Stability estimates for the inverse boundary value problem by partial Cauchy data

Heck, Horst; Wang, Jenn‐Nan

doi:10.1088/0266-5611/22/5/015

Cited by 74 publications

(109 citation statements)

References 16 publications

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“…We point out that, concerning the elliptic case, a stability estimate corresponding to the uniqueness result by Bukhgeim and Uhlmann [6] was established by Heck and Wang [27]. This result was improved by the authors and Soccorsi in [17].…”

Section: 3)mentioning

confidence: 70%

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

Choulli

Kian

2018

Journal de Mathématiques Pures et Appliquées

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Abstract. We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding "linearized" Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann [6]. This parabolic Carleman inequality enters in an essential way in the construction of CGO solutions that vanish at a part of the lateral boundary.

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Section: 3)mentioning

confidence: 70%

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

Choulli

Kian

2018

Journal de Mathématiques Pures et Appliquées

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“…In [20] the regularity assumption on the conductivity was relaxed to C 3/2+α with some α > 0. The corresponding stability estimates are proved in [15]. In [19], the result in [8] was generalized to show that by measuring all possible pairs of Dirichlet data on a possibly very small subsets of the boundary Γ + and Neumann data on a slightly larger open domain than ∂Ω \ Γ + , one can uniquely determine the potential.…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

139

193

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We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.

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“…The regularity assumption on the conductivity was relaxed to C 1+ , > 0 in [108]. Stability estimates for the uniqueness result of [35] were given in [76]. Stability estimates for the magnetic Schrödinger operator with partial data in the setting of [35] can be found in [198].…”

Section: The Partial Data Problemmentioning

confidence: 99%

Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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Stability estimates for the inverse boundary value problem by partial Cauchy data

Cited by 74 publications

References 16 publications

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

The Calderón problem with partial data in two dimensions

Inverse problems: seeing the unseen

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