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We introduce an output least-squares 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

Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts. 1.

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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 MUSIC-type algorithm. In particular, we present a number of numerical results to highlight the potential and the limitations of these two methods.

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 non-overdetermined (”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].

. The recent research of the author and his collaborators on multiscale computational methods is reported, emphasizing main ideas and inter-relations between various fields, and listing the relevant bibliography. The reported areas include: top-efficiency multigrid methods in fluid dynamics; atmospheric data assimilation; PDE solvers on unbounded domains; wave/ray methods for highly indefinite equations; many-eigenfunction problems and ab-initio quantum chemistry; fast evaluation of integral transforms on adaptive grids; multigrid Dirac solvers; fast inverse-matrix and determinant updates; multiscale Monte-Carlo methods in statistical physics; molecular mechanics (including fast force summation, fast macromolecular energy minimization, Monte-Carlo methods at equilibrium and the combination of small-scale equilibrium with large-scale dynamics); image processing (edge detection and segmentation); and tomography. Key words. scientific computation, multiscale, multi-resolution, multigrid,...

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 well-known reduction to a first order system, for which the ∂-method of inverse scattering theory can be applied. 1.

We consider the Cauchy data associated to the Schrödinger equation with a potential on a bounded domain � ⊂ R n, n ≥ 3. We show that the integral of the potential over a two-plane � is determined by the Cauchy data of certain exponentially growing solutions on any open subset U ⊂ ∂ � which contains � ∩ ∂�. For �, a bounded domain in R n with Lipschitz boundary, ∂�, and real-valued q(x) ∈ L ∞ (�), let �q: H 1/2 (∂�) − → H −1/2 (∂�) (0.1) be the Dirichlet-to-Neumann map associated with the operator � + q on �, which

Electrical Impedance Tomography of closed conductive media is an ill-posed inverse problem. Using the Finite Elements Method to solve the corresponding direct problem allows to preserve the nonlinear dependence of the observation set upon the conductivity distribution. In this paper, we show that the Bayesian ap- proach presented in [1] for linear inverse imaging prob- lems is still valid for such a non linear inverse pvblem. Our contribution is based on an edge-preserving Markov model as prior for conductivity distribution. Maximum A PosterJori reconstruction results from 40dB noisy measurements (simulated with a finer mesh) yield significant resolution improvement compared to classical methods.