Results 1  10
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61
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 44 (6 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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Exponential instability in an inverse problem for the Schrödinger equation
 Inverse Problems 17:5 (2001), 1435–1444. MR 2002h:35339 Zbl 0985.35110
"... Abstract. We consider the problem of the determination of the potential from the Dirichlet to Neumann map of the Schrödinger operator.We show that this problem is severely ill posed.The results extend to the electrical impedance tomography.They show that the logarithmic stability results of Alessand ..."
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Cited by 37 (0 self)
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Abstract. We consider the problem of the determination of the potential from the Dirichlet to Neumann map of the Schrödinger operator.We show that this problem is severely ill posed.The results extend to the electrical impedance tomography.They show that the logarithmic stability results of Alessandrini are optimal. 1
Boundary Regularity for the Ricci Equation, Geometric Convergence, and Gel'fand's Inverse Boundary Problem
"... Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The secon ..."
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Cited by 14 (13 self)
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Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts. 1.
A global stability estimate for the Gel’fandCalderón inverse problem in two dimensions
, 2010
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Formulae and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential
 Inverse Problems
"... For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Nov ..."
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Cited by 14 (8 self)
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For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Novikov, Multidimensional inverse spectral problem for the equation −∆ψ + (v(x) − Eu(x))ψ = 0, Funkt. Anal. i Ego Prilozhen
Recent Progress in Electrical Impedance Tomography
 Inverse Problems, 19, S65S90
, 2003
"... We consider the inverse problem of finding cavities within some body from electrostatic measurements on the boundary. By a cavity we understand any object with a different electrical conductivity than the background material of the body. We survey two algorithms for solving this inverse problem, nam ..."
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Cited by 9 (2 self)
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We consider the inverse problem of finding cavities within some body from electrostatic measurements on the boundary. By a cavity we understand any object with a different electrical conductivity than the background material of the body. We survey two algorithms for solving this inverse problem, namely the factorization method and a MUSICtype algorithm. In particular, we present a number of numerical results to highlight the potential and the limitations of these two methods.
Inverse problems for nonsmooth first order perturbations of the Laplacian
, 2004
"... We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of Sylves ..."
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Cited by 8 (4 self)
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We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of SylvesterUhlmann type to achieve this. A main tool is an extension of the NakamuraUhlmann intertwining method to operators which have continuous coefficients. For the inverse conductivity problem for a C 1+ε conductivity, we construct complex geometrical optics solutions whose properties depend explicitly on ε. This implies the uniqueness result of PäivärintaPanchenkoUhlmann for C 3/2 conductivities. For the magnetic Schrödinger equation, the result is that the DirichlettoNeumann map uniquely determines the magnetic field corresponding to a Dini continuous magnetic potential in C 1,1 domains. For the steady state heat equation with a convection term, we obtain global uniqueness of Lipschitz continuous convection terms in Lipschitz
On nonoverdetermined inverse scattering at zero energy in three dimensions
 Ann. Scuola Norm. Sup. Pisa Cl. Sci
"... Abstract. We develop the ¯ ∂ approach to inverse scattering at zero energy in dimensions d ≥ 3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruc ..."
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Cited by 7 (6 self)
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Abstract. We develop the ¯ ∂ approach to inverse scattering at zero energy in dimensions d ≥ 3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruction for the problem of finding a sufficiently small potential v in the Schrödinger equation from a fixed nonoverdetermined (”backscattering” type) restriction h ∣ ∣ Γ of the Faddeev generalized scattering amplitude h in the complex domain at zero energy in dimension d = 3. For sufficiently small potentials v we formulate also a characterization theorem for the aforementioned restriction h ∣ ∣ Γ and a new characterization theorem for the full Faddeev function h in the complex domain at zero energy in dimension d = 3. We show that the results of the present work have direct applications to the electrical impedance tomography via a reduction given first in [Novikov, 1987, 1988].