Lipschitz stability in determining density and two Lamé coefficients

Bellassoued, Mourad; Yamamoto, Masahiro

doi:10.1016/j.jmaa.2006.06.094

Cited by 37 publications

(23 citation statements)

References 41 publications

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“…As for Carleman estimates for functions without compact supports and applications to inverse problems for the Lamé system without the integral terms, we refer to Imanuvilov and Yamamoto [19] - [22]. For the Carleman estimate for with Lamé system with L λ, µ (x, x 0 , D ′ ) = 0, see Bellassoued, Imanuvilov and Yamamoto [2], Bellassoued and Yamamoto [3], Imanuvilov, Isakov and Yamamoto [24]. In this paper, we modify the arguments in those papers and establish a Carleman estimate for (1.1) for u not having compact supports.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Carleman Estimate for Linear Viscoelasticity Equations and an Inverse Source Problem

Imanuvilov¹,

Yamamoto²

2020

SIAM J. Math. Anal.

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We consider the linear system of viscoelasticity with the homogeneous Dirichlet boundary condition. First we prove a Carleman estimate with boundary values of solutions of viscoelasticity system. Since a solution u under consideration is not assumed to have compact support, in the decoupling of the Lamé operator by introducing div u and rot u, we have no boundary condition for them, so that we have to carry out arguments by a pseudodifferential operator. Second we apply the Carleman estimate to an inverse source problem of determining a spatially varying factor of the external source in the linear viscoelastitiy by extra Neumann data on the lateral subboundary over a sufficiently long time interval and establish the stability estimate.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Carleman Estimate for Linear Viscoelasticity Equations and an Inverse Source Problem

Imanuvilov¹,

Yamamoto²

2020

SIAM J. Math. Anal.

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“…The original idea of using a Carleman estimate to solve inverse problems goes back to the pioneering article [8] by Bukhgeim and Klibanov. Since then this technique has then been widely and succesfully used by numerous authors, see e.g. [3,4,6,7,12,15,16,19,23,25], in the study of inverse wave propagation, elasticity or parabolic problems.…”

Section: Existing Papersmentioning

confidence: 99%

A Carleman estimate for infinite cylindrical quantum domains and the application to inverse problems

2014

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We consider the inverse problem of determining the time independent scalar potential q of the dynamic Schrödinger equation in an infinite cylindrical domain Ω, from one Neumann boundary observation of the solution. Assuming that q is known outside some fixed compact subset of Ω, we prove that q may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schrödinger operator acting in an unbounded cylindrical domain, the Neumann data is measured on an infinitely extended subboundary of the cylinder.

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“…Several recent works have addressed uniqueness and Lipschitz stability questions for the problem of determining finitely many parameters (e.g., by assuming piecewiseconstantness) from finitely or infinitively many measurements in inverse coefficient problems, cf., [2][3][4][5][6][7]11,15,[18][19][20][21][22]32,48,51,53,68,69,71,76,77,87,93,96,104,105]. To the knowledge of the author, the results presented herein, is the first on explicitly calculating those measurements that uniquely determine the unknown parameters, and, together with [53], it is the first result to explicitly calculate the Lipschitz constant for a given setting.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem

Harrach

2020

Numer. Math.

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We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton’s method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.

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Lipschitz stability in determining density and two Lamé coefficients

Cited by 37 publications

References 41 publications

Carleman Estimate for Linear Viscoelasticity Equations and an Inverse Source Problem

Carleman Estimate for Linear Viscoelasticity Equations and an Inverse Source Problem

A Carleman estimate for infinite cylindrical quantum domains and the application to inverse problems

Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem

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