Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary

Bacchelli, Valeria; Vessella, Sergio

doi:10.1088/0266-5611/22/5/007

Cited by 35 publications

(26 citation statements)

References 15 publications

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“…In this section we prove the main result that consists of showing the uniform continuity for DF and F −1 , and establishing a lower bound for DF . These results together with the Fréchet differentiability of F establish Theorem 2.3 by Proposition 5 of [6]. 4.1.…”

Section: Proof Of the Main Resultssupporting

confidence: 61%

Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves

et al. 2017

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Section: Proof Of the Main Resultssupporting

confidence: 61%

Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves

et al. 2017

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“…Therefore, asymptotically, which in practice means paying the price of ensuring a very small error on the measurement, the exponential ill-posedness of the problem may be kept under control. This is in accord with several results in which the illposedness of an inverse boundary value problem is tamed if the unknown features to be recovered can be described in a discrete way, see for example [2,4,5]. With respect to the unknown discrete boundaries considered in [2,5], the main novelty here is the fact that we deal with a three-dimensional, instead of two-dimensional, problem and that the number of pieces forming the unknown boundary, namely the number of cells, is not fixed and is not a priori known.…”

Section: Introductionsupporting

confidence: 90%

“…However, such a result has been proved only for scatterers of a special type. Following previous results by Liu and Nachman, [16], and Cheng and Yamamoto, [7,8], it has been proved in [3] that any polyhedral scatterer is determined in a unique way by a single far-field measurement. By a polyhedral scatterer we mean a scatterer whose boundary is the union of a finite number of cells, each cell being the closure of a domain contained in a hypersurface.…”

Section: Introductionmentioning

confidence: 69%

Stable determination of sound-soft polyhedral scatterers by a single measurement

Rondi¹

2008

Indiana Univ. Math. J.

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We prove optimal stability estimates for the determination of a finite number of sound-soft polyhedral scatterers in R^3 by a single far-field measurement. The admissible multiple polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time obstacles, screens and even more complicated scatterers. We characterize any multiple\ud polyhedral scatterer by a size parameter h which is related to the minimal size of the cells of its boundary. In a first step we show that, provided the error epsilon on the far-field measurement is small enough with respect to h, then the corresponding error, in the Hausdorff distance, on the multiple polyhedral scatterer can be controlled by an explicit function of epsilon which approaches zero,\ud as epsilon goes to 0, in an essentially optimal, although logarithmic, way. Then, we show how to improve this stability estimate, provided we restrict our attention to multiple polyhedral obstacles and epsilon is even smaller with respect to h. In this case we obtain an explicit estimate essentially of Hoelder type

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“…The numerical solution represents reasonable approximation and it is stable with respect to a small amount of noise as shown in figures 6,9. Figures 4,5,7,8,10,11, shown the decrease of cost function J and the L ∞ -norm of DJ(γ) in the course of the optimization process. 7.1.2.…”

Section: Numerical Examplesmentioning

confidence: 99%

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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Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary

Cited by 35 publications

References 15 publications

Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves

Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves

Stable determination of sound-soft polyhedral scatterers by a single measurement

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

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