arXiv subjects

Govanni Granados

Publications and source records attributed to Govanni Granados.

6 recordsLinked to original sources

arXiv2026

Qualitative reconstruction methods for imaging interior Robin interfaces in EIT from Robin-to-Dirichlet data

We consider an inverse shape problem arising in electrical impedance tomography (EIT) for nondestructive testing, in which interior defects are modeled through Robin transmission conditions. Unlike classical formulations, we impose Robin boundary conditions on both the exterior measurement surface and the interior interface, and use the Robin-to-Dirichlet (RtD) map as the available data. Within this setting, we develop qualitative (non-iterative) reconstruction methods based on the Linear Sampling Method (LSM) and the Regularized Factorization Method (RFM), and derive new analytical characterizations that enable these methods to identify interior regions. We further propose a numerical implementation that incorporates regularization strategies and demonstrate, through experiments, that the methods reliably reconstruct interior regions of interest.

math.NA↗

arXiv2025

Recovering elastic subdomains with strain-gradient elastic interfaces from force measurements: the antiplane shear setting

We introduce and study a new inverse problem for antiplane shear in elastic bodies with strain-gradient interfaces. The setting is a homogeneous isotropic elastic body containing an inclusion separated by a thin interface endowed with higher-order surface energy. Using displacement-stress measurements on the exterior boundary, expressed through a certain Dirichlet-to-Neumann map, we show uniqueness in recovering both the shear and interface parameters, as well as the shape of the inclusion. To address the inverse shape problem, we adapt the factorization method to account for the complications introduced by the higher-order boundary operator and its nontrivial null space. The resulting characterization relies on pairs of sampling points rather than a single-point test used in classical factorization methods. After fixing an interior reference point, the reconstruction procedure reduces to a single-point sampling algorithm. Focusing on the latter stage, numerical experiments illustrate the feasibility of the proposed reconstruction method and suggest that the framework has potential for the nondestructive detection of interior inhomogeneities, including damaged subvolumes.

math.AP↗

arXiv2025

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

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.

math.AP↗

arXiv2023

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

We consider an inverse shape problem coming from electrical impedance tomography with a generalized Robin transmission condition. We will derive an algorithm in order to detect whether two materials that should be in contact are separated or delaminated. More precisely, we assume that the undamaged material or background state is known and shares an interface or boundary with the damaged subregion. The Robin transmission condition on this boundary asymptotically models delamination. We assume that the DtN operator is given from measuring the current on the surface of the material from an imposed voltage. We show that this mapping uniquely recovers the boundary parameters. Furthermore, using this electrostatic Cauchy data as physical measurements, we can determine if all of the coefficients from the Robin transmission condition are real-valued or complex-valued. We study these two cases separately and show that the regularized factorization method can be used to detect whether delamination has occurred and recover the damaged subregion. Numerical examples will be presented for both cases in two dimensions in the unit circle.

math.AP↗

arXiv2023

Reciprocity gap functional methods for potentials/sources with small volume support for two elliptic equations

In this paper, we consider inverse shape problems coming from diffuse optical tomography and inverse scattering. In both problems, our goal is to reconstruct small volume interior regions from measured data on the exterior surface of an object. In order to achieve this, we will derive an asymptotic expansion of the reciprocity gap functional associated with each problem. The reciprocity gap functional takes in the measured Cauchy data on the exterior surface of the object. In diffuse optical tomography, we prove that a MUSIC-type algorithm can be used to recover the unknown subregions. This gives an analytically rigorous and computationally simple method for recovering the small volume regions. For the problem coming from inverse scattering, we recover the subregions of interest via a direct sampling method. The direct sampling method presented here allows us to accurately recover the small volume region from one pair of Cauchy data. We also prove that the direct sampling method is stable with respect to noisy data. Numerical examples will be presented for both cases in two dimensions where the measurement surface is the unit circle.

math.AP↗

arXiv2022

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

We consider an inverse shape problem coming from electrical impedance tomography with a Robin transmission condition. In general, a boundary condition of Robin type models corrosion. In this paper, we study two methods for recovering an interior corroded region from electrostatic data. We consider the case where we have small volume and extended regions. For the case where the region has small volume, we will derive an asymptotic expansion of the current gap operator and prove that a MUSIC-type algorithm can be used to recover the region. In the case where one has an extended region, we will show that the regularized factorization method can be used to recover said region. Numerical examples will be presented for both cases in two dimensions in the unit circle.

math.AP↗