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Merońo Moreno, CristobalAutor o coautorMaciá Lang, FabricioAutor o coautorCastro Barbero, Carlos ManuelAutor o coautorBarceló Ja, Castro C, Macià F, Meroño CjAutor o coautor

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THE BORN APPROXIMATION IN THE THREE-DIMENSIONAL CALDERON PROBLEM II: NUMERICAL RECONSTRUCTION IN THE RADIAL CASE

Publicat a:Inverse Problems And Imaging. 18 (1): 183-207 - 2024-02-01 18(1), DOI: 10.3934/ipi.2023029

Autors: Barceló JA; Castro C; Macià F; Meroño CJ

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In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón 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 BR ⊂ R3 of radius R, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for L∞ conductivities, and 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 ∂BR. We then investigate how increasing the radius R affects the quality of the Born approximation, and the existence of a scattering limit as R → ∞. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator −∆ + q, and strong recovery of singularity results are observed in this case.

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