Results 11  20
of
98
Global uniqueness from partial Cauchy data in two dimensions. Arxiv preprint arXiv:0810.2286
, 2008
"... Abstract. 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 determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an a ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Abstract. 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 determines uniquely 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 determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. 1.
The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction
 Comm. Pure Appl. Math
"... The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than th ..."
Abstract

Cited by 11 (10 self)
 Add to MetaCart
The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function
Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field
"... We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n, n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The me ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n, n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces. 1
Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result
"... A formula is given for recovering the boundary values of the coefficient of an elliptic operator, divr, from the Dirichlet to Neumann map. The main point is that one may recover without any a priori smoothness assumptions. The formula allows one to recover the value of pointwise. Let... ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
A formula is given for recovering the boundary values of the coefficient of an elliptic operator, divr, from the Dirichlet to Neumann map. The main point is that one may recover without any a priori smoothness assumptions. The formula allows one to recover the value of pointwise. Let...
Transparent potentials at fixed energy in dimension two. Fixedenergy dispersion relations for the fast decaying potentials
 Comm. Math. Phys
, 1995
"... Abstract: For the twodimensional Schrödinger equation [− ∆ + v(x)]ψ = Eψ, x ∈ R 2, E = Efixed> 0 at a fixed positive energy with a fast decaying at infinity potential v(x) dispersion relations on the scattering data are given.Under ”small norm ” assumption using these dispersion relations we give ( ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Abstract: For the twodimensional Schrödinger equation [− ∆ + v(x)]ψ = Eψ, x ∈ R 2, E = Efixed> 0 at a fixed positive energy with a fast decaying at infinity potential v(x) dispersion relations on the scattering data are given.Under ”small norm ” assumption using these dispersion relations we give (without a complete proof of sufficiency) a characterization of scattering data for the potentials from the Schwartz class S = C (∞) (R 2). For the potentials with zero scattering amplitude at a fixed energy Efixed (transparent potentials) we give a complete proof of this characterization. As a consequence we construct a family (parameterized by a function of one variable) of twodimensional sphericallysymmetric real potentials from the Schwartz class S transparent at a given energy. For the twodimensional case (without assumption that the potential is small) we show that there are no nonzero real exponentially decreasing at infinity, potentials transparent at a fixed energy. For any dimension greater or equal 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval. We show that KdVtype equations in
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 ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
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.
Matching Pursuit for Imaging High Contrast Conductivity
, 1999
"... We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solu ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solution of the high contrast imaging problem, if the library of functions is constructed carefully and in accordance with the asymptotic theory. We also show how other libraries of functions that at first glance seem reasonable, in fact, do not work well. When the contrast in the conductivity is not so high, we show that wavelets can be used, especially nonorthogonal wavelet libraries. However, the library of functions that is based on the high contrast asymptotic theory is more robust, even for intermediate contrasts, and especially so in the presence of noise. Key words. Impedance tomography, high contrast, asymptotic resistor network, imaging. Contents 1 Introduction 1 2 The Neumann to Dir...
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 ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
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
"... ..."
The Approximate Inverse for Solving an Inverse Scattering Problem for Acoustic Waves in an Inhomogeneous Medium
, 1999
"... An application of the method of approximate inverse to a twodimensional inverse scattering problem is discussed. ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
An application of the method of approximate inverse to a twodimensional inverse scattering problem is discussed.