Results 1  10
of
27
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 ..."
Abstract

Cited by 37 (0 self)
 Add to MetaCart
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
A global stability estimate for the Gel’fandCalderón inverse problem in two dimensions
, 2010
"... ..."
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
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...
Uniqueness in the Inverse Conductivity Problem for Conductivites with 3/2 Derivatives in L^p, p > 2n
"... this paper, we contribute nothing to the analysis of G # . The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L with p > ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
this paper, we contribute nothing to the analysis of G # . The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L with p > 2n/3. However, the straightforward generalization of the argument presented below would require that f # #G # f map functions which are compactly supported to functions which are locally in L with p and r satisfying 1/p 1/r = 1/n. Many such estimates fail, see [2] for further discussion
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 ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
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 ..."
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].
Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements
"... ..."
Reconstruction of less regular conductivities in the plane
 Comm. Partial Differential Equations
"... Abstract. We study the inverse conductivity problem of how to reconstruct an isotropic electrical conductivity distribution γ in an object from static electrical measurements on the boundary of the object. We give an exact reconstruction algorithm for the conductivity γ ∈ C 1+ǫ (Ω) in the plane doma ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. We study the inverse conductivity problem of how to reconstruct an isotropic electrical conductivity distribution γ in an object from static electrical measurements on the boundary of the object. We give an exact reconstruction algorithm for the conductivity γ ∈ C 1+ǫ (Ω) in the plane domain Ω from the associated Dirichlet to Neumann map on ∂Ω. Hence we improve earlier reconstruction results. The method used relies on a wellknown reduction to a first order system, for which the ∂method of inverse scattering theory can be applied. 1.
Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
"... by ..."