Well-posed Questions for Ill-posed Inverse Problems: a Note in Memory of Pierre Sabatier

TL;DR

This work applies Sabatier's idea of seeking well-posed questions to an ill-posed inverse problem in Electrical Impedance Tomography (EIT) by focusing on a heavily constrained 2D model of a small elliptical anomaly. It combines a regularized least squares framework with an optimal experimental design (D-optimality) to select electrode placements that maximize stability of the inverse solution, assessed via the forward map and its Jacobian. The numerical example shows that carefully designed electrode configurations dramatically improve the recoverability of the ellipse's parameters, especially the poorly determined aspect ratio $r$ and orientation $ξ$, bringing estimates closer to the ground truth. The study argues that well-posed questions can be realized in practice by design choices, offering practical guidance for EIT experiments and broader inverse problems.

Abstract

Professor Pierre Sabatier contributed much to the study of inverse problems in theory and practice. Two of these contributions were a focus on theory that actually supports practice, and the identification of well-posed aspects of inverse problems that may quite ill-posed. This paper illustrates these two themes in the context of Electrical Impedance Tomography (EIT), which is both very ill-posed and very practical. We show that for a highly constrained version of this inverse problem, in which a small elliptical inclusion in a homogeneous background is to be identified, optimization of the experimental design (that is, electrode locations) vastly improves the stability of the solution.

Figures (4)

• Figure 1: In this schematic, the source-sink pair is $A_+$-$A_-$, i.e., current flows from $A_+$ to $A_-$. The voltage measurement is taken at $B_+$-$B_-$, i.e., the voltage difference between $B_+$ and $B_-$. The small elliptical anomaly is indicated by $D$.
• Figure 2: The optimal electrodes found by maximizing the determinant of $\left[ \, \mathcal{J}(\Phi)^T \mathcal{J}(\Phi) + \lambda R^T R \, \right]$. The electrodes and the center of estimated anomaly center are indicated.
• Figure 3: Comparison of the recoveries from two electrode placements with the same relative noise in the data. (Left) The recovery from electrodes placed at $(0,\pi/2,\pi,3\pi/2)$ and (Right) the recovery from optimally placed electrodes (see Figure \ref{['optimal electrodes']}). Note that in the case of optimally placed electrodes regularization is below the threshold of sensitivity and is basically ignored.
• Figure 4: These box plots summarizes the 100 inversions using electrodes at initial locations $(0,\pi/2,\pi,3\pi/2)$ (in blue) and those using optimal electrode locations (in orange - see Figure \ref{['optimal electrodes']}). Notice how much closer the means of the ellipse parameters for the estimates with optimal electrode placement are to the ground truth values ($A=0.025, b_1=0.452, b_2=-0.165, r=1/2.323, \xi=0.864-\pi/2$). The result suggests that by optimizing the electrode locations, the problem becomes much more well-posed.