Linearised Calder\'on problem: Reconstruction of unbounded perturbations in 3D. Garde, H. & Hirvensalo, M. March, 2024. arXiv:2403.16588 [cs, math] version: 1
Linearised Calder\'on problem: Reconstruction of unbounded perturbations in 3D [link]Paper  doi  abstract   bibtex   
Recently an algorithm was given in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L{\textasciicircum}2$ 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 $L{\textasciicircum}3$ 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 $L{\textasciicircum}2$ basis of current densities.
@misc{garde_linearised_2024,
	title = {Linearised {Calder}{\textbackslash}'on problem: {Reconstruction} of unbounded perturbations in {3D}},
	shorttitle = {Linearised {Calder}{\textbackslash}'on problem},
	url = {http://arxiv.org/abs/2403.16588},
	doi = {10.48550/arXiv.2403.16588},
	abstract = {Recently an algorithm was given in [Garde \& Hyv{\textbackslash}"onen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any \$L{\textasciicircum}2\$ perturbation from linearised data in the two-dimensional linearised Calder{\textbackslash}'on problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calder{\textbackslash}'on problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any \$L{\textasciicircum}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{\textasciicircum}2\$ basis of current densities.},
	urldate = {2024-06-27},
	publisher = {arXiv},
	author = {Garde, Henrik and Hirvensalo, Markus},
	month = mar,
	year = {2024},
	note = {arXiv:2403.16588 [cs, math]
version: 1},
	keywords = {Calderón problem, inverse problems, mentions sympy, numerical analysis, partial differential equations},
}

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