Results 1  10
of
12
LOOPERASED WALKS AND TOTAL POSITIVITY
, 2000
"... We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves looperased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.
Discrete And Continuous DirichletToNeumann Maps In The Layered Case
"... . Every sufficiently regular nonnegative function fl (conductivity) on the closed unit disk D induces the DirichlettoNeumann map fl on functions on @D . The main forward problem is to give a characterization of the maps fl . The main inverse problem is to find out when fl uniquely determines ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
. Every sufficiently regular nonnegative function fl (conductivity) on the closed unit disk D induces the DirichlettoNeumann map fl on functions on @D . The main forward problem is to give a characterization of the maps fl . The main inverse problem is to find out when fl uniquely determines fl. In this paper we consider the case of conductivities that are constant on circles centered at the origin, and a discrete analog of this, so called, layered case. We characterize the set of the layered DirichlettoNeumann maps in terms of their kernels and spectra. We also give sharp conditions on fl for the uniqueness in the inverse problems. The characterization in terms of the spectra shows that continuous DirichlettoNeumann maps can be viewed as limits of the discrete DirichlettoNeumann maps. The characterization in terms of the kernels supports the conjecture in [8] that the alternating property essentially characterizes continuous DirichlettoNeumann maps. The chracterization...
On the Parametrization of Illposed Inverse Problems Arising from Elliptic Partial Differential Equations
, 2006
"... ..."
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 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
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
"... ..."
Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
"... by ..."
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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.