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16
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares 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 ..."
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Cited by 59 (6 self)
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We introduce an output leastsquares 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
A global stability estimate for the Gel’fandCalderón inverse problem in two dimensions
, 2010
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On the Parametrization of Illposed Inverse Problems Arising from Elliptic Partial Differential Equations
, 2006
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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.
Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements
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Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
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Resistor network approaches to electrical impedance tomography
 Inside Out, Mathematical Sciences Research Institute Publications
, 2011
"... We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call ..."
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Cited by 2 (1 self)
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We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call optimal. The name refers to the fact that the grids give spectrally accurate approximations of the Dirichlet to Neumann map, the data in EIT. The fundamental feature of the optimal grids in inversion is that they connect the discrete inverse problem for resistor networks to the continuum EIT problem. 1.
INVERSE PROBLEMS
, 2008
"... doi:10.1088/02665611/24/3/035013 Electrical impedance tomography with resistor networks ..."
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doi:10.1088/02665611/24/3/035013 Electrical impedance tomography with resistor networks
On the Continuum Limit of a . . .
, 2004
"... We consider finite difference approximations of solutions of inverse SturmLiouville problems in bounded intervals. Using threepoint finite difference schemes, we discretize the equations on socalled optimal grids constructed as follows: For a staggered grid with 2k points, we ask that the finite ..."
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We consider finite difference approximations of solutions of inverse SturmLiouville problems in bounded intervals. Using threepoint finite difference schemes, we discretize the equations on socalled optimal grids constructed as follows: For a staggered grid with 2k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the SturmLiouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal
A nonlinear multigrid . . . conductivity and permittivity at low frequency
, 2001
"... We propose a nonlinear multigrid approach for imaging the electrical conductivity and permittivity of a body �, given partial, usually noisy knowledge of the NeumanntoDirichlet map at the boundary. The algorithm is a nested iteration, where the image is constructed on a sequence of grids in �, sta ..."
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We propose a nonlinear multigrid approach for imaging the electrical conductivity and permittivity of a body �, given partial, usually noisy knowledge of the NeumanntoDirichlet map at the boundary. The algorithm is a nested iteration, where the image is constructed on a sequence of grids in �, starting from the coarsest grid and advancing towards the finest one. We show various numerical examples that demonstrate the effectiveness and robustness of the algorithm and prove local convergence. (Some figures in this article are in colour only in the electronic version; see www.iop.org)