The Born approximation in the three-dimensional Calder\'on problem II: Numerical reconstruction in the radial case

Abstract

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calder\'on inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $B_R \subset \mathbb{R}^3 $ of radius $R$, in which the linearization of the Calder\'on problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $L^{\infty}$ conductivities, and we present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $\partial B_R$. We then investigate how increasing the radius $R$ affects the quality of the Born approximation, and the existence of a scattering limit as $R\to \infty$. Similar properties are also illustrated in the inverse boundary problem for the Schr\"odinger operator $-\Delta +q$, and strong recovery of singularity results are observed in this case.