part of the boundary $\Gamma_0$ is shown as thick parts on $\partial\Omega$. The support of the perturbation is here composed of three parts, delimited by dotted lines. This support does not touch ${\mathcal{V}}$, the tubular neighborhood where $\kappa$ is assumed to be known, represented here by the part filled with vertical lines. Finally, $\Gamma_\text{int}$ is an artificial boundary included in ${\mathcal{V}}$, delimiting the subdomain $U$ represented by the gray part.
• Figure 2: Configuration of the domain $\Omega$ for the first numerical test. The thicker parts on the boundary is $\Gamma_0$. The artificial boundary is the dashed line, while the dotted line represents the boundary of the support of the perturbation.
• Figure 3: Real part of the reconstructed refractive index in the unit disk. The boundary of the exact support of the perturbation is shown as a white line.
• Figure 4: Configuration of the domain $\Omega$ in 3D. The accessible part $\Gamma_0$ is represented by the patches on the sphere.
• Figure 5: Real part of the reconstructed refractive index in the 3D unit ball. The boundary of the exact support of the perturbation is shown as a white line.