Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915800472879104 |
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| author | Garde, Henrik Hirvensalo, Markus |
| author_facet | Garde, Henrik Hirvensalo, Markus |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D Garde, Henrik Hirvensalo, Markus Analysis of PDEs Numerical Analysis 35R30, 35R25 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. |
| title | Linearised Calderón problem: Reconstruction of unbounded perturbations in 3D |
| topic | Analysis of PDEs Numerical Analysis 35R30, 35R25 |
| url | https://arxiv.org/abs/2403.16588 |