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33
The ∂approach to approximate inverse scattering at fixed energy in three dimensions
 IMRP Int. Math. Res. Pap
, 2005
"... We develop the ¯ ∂approach to inverse scattering at fixed energy in dimensions d ≥ 3 of [Beals, Coifman 1985] and [Henkin, Novikov 1987]. As a result we propose a stable method for nonlinear approximate finding a potential v from its scattering amplitude f at fixed energy E> 0 in dimension d = 3. I ..."
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Cited by 18 (10 self)
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We develop the ¯ ∂approach to inverse scattering at fixed energy in dimensions d ≥ 3 of [Beals, Coifman 1985] and [Henkin, Novikov 1987]. As a result we propose a stable method for nonlinear approximate finding a potential v from its scattering amplitude f at fixed energy E> 0 in dimension d = 3. In particular, in three dimensions we stably reconstruct ntimes smooth potential v with sufficient decay at infinity, n> 3, from its scattering amplitude f at fixed energy E up to O(E −(n−3−ε)/2) in the uniform norm as E → + ∞ for any fixed arbitrary small ε> 0 (that is with almost the same decay rate of the error for E → + ∞ as in the linearized case near zero potential). where where where
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.
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].
On the Parametrization of Illposed Inverse Problems Arising from Elliptic Partial Differential Equations
, 2006
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An effectivization of the global reconstruction in the Gel’fandCalderon inverse problem in three dimensions
, 2008
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Quantitative estimates of unique continuation for parabolic equations, determination of unknown timevarying boundaries and optimal stability estimates
, 2007
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Circular resistor networks for electrical impedance tomography with partial boundary measurements.
"... Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in ..."
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Cited by 5 (4 self)
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Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on socalled optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting tradeoff between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary. 1.