Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D

Abstract

Recently an algorithm was given in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L^2$ 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 $L^3$ 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 $L^2$ basis of current densities.

arXiv comment11 pages, 3 figures

References in corpus (4)

• Zernike Basis to Cartesian Transformations
• Local analytic regularity in the linearized Calderón problem
• Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations
• Some properties of Wigner $3j$ coefficients: non-trivial zeros and connections to hypergeometric functions