Inverse Problems for Electrical Networks

Overdetermined partial boundary value problems on finite networks

“…[14,Corollary 3.14] that for a circular planar network Λ A,B is non-singular iff the pair (A; B) is connected through Γ. In particular, this happens if Γ is well-connected.…”

“…In the same reference they found the 4 types of basic well-connected planar networks and then, described an algorithm for recovering the conductances. One of the keys of the algorithm is the boundary spike formula, see [14,Corollary 3.16]; that allows to recover the conductance on a boundary spike edge. The proof of this result can be easily adapted to the case of Schrödinger operators and Dirichlet-to-Neumann maps.…”

“…For instance, Sylvester and Uhlmann treated in [9,18] the uniqueness of solution; Curtis, Ingerman and Morrow have worked on critical circular planar networks conductivity reconstruction [12,13,14,16]; Borcea, Druskin, Guevara and Mamonov have gone into EIT problems in depth and their last works on the subject treat numerical conductivity reconstruction [6,7,8].…”

“…In fact, this type of problems were implicitly considered in some previous works, but only for specific networks and boundary data, see [13,14]. We analyze the uniqueness and existence of solution of overdetermined partial boundary value problems through the non-singularity of the partial Dirichlet-to-Neumann maps.…”

“…Afterwards, we give explicit formulae for the acquirement of boundary spike conductances on critical planar networks and execute a full conductance recovery for spider networks. This algorithm is an adaptation of the one proposed in [14] for the Combinatorial Laplacian and when the corresponding Dirichlet-to-Neumann map is singular.…”