The Born approximation in the Calderón problem

Nom de l'orateur
Cristóbal Meroño
Etablissement de l'orateur
Universidad Politécnica de Madrid
Date et heure de l'exposé
Lieu de l'exposé
salle des séminaires

Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem of a Schrödinger operator with a real potential. Due to the lack of a rigorous definition, the Born approximation has been relegated to a marginal place in the reconstruction problem. In this talk we will introduce the Born approximation for Schrödinger operators in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension >2. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map, and which is closely related to a particular moment problem. We will also show that these formulas can be extended to the conductivity problem very easily. We then turn to real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem, and obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis of spherical harmonics. Then we will present some numerical evidences that in the radial case the Born approximation is well defined for discontinuous conductivities and has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $\partial B_R$, as well as strong recovery of singularity capabilities.