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31
Total positivity: tests and parametrizations
 Math. Intelligencer
"... A matrix is totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative) real numbers. The first systematic study of these classes of matrices was undertaken in the 1930s by F. R. Gantmacher and M. G. Krein [20, 21, 22], who established their remarkable spectral pr ..."
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Cited by 42 (9 self)
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A matrix is totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative) real numbers. The first systematic study of these classes of matrices was undertaken in the 1930s by F. R. Gantmacher and M. G. Krein [20, 21, 22], who established their remarkable spectral properties (in particular,
Total positivity, Grassmannians, and networks
, 2007
"... The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells a ..."
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Cited by 38 (2 self)
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The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on
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 ..."
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Cited by 24 (0 self)
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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 ..."
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Cited by 11 (0 self)
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. 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...
Matching Pursuit for Imaging High Contrast Conductivity
, 1999
"... We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solu ..."
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Cited by 9 (4 self)
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We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solution of the high contrast imaging problem, if the library of functions is constructed carefully and in accordance with the asymptotic theory. We also show how other libraries of functions that at first glance seem reasonable, in fact, do not work well. When the contrast in the conductivity is not so high, we show that wavelets can be used, especially nonorthogonal wavelet libraries. However, the library of functions that is based on the high contrast asymptotic theory is more robust, even for intermediate contrasts, and especially so in the presence of noise. Key words. Impedance tomography, high contrast, asymptotic resistor network, imaging. Contents 1 Introduction 1 2 The Neumann to Dir...
On the Parametrization of Illposed Inverse Problems Arising from Elliptic Partial Differential Equations
, 2006
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Borgne. Topology and static response of interaction networks in molecular biology
 J R Soc Interface
, 2006
"... molecular biology ..."
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.