On the required number of electrodes for uniqueness and convex reformulation in an inverse coefficient problem

TL;DR

The paper tackles the inverse problem of recovering a Robin boundary transmission coefficient $\gamma$ on an interior boundary from electrode-based current-voltage measurements in a shunt electrode model. It introduces a computer-assisted criterion to determine the minimum number and layout of electrodes required for uniqueness and a Lipschitz stability bound, and it reformulates the nonlinear inverse problem as a convex semidefinite program to achieve global convergence. The forward map $\mathcal{F}: \mathbb{R}^n_+ \to \mathbb{S}^m_+$ is analyzed for monotonicity and convexity, with explicit finite-test criteria that validate unique solvability using finitely many forward solves. Numerical experiments demonstrate that a finite electrode configuration can guarantee uniqueness and enable accurate reconstructions via the SDP approach, even in the presence of noise, while highlighting the intrinsic ill-posedness that persists as the resolution grows.

Abstract

We introduce a computer-assisted proof for the required number of electrodes for uniqueness and global reconstruction for the inverse Robin transmission problem, where the corrosion function on the boundary of an interior object is to be determined from electrode current-voltage measurements. We consider the shunt electrode model where, in contrast to the standard Neumann boundary condition, the applied electrical current is only partially known. The aim is to determine the corrosion coefficient with a finite number of measurements. In this paper, we present a numerically verifiable criterion that ensures unique solvability of the inverse problem, given a desired resolution. This allows us to explicitly determine the required number and position of the electrodes. Furthermore, we will present an error estimate for noisy data. By rewriting the problem as a convex optimization problem, our aim is to develop a globally convergent reconstruction algorithm.

Figures (9)

• Figure 1: The domain $\Omega=\Omega_1 \cup \Omega_2$.
• Figure 2: Resolution dimension $n=3$ and $m=5$ electrodes.
• Figure 3: Plot of the residual function with $\hat{\gamma} =(2, 2)$ and the logarithmic error $\log_{10}(\Vert \gamma^{(N)}- \hat{\gamma}\Vert_2)$ for the least squares data fitting approach with varying corrosion parameters $\hat{\gamma}=(\hat{\gamma}_1,\hat{\gamma}_2)$. The initial value is set to $\gamma^{(0)} =(2, 2)$. The maximum error $\Vert \gamma^{(N)}- \hat{\gamma}\Vert_2 \approx 0.72$ occurs at $\hat{\gamma}=(1.09,2.68)$.
• Figure 4: Plot of the equivalent approach for varying corrosion parameters $\hat{\gamma}=(\hat{\gamma}_1,\hat{\gamma}_2)$, with the initial value set to $\gamma^{(0)} =(2, 2)$. The maximum error is $\Vert \gamma^{(N)}- \hat{\gamma}\Vert_2\leq 5.5\cdot 10^{-7}$.
• Figure 5: Plot of the required number of electrodes $m$ for Criterion \ref{['criterion1']} and the stability constant $\lambda$ for different resolution dimensions $n$ with $a=1$ and $b=3$. The dashed line describes $\lambda$ calculated with five extra electrodes.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1
• proof