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Discrete inverse problems with internal functionals

Marcus Corbett, Fernando Guevara Vasquez, Alexander Royzman, Guang Yang

TL;DR

This work develops a systematic framework for discrete inverse problems on graphs where internal functionals, specifically dissipated power measurements, are used to recover edge conductivities or interior Schrödinger potentials. The authors recast the problem as a non-linear redundant system $\mathcal{L}(\gamma,u^{(j)})=b^{(j)}$, $\mathcal{M}(\gamma,u^{(j)})=H^{(j)}$, and establish local uniqueness by proving injectivity of the linearized system under concrete conditions that couple forward solvability, support coverage by boundary data, and invertibility of certain graph-based operators. They treat multiple models, including real and complex conductivities, a discrete Schrödinger problem, and a two-frequency approach to overcome under-determinacy, with explicit injectivity criteria expressed through signed/diagonalized Laplacian matrices and their invertibility. The paper also provides a Gauss-Newton reconstruction scheme, and demonstrates highly accurate noiseless recovery and robustness to moderate noise on a $10\times10$ grid, highlighting practical viability for using thermal-noise–based measurements to infer dissipated power and material properties in networks. Overall, the results bridge continuum internal-functional methods with discrete graph problems, enabling local uniqueness and tractable numerics for a family of hybrid inverse problems on networks.

Abstract

We study the problem of finding the resistors in a resistor network from measurements of the power dissipated by the resistors under different loads. We give sufficient conditions for local uniqueness, i.e. conditions that guarantee that the linearization of this non-linear inverse problem admits a unique solution. Our method is inspired by a method to study local uniqueness of inverse problems with internal functionals in the continuum, where the inverse problem is reformulated as a redundant system of differential equations. We use our method to derive local uniqueness conditions for other discrete inverse problems with internal functionals including a discrete analogue of the inverse Schrödinger problem and problems where the resistors are replaced by impedances and dissipated power at the zero and a positive frequency are available. Moreover, we show that the dissipated power measurements can be obtained from measurements of thermal noise induced currents.

Discrete inverse problems with internal functionals

TL;DR

This work develops a systematic framework for discrete inverse problems on graphs where internal functionals, specifically dissipated power measurements, are used to recover edge conductivities or interior Schrödinger potentials. The authors recast the problem as a non-linear redundant system , , and establish local uniqueness by proving injectivity of the linearized system under concrete conditions that couple forward solvability, support coverage by boundary data, and invertibility of certain graph-based operators. They treat multiple models, including real and complex conductivities, a discrete Schrödinger problem, and a two-frequency approach to overcome under-determinacy, with explicit injectivity criteria expressed through signed/diagonalized Laplacian matrices and their invertibility. The paper also provides a Gauss-Newton reconstruction scheme, and demonstrates highly accurate noiseless recovery and robustness to moderate noise on a grid, highlighting practical viability for using thermal-noise–based measurements to infer dissipated power and material properties in networks. Overall, the results bridge continuum internal-functional methods with discrete graph problems, enabling local uniqueness and tractable numerics for a family of hybrid inverse problems on networks.

Abstract

We study the problem of finding the resistors in a resistor network from measurements of the power dissipated by the resistors under different loads. We give sufficient conditions for local uniqueness, i.e. conditions that guarantee that the linearization of this non-linear inverse problem admits a unique solution. Our method is inspired by a method to study local uniqueness of inverse problems with internal functionals in the continuum, where the inverse problem is reformulated as a redundant system of differential equations. We use our method to derive local uniqueness conditions for other discrete inverse problems with internal functionals including a discrete analogue of the inverse Schrödinger problem and problems where the resistors are replaced by impedances and dissipated power at the zero and a positive frequency are available. Moreover, we show that the dissipated power measurements can be obtained from measurements of thermal noise induced currents.
Paper Structure (15 sections, 6 theorems, 55 equations, 3 figures)

This paper contains 15 sections, 6 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation as a redundant system of non-linear equations
  3. Related work
  4. Contents
  5. The discrete inverse conductivity problem from thermal noise
  6. The real conductivity problem
  7. Parallels with the continuum
  8. Examples
  9. The complex conductivity problem
  10. Inverse problem from dissipated power measurements
  11. The inverse Schrödinger problem
  12. Real Schrödinger problem
  13. The complex Schrödinger problem
  14. Numerical reconstruction with Gauss-Newton method
  15. Summary and perspectives

Key Result

Proposition 1

Let $g = (L_\sigma u)_{\mathcal{B}}$ be the net currents at the boundary nodes for a voltage $u: \mathcal{V} \to \mathbb{R}$ satisfying For random currents $J_{\text{noise}}$ satisfying eq:jdist and invertible $(L_\sigma)_{\mathcal{I}\mathcal{I}}$ we have

Figures (3)

  • Figure 1: Comparison of power dissipated (left) to its estimation from thermal noise induced currents (right) using \ref{['thm:det']}. Here all the edges on edges of the outer square are boundary nodes and the conductivities are all equal to one. The boundary condition corresponds to setting the nodes on the top and right edges to a unit voltage and the bottom and left edges to a zero voltage.
  • Figure 2: Examples of weighted graphs illustrating different cases of \ref{['thm:realcond']}, where the conductivities are indicated on each edge. For each graph $\mathcal{V} \subset \mathbb{N}$, and the nodes are ordered so that the boundary nodes appear first (in red) and the interior nodes last (in white). To ensure positive edge weights or conductivities it is assumed that $\mu > 0$ and $|\varepsilon|<1$. For more details see \ref{['sec:examples']}.
  • Figure 3: The Gauss-Newton method was used to find a real conductivity from dissipated power data (a) and (b). The reconstructions without noise are given in (c) and those with 5% additive noise are given in (d).

Theorems & Definitions (17)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Theorem 3
  • ...and 7 more