Shape and parameter reconstruction for the Robin transmission inverse problem

Laurain, Antoine; Meftahi, Houcine

doi:10.1515/jiip-2015-0008

Cited by 7 publications

(5 citation statements)

References 44 publications

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“…Such problem arises in many physical situations such as electrical impedance tomography (for instance, ). In , a partial differential equation with Robin‐type transmission conditions, which models the situation where the corrosion takes place between two layers of a non‐homogenous medium, is considered. The authors provide an algorithm for the recovery of the Robin parameter and the jump set of the conductivity, either independently or simultaneously.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis in the inverse Robin transmission problem

Meftahi

2016

Math Methods in App Sciences

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In this paper, we consider the conductivity problem with piecewise-constant conductivity and Robin-type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non-monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn-Vogelius-type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first-order and second-order shape derivative of the proposed cost functional, which is performed rigorously by means of shape-Lagrangian formulation without using the shape sensitivity of the states variables.

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Section: Introductionmentioning

confidence: 99%

Stability analysis in the inverse Robin transmission problem

Meftahi

2016

Math Methods in App Sciences

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“…Thus, we assume that we have full knowledge of the Dirichlet-to-Neumann (DtN) mapping on the exterior boundary for Laplace's equation in the domain with a delaminated subregion. In [25,24,32], the authors studied the inverse parameter problem for the EIT problem with a Robin transmission condition. In the aforementioned papers, the authors studied the uniqueness, stability and numerical reconstruction for the inverse parameter problem using the Neumann-to-Dirichlet mapping.…”

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confidence: 99%

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

Granados,

Harris,

Lee

2023

CAC

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In this paper, we consider an inverse shape problem coming from electrical impedance tomography (EIT) 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 Dirichlet-to-Neumann (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.

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“…The corrosion may occur in many different forms and several models are considered in the literature [10,62,60,21,23,17,55]. Identifying the Robin parameter from boundary measurements turns out to be a way to locate the corroded part in a given medium and possibly evaluate the damage level by electrical impedance tomography process.…”

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confidence: 99%

“…In this work, we consider the mathematical model problem where the corrosion takes place between two layers of a non-homogenous medium [10,62,60]. The geometry of the corrosion boundary is assumed to be known in advance, but the coefficient of corrosion is unknown and is subject to be reconstructed from the Neumann-to-Dirichlet map.…”

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Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission Problem

Harrach¹,

Meftahi²

2019

SIAM J. Appl. Math.

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In this paper, we consider the inverse problem of detecting a corrosion coefficient between two layers of a conducting medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the description of the index of corrosion in non-destructive testing. We show a monotonicity estimates between the Robin coefficient and the Neumann-to-Dirichlet operator. We prove a global uniqueness result and Lipschitz stability estimate, and show how to quantify the Lipschitz stability constant for a given setting.Our quantification of the Lipschitz constant does not rely on quantitative unique continuation or analytic estimates of special functions. Instead of deriving an analytic estimate, we show that the Lipschitz constant for a given setting can be explicitly calculated from the a priori data by solving finitely many well-posed PDEs. Our arguments rely on standard (non-quantitative) unique continuation, a Runge approximation property, the monotonicity result and the method of localized potentials.To solve the problem numerically, we reformulate the inverse problem into a minimization problem using a least square functional. The reformulation of the minimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup formulation. The reconstruction is then performed by means of the BFGS algorithm. Finally, numerical results are presented to illustrate the efficiency of the proposed alogorithm.

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Shape and parameter reconstruction for the Robin transmission inverse problem

Cited by 7 publications

References 44 publications

Stability analysis in the inverse Robin transmission problem

Stability analysis in the inverse Robin transmission problem

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

Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission Problem

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