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### Open Access Journals

IPI

A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm
based on the global uniqueness proof of Nachman [Ann. Math.

**143**(1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.
IPI

We extend results of Dos Santos Ferreira-Kenig-Sjöstrand-Uhlmann
(Comm. Math. Phys., 2007) to less smooth coefficients, and we show
that measurements on part of the boundary for the magnetic
Schrödinger operator determine uniquely the magnetic field related
to a Hölder continuous potential. We
give a similar result for determining a convection term. The
proofs involve Carleman estimates, a smoothing procedure, and an
extension of the Nakamura-Uhlmann pseudodifferential
conjugation method to logarithmic Carleman weights.

IPI

Complex Geometrical Optics (CGO) solutions have, for almost three decades, played a large role in the rigorous analysis of nonlinear inverse problems. They have the added bonus of also being useful in practical reconstruction algorithms. The main benefit of CGO solutions is to provide solutions in the form of almost-exponential functions that can be used in a variety of ways, for example for defining tailor-made nonlinear Fourier transforms to study the unique solvability of a nonlinear inverse problem.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

keywords:

IPI

The Calderón problem is the mathematical formulation of the inverse problem in Electrical Impedance Tomography and asks for the uniqueness and reconstruction of an electrical conductivity distribution in a bounded domain from the knowledge of the Dirichlet-to-Neumann map associated to the generalized Laplace equation. The 3D problem was solved in theory in late 1980s using complex geometrical optics solutions and a scattering transform. Several approximations to the reconstruction method have been suggested and implemented numerically in the literature, but here, for the first time, a complete computer implementation of the full nonlinear algorithm is given. First a boundary integral equation is solved by a Nyström method for the traces of the complex geometrical optics solutions, second the scattering transform is computed and inverted using fast Fourier transform, and finally a boundary value problem is solved for the conductivity distribution. To test the performance of the algorithm highly accurate data is required, and to this end a boundary element method is developed and implemented for the forward problem. The numerical reconstruction algorithm is tested on simulated data and compared to the simpler approximations. In addition, convergence of the numerical solution towards the exact solution of the boundary integral equation is proved.

PROC

Two algorithms for the direct reconstruction of conductivities in
a bounded domain in $\mathbb{R}^3$ from surface measurements of the solutions to the
conductivity equation are presented. The algorithms are based on complex
geometrical optics solutions and a nonlinear scattering transform. We test the
algorithms on three numerically simulated examples, including an example with
a complex coefficient. The spatial resolution and amplitude of the examples
are well-reconstructed.

PROC

In electrical impedance tomography the electrical
conductivity inside a physical body is computed from electro-static
boundary
measurements.
The focus of this paper is to extend recent results for the 2D
problem to 3D: prior information about the sparsity and spatial distribution of the conductivity is used to improve
reconstructions for the partial data problem with Cauchy
data measured only on a subset of the boundary. A sparsity prior is enforced
using the $\ell_1$ norm in the penalty term of a Tikhonov functional, and spatial
prior information is incorporated by applying a spatially
distributed regularization parameter. The optimization problem is
solved numerically using a generalized conditional gradient method
with soft thresholding. Numerical examples show the effectiveness of
the suggested method even for the partial data problem with
measurements affected by noise.

keywords:
sparsity
,
numerical
reconstruction.
,
partial data
,
prior information
,
Impedance tomography

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