On the non-convexity issue in the radial Calder\'on problem

TL;DR

This paper challenges the common belief that the nonlinear least-squares approach to the Calderón problem is plagued by bad local minima, showing that for certain radial conductivities, no spurious critical points exist, and compares convexification methods.

Contribution

It proves the absence of spurious critical points for piecewise constant radial conductivities with two unknowns and provides practical implementation insights for convexification methods.

Findings

No spurious critical points for two scalar unknowns in radial case

Least squares solvers outperform convexification in speed and measurement efficiency

Partial proof of no spurious critical points in general setting under verifiable assumptions

Abstract

A classical approach to the Calder\'on problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local minimums. We revisit this issue in the case of piecewise constant radial conductivities and prove that, contrary to previous claims, there are no spurious critical points in the case of two scalar unknowns with no measurement noise. We also provide a partial proof of this result in the general setting which holds under a numerically verifiable assumption. Finally, we investigate whether a recently proposed approach based on convexification yields better reconstructions. For the first time, we propose a way to implement it in practice and show that it is consistently outperformed by some least squares solvers, which are also faster and require less measurements.