Identification of Resistors in Electrical Networks

Chung, Soon‐Yeong

doi:10.4134/jkms.2010.47.6.1223

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…Discussion and future work. Our results generalize existing solvability results for the conductivity and Schrödinger problems on graphs [20,18,15,19,14,13,32,3,5,4] by relaxing what is meant by solvability and allowing conductivities (or Schrödinger potentials) in zero measure sets to have the same boundary data. Thus the problem is ill-posed.…”

Section: Schrödinger Problemsupporting

confidence: 76%

“…In a nutshell, the result of Chung [14] guarantees that if two conductivities γ 1 ≤ γ 2 (the inequality being componentwise) agree in a neighborhood of the boundary nodes and one measurement of the potential and the corresponding current at the boundary is identical for both conductivities then γ 1 = γ 2 . The result [13] is an extension to other kinds of measurements.…”

Section: Discrete Inverse Conductivity Problemmentioning

confidence: 91%

“…Other results by Chung and Berenstein [14] and Chung [13] are not limited to circular planar graphs, but assume monotonicity and positive real conductivities. These results are discrete analogous of the uniqueness result of Alessandrini [1] for the continuum conductivity problem (see also [30, §4.3]).…”

Section: 1mentioning

confidence: 99%

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On the Solvability of the Discrete Conductivity and Schrödinger Inverse Problems

Boyer¹,

Garzella²,

Vasquez³

2016

SIAM J. Appl. Math.

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Abstract. We study the uniqueness question for two inverse problems on graphs. Both problems consist in finding (possibly complex) edge or nodal based quantities from boundary measurements of solutions to the Dirichlet problem associated with a weighted graph Laplacian plus a diagonal perturbation. The weights can be thought of as a discrete conductivity and the diagonal perturbation as a discrete Schrödinger potential. We use a discrete analogue to the complex geometric optics approach to show that if the linearized problem is solvable about some conductivity (or Schrödinger potential) then the linearized problem is solvable for almost all conductivities (or Schrödinger potentials) in a suitable set. We show that the conductivities (or Schrödinger potentials) in a certain set are determined uniquely by boundary data, except on a zero measure set. This criterion for solvability is used in a statistical study of graphs where the conductivity or Schrödinger inverse problem is solvable.

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Section: Schrödinger Problemsupporting

confidence: 76%

Section: Discrete Inverse Conductivity Problemmentioning

confidence: 91%

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On the Solvability of the Discrete Conductivity and Schrödinger Inverse Problems

Boyer¹,

Garzella²,

Vasquez³

2016

SIAM J. Appl. Math.

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“…planar graphs that can be embedded in a disk and where the nodes at which measurements are made can be laid on the disk boundary) and real conductivities. A different approach is taken in [9] where a monotonicity property inspired from the continuum [1] is used to show that if the conductivities satisfy a certain inequality then they can be uniquely determined from measurements, without specific assumptions on the underlying graph. The lack of uniqueness is shown for cylindrical graphs in [21].…”

mentioning

confidence: 99%

Matrix valued inverse problems on graphs with application to mass-spring-damper systems

Draper¹,

Vasquez²,

Tse³

et al. 2020

Networks &Amp; Heterogeneous Media

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We consider the inverse problem of finding matrix valued edge or node quantities in a graph from measurements made at a few boundary nodes. This is a generalization of the problem of finding resistors in a resistor network from voltage and current measurements at a few nodes, but where the voltages and currents are vector valued. The measurements come from solving a series of Dirichlet problems, i.e. finding vector valued voltages at some interior nodes from voltages prescribed at the boundary nodes. We give conditions under which the Dirichlet problem admits a unique solution and study the degenerate case where the edge weights are rank deficient. Under mild conditions, the map that associates the matrix valued parameters to boundary data is analytic. This has practical consequences to iterative methods for solving the inverse problem numerically and to local uniqueness of the inverse problem. Our results allow for complex valued weights and give also explicit formulas for the Jacobian of the parameter to data map in terms of certain products of Dirichlet problem solutions. An application to inverse problems arising in networks of springs, masses and dampers is presented.

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Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

Bezuglyi

Jørgensen

2018

Acta Appl Math

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In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.

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Identification of Resistors in Electrical Networks

Cited by 4 publications

References 16 publications

On the Solvability of the Discrete Conductivity and Schrödinger Inverse Problems

On the Solvability of the Discrete Conductivity and Schrödinger Inverse Problems

Matrix valued inverse problems on graphs with application to mass-spring-damper systems

Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

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