On uniqueness in the inverse conductivity problem

Sever, Ali

doi:10.1002/(sici)1099-1476(199908)22:12<953::aid-mma10>3.0.co;2-j

Cited by 4 publications

(6 citation statements)

References 12 publications

(7 reference statements)

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“…(2.1) is ill-posed.For the integral Eq. (2.1), with a Riesz-type kernel and non-intersecting domains O and O k , there is uniqueness in L 2 (O)[9]…”

supporting

confidence: 74%

A neural network algorithm to pattern recognition in inverse problems

Sever¹

2013

Applied Mathematics and Computation

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“…(2.1) is ill-posed.For the integral Eq. (2.1), with a Riesz-type kernel and non-intersecting domains O and O k , there is uniqueness in L 2 (O)[9]…”

supporting

confidence: 74%

A neural network algorithm to pattern recognition in inverse problems

Sever¹

2013

Applied Mathematics and Computation

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“…This idea was introduced and used in [17] to prove the uniqueness of transparent obstacles from the Dirichlet-to-Neumann map. The uniqueness of some analytic transparent obstacles follows from the methods and the results of Kohn and Vogelius [28], and piecewise analytic conductivities with Lipschitz interfaces is shown in [35]. In the anisotropic case, the most general results on the identification of inclusion of different conductivities are obtained in [23].…”

Section: Many Boundary Measurementsmentioning

confidence: 98%

“…with the transmission condition (5) on the interface and the Neumann data 35) it follows that v 0 on 0 is given as well (for any Neumann data G 1 ). Hence we are given a partial Dirichlet-to-Neumann map for the elliptic equations (with any large τ ).…”

Section: Hyperbolic Equationsmentioning

confidence: 98%

Inverse obstacle problems

Isakov

2009

Inverse Problems

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In this review, we describe the main results on the uniqueness and stability of penetrable or impenetrable obstacles from various boundary data. We present basic ideas of proofs in the case of single and many boundary measurements. We discuss some methods of reconstruction. We consider the inverse problem of gravimetry, inverse conductivity and scattering problems, and similar inverse obstacle problems for hyperbolic and parabolic equations. We expose mostly known results and some new ideas and approaches. In conclusion, we list several open research problems of significance for theory and applications.

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“…Hence we are given a partial Dirichlet-to-Neumann map for all elliptic equations (55). Due to analyticity we can take τ = 0, then uniqueness of piecewise Lipschitz constant a follows from the generalization of results of of Kohn and Vogelius [20] given by Sever [25]. Similarly one can show uniqueness of a 0 .…”

Section: Compressible Fluidsmentioning

confidence: 85%

“…Given all coefficients and source terms, we have an elliptic free boundary problem (23), (24), (25). This free boundary problem is not overdetermined in R 2 and it is overdetermined in R 3 .…”

Section: Muskat Free Boundary Modelmentioning

confidence: 99%

On inverse problems in secondary oil recovery

Isakov¹

2008

Eur. J. Appl. Math

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We review simple models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first-order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational difficulties mainly due to the degeneracy of the system. The practical value of the problem justifies efforts to improve the methods for its solution. We formulate ‘history matching’ as a problem of identification of two coefficients of this system. We consider global and local versions of this inverse problem and propose some approaches, including the use of the inverse conductivity problem and the structure of fundamental solutions. The global approach looks for properties of the ground in the whole domain, while the local one is aimed at recovery of these properties near wells. We discuss the use of the model proposed by Muskat which is a difficult free boundary problem. The inverse Muskat problem combines features of inverse elliptic and hyperbolic problems. We analyse its linearisation about a simple solution and show uniqueness and exponential instability for the linearisation.

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On uniqueness in the inverse conductivity problem

Cited by 4 publications

References 12 publications

A neural network algorithm to pattern recognition in inverse problems

A neural network algorithm to pattern recognition in inverse problems

Inverse obstacle problems

On inverse problems in secondary oil recovery

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