Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

Abstract

In this paper, we derive a sampling method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet-to-Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. A simple numerical example is given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of determining the impedance parameters for a known material from the Dirichlet-to-Neumann mapping assuming the inclusion has been reconstructed where uniqueness for the reconstruction of the coefficients is proven. KEYWORDS inverse boundary value problems, sampling methods, second order boundary condition, shape reconstruction MSC CLASSIFICATION 35J05; 31A25; 78A30

Reconstruction of small and extended regions in EIT with a Robin transmission condition

“…Using the sesquilinear form A(•, •), we can show that the solution u for equation (1) is unique just as in [27], which implies that equation (1) is well-posed. The above analysis gives the following result.…”

Reconstruction of extended regions in EIT with a generalized Robin transmission condition

“…In this section, we study the case when the boundary parameters µ and γ are complex-valued. The methodology used here is influenced by the work in [27]. The analysis is based on the factorization of the current-gap operator (Λ − Λ 0 ).…”

“…Hence, one needs to employ a regularization technique to find an approximate solution to the discretized equation. In our experiments, we use the Spectral cut-off as the regularization scheme and follow a similar procedure demonstrated in [27] where f α z represents the regularized solution to Im(A δ )f z = b z , and α > 0 denotes the regularization parameter. To define the imagining functional, we follow [26] to have the following:…”

“…This generalized Robin condition has that there is a jump in the normal derivative of the electrostatic potential across the delaminated subregion's boundary (i.e., the boundary between the healthy region and the unknown defective subregion) and is quasi-proportional to the electrostatic potential itself. This is a generalization to the boundary condition of the inverse shape problem studied in [21] and [27]. In non-invasive and non-destructive testing, one wishes to recover the location of all possible subregions of interest in a given material using data on the 348 GOVANNI GRANADOS, ISAAC HARRIS AND HEEJIN LEE material's surface.…”

“…It is clear that the sesquilinear form is bounded, whereas the coercivity on H 1 0 (Ω) can be shown by the assumptions on µ and γ as well as the Poincaré inequality. We also have that A(u 0 , φ) is a conjugate linear and bounded functional acting on H 1 0 (Ω), and using the Trace Theorem just as in [27], we have that…”