Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D

2024/03/25 by Garde, Henrik, Hirvensalo, Markus · 1 citation
#35R25 #35R30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2403.16588

Abstract

Recently an algorithm was given in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any L² perturbation from linearised data in the two-dimensional linearised Calderón problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calderón 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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