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23
On a characterization of the kernel of the DirichlettoNeumann map for a planar region
 SIAM Journal on Applied Mathematics
, 1998
"... Abstract. We will show that the DirichlettoNeumann map Λ for the electrical conductivity equation on a simply connected plane region has an alternating property, which may be considered as a generalized maximum principle. Using this property, we will prove that the kernel, K, of Λ satisfies a set ..."
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Abstract. We will show that the DirichlettoNeumann map Λ for the electrical conductivity equation on a simply connected plane region has an alternating property, which may be considered as a generalized maximum principle. Using this property, we will prove that the kernel, K, of Λ satisfies a set of inequalities of the form (−1)n(n+1)2 detK(xi, yj)> 0. We will show that these inequalities imply Hopf’s lemma for the conductivity equation. We will also show that these inequalities imply the alternating property of a kernel. 1.
Electrical networks, symplectic reductions, and applications to the renormalization map of selfsimilar lattices
 PROC. OF SYMP. IN PURE MATH., MANDELBROT JUBILEE. ARXIV/MATHPH/0304015
, 2004
"... The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the “trace map ” and the “gluing map”) and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductio ..."
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Cited by 6 (2 self)
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The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the “trace map ” and the “gluing map”) and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductions, in particular we study the singularities of symplectic reductions when considered as rational maps on Lagrangian Grassmannians. This is motivated by [23] where a renormalization map was introduced in order to describe the spectral properties of selfsimilar lattices. In this text, we show that this renormalization map can be expressed in terms of symplectic reductions and that some of its key properties are direct consequences of general properties of symplectic reductions (and the singularities of the symplectic reduction play an important role in relation with the spectral properties of our operator). We also present new examples where we can compute the renormalization map.
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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FourTerminal Reducibility and ProjectivePlanar WyeDeltaWyeReducible Graphs
 J. GRAPH THEORY
, 2000
"... A graph is Y∆Y reducible if it can be reduced to a vertex by a sequence of seriesparallel reductions and Y∆Ytransformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that fourterminal planar graphs are Y∆Yreducible w ..."
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A graph is Y∆Y reducible if it can be reduced to a vertex by a sequence of seriesparallel reductions and Y∆Ytransformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that fourterminal planar graphs are Y∆Yreducible when at least three of the vertices lie on the same face. Using this result we characterize Y∆Yreducible projectiveplanar graphs. We also consider terminals in projectiveplanar graphs, and establish that graphs of crossingnumber one are Y∆Yreducible.
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.
Discrete period matrices and related topics
, 2001
"... Abstract. We continue our investigation of Discrete Riemann Surfaces with the discussion of the discrete analogs of period matrices, Riemann’s bilinear relations, the Jacobian variety, exponential of constant argument, series and electrical moves. 1. ..."
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Abstract. We continue our investigation of Discrete Riemann Surfaces with the discussion of the discrete analogs of period matrices, Riemann’s bilinear relations, the Jacobian variety, exponential of constant argument, series and electrical moves. 1.
Spectre D'Opérateurs Differentiels Sur Les Graphes
, 2006
"... Dans cet exposé de survol, nous commençons par présenter des ensembles naturels d’opérateurs de type Schrödinger associés à un graphe fini. Nous étudions ensuite les limites singulières (au sens de la Γ−convergence) de tels opérateurs et montrons qu’elles sont associées à des relations naturelles en ..."
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Dans cet exposé de survol, nous commençons par présenter des ensembles naturels d’opérateurs de type Schrödinger associés à un graphe fini. Nous étudions ensuite les limites singulières (au sens de la Γ−convergence) de tels opérateurs et montrons qu’elles sont associées à des relations naturelles entre graphes (mineurs, transformation étoiletriangle) ou à des limites d’un intérêt indépendant (processus de Markov (recuit simulé), réseaux électriques). Cela conduit à introduire la notion de stabilité structurelle pour une valeur propre multiple d’un opérateur d’une famille en utilisant la transversalité dans les espaces d’opérateurs symétriques. Les invariants numériques de graphes ainsi construits sont liés à des problèmes classiques de la combinatoire des graphes: planarité, genre, plongement nonnoué dans R³, largeur d’arbre.
STUDY OF NOISE EFFECTS IN ELECTRICAL IMPEDANCE TOMOGRAPHY WITH RESISTOR NETWORKS
, 1105
"... Abstract. We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite vol ..."
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Abstract. We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum aposterioriestimatesoftheconductivity,onoptimalgrids. Forsmallnoise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the CramérRao bound. For larger noise we use regularization and quantify the tradeoff between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered. 1.