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

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