Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition

Bacchelli, Valeria

doi:10.1088/0266-5611/25/1/015004

Cited by 37 publications

(44 citation statements)

References 19 publications

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“…The question is the following: does a measurement (or several measurements) determine uniquely the domain ω? This kind of result was recently proved by Bacchelli in [5] for Robin boundary conditions. In the case of generalized impedance boundary condition, the literature is reduced to the discussion by Cakoni et al in [11].…”

Section: The Studied Problemmentioning

confidence: 55%

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Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

2013

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We aim to reconstruct an inclusion ω immersed in a perfect fluid flowing in a larger bounded domain via boundary measurements on ∂ . The obstacle ω is assumed to have a thin layer and is then modeled using generalized boundary conditions (precisely Ventcel boundary conditions). We first obtain an identifiability result (i.e. the uniqueness of the solution of the inverse problem) for annular configurations through explicit computations. Then, this inverse problem of reconstructing ω is studied, thanks to the tools of shape optimization by minimizing a least-squares-type cost functional. We prove the existence of the shape derivatives with respect to the domain ω and characterize the gradient of this cost functional in order to make a numerical resolution. We also characterize the shape Hessian and prove that this inverse obstacle problem is unstable in the following sense: the functional is degenerate for highly oscillating perturbations. Finally, we present some numerical simulations in order to confirm and extend our theoretical results.

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Section: The Studied Problemmentioning

confidence: 55%

“…To solve this inverse problem, we consider, for ω ∈ O δ , the least squares functional 5) measuring in the misfit to data in the L 2 sense. Notice that, according to Theorem A.2, the boundary value problem (2.5) admits a unique solution u ∈ H 3 (Ω\ω).…”

Section: The Studied Problemmentioning

confidence: 99%

Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

2013

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“…Even when α is known, one set of Cauchy data (2.1) and (2.3) may not be enough to determine uniquely the corroded boundary Γ 2 , as shown by the counterexamples given in [7,9,35] and some thorough numerical investigation reported in [15]. However, it turns out that two linearly independent boundary data f 1 and f 2 , one of which is positive, inducing, via (2.3), two corresponding flux measurements g 1 and g 2 , are sufficient to provide a unique solution for the pair (Γ 2 , α), [1,34,35]. The stability issue has also been recently addressed in [36] and numerical results based either on a potential approach or on a Green's integral formulation have been reported in [10].…”

Section: Mathematical Formulationmentioning

confidence: 94%

Simultaneous numerical determination of a corroded boundary and its admittance

Karageorghis

Bin‐Mohsin

Lesnic

et al. 2014

Inverse Problems in Science and Engineering

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Abstract. In this paper, an inverse geometric problem for Laplace's equation arising in boundary corrosion detection is considered. This problem, which consists of determining an unknown corroded portion of the boundary of a bounded domain and its admittance Robin coefficient from two pairs of boundary Cauchy data (boundary temperature and heat flux), is solved numerically using the meshless method of fundamental solutions. A nonlinear minimisation of the objective function is regularised, and the stability of the numerical results is investigated with respect to noise in the input data and various values of the regularisation parameters involved.

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“…(1.1) Boundary and parameter identification problems related to this elliptic equation have been considered by many authors even recently, see e.g., [3,5,7,8,11,15,18]. According to this model, Ω represents a conductor which contains no sources and no sinks, so that the potential u is harmonic.…”

Section: Introductionmentioning

confidence: 99%

Linearization of a free boundary problem in corrosion detection

Cabib

Fasino

Sincich

2011

Journal of Mathematical Analysis and Applications

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We consider a boundary identification problem arising in nondestructive testing of materials. The problem is to recover a part Gamma(1) subset of partial derivative Omega of the boundary of a bounded, planar domain Omega from one Cauchy data pair (u, partial derivative u/partial derivative v) of a harmonic potential u in Omega collected on an accessible boundary subset Gamma(A) subset of partial derivative Omega. We prove Frechet differentiability of a suitably defined forward map, and discuss local uniqueness and Lipschitz stability results for the linearized problem. (C) 2011 Elsevier Inc. All rights reserved

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Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition

Abstract: We consider the problem of determining the corroded portion of the boundary of a n-dimensional body (n=2, 3) and the impedance by two measures on the accessible portion of the boundary. On the unknown boundary part it is assumed the Robin homogeneous condition.

Cited by 37 publications

References 19 publications

Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

Simultaneous numerical determination of a corroded boundary and its admittance

Linearization of a free boundary problem in corrosion detection

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