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43
Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements
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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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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.
Combinatorics of Tripartite Boundary Connections for Trees and Dimers
"... A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular ..."
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A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for “tripartite ” pairings of the nodes, the probability can be computed as a Pfaffian in the entries of the DirichlettoNeumann matrix (discrete Hilbert transform) of the graph. These formulas generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings. These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements. We prove similar theorems for the doubledimer model on bipartite planar graphs. 1
Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
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Discrete geometric homogenisation and inverse homogenisation of an elliptic operator
"... iii We show how to parameterise a homogenised conductivity in R2 by a scalar function s(x), despite the fact that the conductivity parameter in the related upscaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of s(x), and with consider ..."
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iii We show how to parameterise a homogenised conductivity in R2 by a scalar function s(x), despite the fact that the conductivity parameter in the related upscaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of s(x), and with consideration to mesh connectivity, this equivalence extends to discrete parameterisations over triangulated domains. We apply the parameterisation in three contexts: (i) sampling s(x) produces a family of stiffness matrices representing the elliptic operator over a hierarchy of scales; (ii) the curvature of s(x) directs the construction of meshes welladapted to the anisotropy of the operator, improving the conditioning of the stiffness matrix and interpolation properties of the mesh; and (iii) using electric impedance tomography to reconstruct s(x) recovers the upscaled conductivity, which while anisotropic, is unique. Extensions of the parameterisation to R3 are introduced. iv Preface
Doubledimer pairings and skew Young diagrams
 Electron. J. Combin
"... We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are “vertically decreasing”. We use these quantities to compute pairing probabilities in the doubledimer model: Given a planar bipartite graph G with special vertices, called nodes, on t ..."
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We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are “vertically decreasing”. We use these quantities to compute pairing probabilities in the doubledimer model: Given a planar bipartite graph G with special vertices, called nodes, on the outer face, the doubledimer model is formed by the superposition of a uniformly random dimer configuration (perfect matching) of G together with a random dimer configuration of the graph formed from G by deleting the nodes. The doubledimer configuration consists of loops, doubled edges, and chains that start and end at the boundary nodes. We are interested in how the chains connect the nodes. An interesting special case is when the graph is ε(Z × N) and the nodes are at evenly spaced locations on the boundary R as the grid spacing ε → 0. 1
2010a). Spectral analysis of synchronization in a lossless structurepreserving power network model
 In IEEE International Conference on Smart Grid Communications. Gaithersburg, MD. Submitted
"... Abstract—This paper considers the synchronization and transient stability analysis in a simple model of a structurepreserving power system. We derive sufficient conditions relating synchronization in a power network directly to the underlying network state, parameters, and topology. In particular, ..."
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Abstract—This paper considers the synchronization and transient stability analysis in a simple model of a structurepreserving power system. We derive sufficient conditions relating synchronization in a power network directly to the underlying network state, parameters, and topology. In particular, we provide a spectral condition based on the algebraic connectivity of the network and a second condition based on the effective resistance among generators. These conditions build upon the authors’ earlier results on synchronization in networkreduced power system models. Central to our analysis is the reduced admittance matrix of the network, which is obtained by a Schur complement of the network’s topological admittance matrix with respect to its bus nodes. This networkreduction process, termed Kron reduction, relates the structurepreserving and the networkreduced power system model. We provide a detailed graphtheoretic, algebraic, and spectral analysis of the Kron reduction process leading directly to the novel synchronization conditions. I.
Low Frequency Electromagnetic Fields in High Contrast Media
"... . Using variational principles we construct discrete network approximations for the Dirichlet to Neumann or Neumann to Dirichlet maps of high contrast, low frequency electromagnetic media. 1 Introduction Imaging of the electrical conductivity and permittivity of a heterogeneous body by means of low ..."
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. Using variational principles we construct discrete network approximations for the Dirichlet to Neumann or Neumann to Dirichlet maps of high contrast, low frequency electromagnetic media. 1 Introduction Imaging of the electrical conductivity and permittivity of a heterogeneous body by means of lowfrequency electrical or electromagnetic field measurements is an inverse problem, often called "impedance tomography", "electromagnetic induction tomography", "magnetotellurics" and so on. Applications arise in many areas, for example in medicine with diagnostic imaging, in nondestructive testing, in oil recovery, in subsurface flow monitoring, in underground contaminant detection, etc. In this paper we will focus attention on imaging heterogeneous media with large variations in the magnitude of their electrical properties. This is relevant in many geophysical applications where the conductivity can vary over several orders of magnitude. For example, a dry rock matrix is insulating compared...
Synchronization of Power Networks: Network Reduction and Effective Resistance
"... Abstract: In transient stability studies in power networks two types of mathematical models are commonly used – the differentialalgebraic structurepreserving model and the reduced dynamic model of interconnected swing equations. This paper analyzes the reduction process relating the two power netw ..."
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Abstract: In transient stability studies in power networks two types of mathematical models are commonly used – the differentialalgebraic structurepreserving model and the reduced dynamic model of interconnected swing equations. This paper analyzes the reduction process relating the two power network models. The reduced admittance matrix is obtained by a Schur complement of the topological network admittance matrix with respect to its bus nodes. We provide a detailed spectral, algebraic, and graphtheoretic analysis of this network reduction process, termed Kron reduction, with particular focus on the effective resistance. As an application of this analysis, we are able to state concise conditions relating synchronization in the considered structurepreserving power network model directly to the state, parameters, and topology of the underlying network. In particular, we provide a spectral condition based on the algebraic connectivity of the network and a second condition based on the effective resistance among generators.