Reseaux électriques planaires II

Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We first construct a poset EP n of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. We then answer various enumerative questions related to EP n , adapting methods of Callan and Stein-Everett. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of Postnikov, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy.

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“…Theorem 2.1.5 ( [dVGV,Théorème 4]). Two electrical networks are equivalent if and only if they are related by a sequence of local equivalences.…”

Section: Y-∆ Transformations (Seementioning

confidence: 99%

Circular planar electrical networks: Posets and positivity

Alman

Lian

Tran

2015

Journal of Combinatorial Theory, Series A

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“…The dimension of the space of possible transfer admittance matrices is clearly no bigger than L(L − 1)/2, and so it is unrealistic to expect to recover more unknown parameters than this. In the case of planar resistor networks the possible transfer admittance matrices can be characterized completely [42], a characterization which is known at least partly to hold in the planar continuum case [77]. A typical electrical imaging system applies current or voltage patterns which form a basis of the space S, and measures some subset of the resulting voltages which as they are only defined up to an additive constant can be taken to be in S.…”

Section: Measurements and Electrodesmentioning

confidence: 99%

“…For example there may be no consistent allocation of angles θ j so that around any given vertex (or edge in 3D) they sum to 2π. The question of uniqueness of solution, as well as the structure of the transconductance matrix for real planar resistor networks well is understood [42,43].…”

Section: The Corresponding Resistor Network For a 2d Fem Meshmentioning

confidence: 99%

“…In addition the real part of Z| S is positive definite, otherwise there would be direct current patterns which dissipate no power. There are other conditions on Z, given in the plane case by [42], associated with Ω being connected, and its is shown in the planar case that the the set of feasible Z is an open subset of the vector space descried above. This confirms that we can measure up to L(L − 1)/2 independent parameters.…”

Section: Sheffield Measurement Protocolmentioning

confidence: 99%

See 1 more Smart Citation

The reconstruction problem

Lionheart¹,

Polydorides²,

Borsic³

2004

Series in Medical Physics and Biomedical Engineering

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“…It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in [14,15,28,17,18]. The circular networks arise in the discretization of equation (1.1) with a five point stencil finite volumes scheme on the optimal grids computed as part of the inverse problem.…”

Section: Introductionmentioning

confidence: 99%

Circular resistor networks for electrical impedance tomography with partial boundary measurements

Borcea¹,

Druskin²,

Mamonov³

2010

Inverse Problems

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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 so-called 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 trade-off 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.

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Reseaux électriques planaires II

Cited by 68 publications

References 6 publications

Circular planar electrical networks: Posets and positivity

Circular planar electrical networks: Posets and positivity

The reconstruction problem

Circular resistor networks for electrical impedance tomography with partial boundary measurements

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