On the Solvability of the Discrete Conductivity and Schrödinger Inverse Problems

Abstract

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.

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

“…In particular if we fix p ∈ R we have shown that M p × M p has zero measure and so M p must have zero measure as well. This is a much simpler way of reaching a result similar to that in [6] and was suggested by Druskin [15].…”

“…If p 1 , p 2 ∈ C m are parameters with identical data Λ p1 = Λ p2 , can we conclude that p 1 = p 2 ? For inverse problems satisfying assumptions 1-3, we can only guarantee uniqueness in a weak sense that we call uniqueness almost everywhere (as in [6]). By this we mean that (a) the linearized problem is injective for almost all parameters p ∈ R and (b) for any p, the sets M p ≡ {q ∈ R | Λ q = Λ p } must have zero measure.…”

“…solutions to the Dirichlet problem). Analyticity can be used to guarantee local uniqueness for such inverse problems, for almost any parameter within a region of interest provided the Jacobian is injective for one parameter (a generalization of the results in [6]). Moreover, we show that Newton's method applied to such problems is very likely to produce valid steps.…”

“…The lack of uniqueness is shown for cylindrical graphs in [21]. For complex conductivities, a condition for "uniqueness almost everywhere" regardless of the underlying graph is given in [6]. Uniqueness almost everywhere means that the set of conductivities that have the same boundary data lie in a zero measure set and that the linearized problem is injective for almost all conductivities in some region.…”

“…A discrete Liouville identity [5] can be used to relate the discrete Schrödinger inverse problem for certain Schrödinger potentials to the discrete conductivity inverse problem, also on circular planar graphs. A condition guaranteeing uniqueness almost everywhere for complex valued potentials without an assumption on the graph is given in [6].…”