Alternatively, we proposed in a recent preprint with Giovanni S. Alberti and Simone Sanna a new reconstruction approach based on lifting and convex relaxation. It seems amenable to regularization, but its computational viability still needs to be investigated.
arxiv.org/abs/2507.00645
Take-home message: the optimization landscape of the least squares objective warrants further investigations. An interesting and challenging future direction is to study this landscape in the noisy regularized setting.
Finally, we provide an open source Julia package called RadialCalderon.jl, which allows to reproduce all our experiments. It might also serve as a benchmark for further studies on this piecewise constant radial version of the Calderón problem.
github.com/rpetit/Radia...
We also investigate the performance of a recently proposed approach based on convexification. For the first time, we propose a way to implement the former and show that it is consistently outperformed by Newton-type least squares solvers, which are also faster and require less measurements.
We revisit this issue in the case of radial piecewise constant conductivities, and prove that there are no local minimums in the case of two scalar unknowns with no measurement noise. We provide a partial proof in the general setting under a numerically verifiable assumption.
This inverse problem is highly ill-posed. The associated forward map (the mapping from the unknown to the measurements) is also nonlinear. For this reason, the classical least squares approach is generally believed to lead to a nonconvex optimization problem which is riddled with bad local minimums.
In electrical impedance tomography, one wishes to recover the conductivity of a medium from boundary current-voltage measurements. Calderón formalized this as an inverse problem for a PDE, whose goal is to estimate a coefficient of the PDE from measurements of its solution.
New preprint: "On the non-convexity issue in the radial Calderón problem" with Giovanni S. Alberti and Clarice Poon. We revisit the non-convexity issue for the least squares approach to the Calderón problem in the case of piecewise constant radial conductivities.
arxiv.org/abs/2507.03379
