On the non-convexity issue in the radial Calderón problem

TL;DR

This work reexamines the nonconvexity of nonlinear least-squares for the Calderón problem in a radial, piecewise-constant conductivity model and shows that, with two scalar unknowns and no noise, there are no spurious critical points and the forward map is identifiable. It proves a unique critical point in the noiseless two-parameter case and extends the result to higher dimensions under verifiable assumptions, challenging the common belief of pervasive nonconvexity. The authors implement and benchmark a convex nonlinear SDP approach against classical least-squares, finding that the LS method is faster, requires fewer measurements, and yields more accurate reconstructions in practice, while the convex approach struggles as problem size grows. They also provide extensive numerical evidence on ill-posedness scaling with boundary distance and annuli count, and release RadialCalderon.jl for reproducible benchmarking. Overall, the paper argues that addressing ill-posedness is more crucial than eliminating nonconvexity and offers practical guidance on method choice for radial Calderón problems.

Abstract

A classical approach to the Calderón 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.

Figures (14)

• Figure 1: plot of the least squares objective (a) and its second derivative (b) in a one-dimensional setting. The objective is not convex (its second derivative is not nonnegative), but it has a single critical point, which is its global minimum.
• Figure 2: schematic description of the piecewise constant radial setting when $n=4$. The conductivity $\gamma:B(0,1)\to \mathbb{R}$ is such that, using the polar coordinates $(r,\theta)$, it holds $\gamma(r,\theta)=\sigma_i$ for every $r_{i}<r<r_{i-1}$ for some $\sigma=(\sigma_i)_{1\leq i\leq n}\in (\mathbb{R}_{>0})^n$.
• Figure 3: mean error on the $i$-th annulus among $100$ pairs $\sigma^\dagger,\hat{\sigma}\in[a,b]^n$ such that $\|\Lambda(\hat{\sigma})-\Lambda(\sigma^\dagger)\|_{\infty}\leq 10^{-15}$ with $(a,b)=(0.5, 1.5)$ and $n=m=10$. The error on the outermost annulus is several orders of magnitude smaller than the error on the innermost annulus.
• Figure 4: mean $\ell^\infty$ error as a function of the number of annuli $n$. The mean is computed on $100$ pairs $\sigma^\dagger,\hat{\sigma}\in[a,b]^n$ such that $\|\Lambda(\hat{\sigma})-\Lambda(\sigma^\dagger)\|_{\infty}\leq 10^{-15}$ with $(a,b)=(0.5,1.5)$ and $m=n$ measurements.
• Figure 5: graph of the functions $g$ and $h$ constructed in the proof of \ref{['prop:2d_inj']}, with $\sigma^\dagger=\mathbf{1}$. The function $h$ is strictly monotone, showing that the forward map is injective.

Theorems & Definitions (15)

• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof