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98
Electrical Impedance Tomography
 SIAM REVIEW
, 1999
"... This paper surveys some of the work our group has done in electrical impedance tomography. ..."
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Cited by 82 (1 self)
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This paper surveys some of the work our group has done in electrical impedance tomography.
The Calderón problem with partial data
 Ann. of Math. (to
"... In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n≥3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but ..."
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Cited by 48 (16 self)
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In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n≥3, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem.
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
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
Uniqueness in the Inverse Conductivity Problem for Nonsmooth Conductivities in Two Dimensions
, 1997
"... this paper) imply, via Sobolev embedding, that the conductivity is continuous. It is interesting to note that the only uniqueness results available for conductivities which are discontinuous are due to Kohn and Vogelius [9] who study conductivities which are piecewise analytic and V. Isakov [8] who ..."
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Cited by 34 (9 self)
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this paper) imply, via Sobolev embedding, that the conductivity is continuous. It is interesting to note that the only uniqueness results available for conductivities which are discontinuous are due to Kohn and Vogelius [9] who study conductivities which are piecewise analytic and V. Isakov [8] who considers a class of conductivities which are piecewise C
Linear convergence of iterative softthresholding
 J. Fourier Anal. Appl
"... ABSTRACT. In this article a unified approach to iterative softthresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis ..."
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Cited by 34 (9 self)
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ABSTRACT. In this article a unified approach to iterative softthresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the socalled finite basis injectivity property or the minimizer possesses a socalled strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method. 1.
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction.
, 1998
"... We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective ..."
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Cited by 23 (5 self)
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We derive an asymptotic formula for the electrostatic voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective computational identification procedure. 1. Introduction 2. The electrostatic problem 3. An energy estimate 4. Some additional preliminary estimates 5. An asymptotic formula for the voltage potential 6. Properties of the polarization tensor 7. The continuous dependence of the inhomogeneities 8. Computational results. 9. References 1 Introduction The nondestructive inspection technique known as electrical impedance imaging has recently received considerable attention in the mathematical as well as in the engineering literature [2, 4, 10, 14, 17]. Using this technique one seeks to determine information about the internal conductivity (or impedance) profile of an object based on boundary i...
A global stability estimate for the Gel’fandCalderón inverse problem in two dimensions
, 2010
"... ..."
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