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Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D (2403.16588v1)

Published 25 Mar 2024 in math.AP, cs.NA, and math.NA

Abstract: Recently an algorithm was given in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L2$ perturbation from linearised data in the two-dimensional linearised Calder\'on problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calder\'on problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any $L3$ perturbation from linearised data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the 3D method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full $L2$ basis of current densities.

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