Eigenstructure of the linearized electrical impedance tomography problem under radial perturbations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908888769495040 |
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| author | Hirvensalo, Markus |
| author_facet | Hirvensalo, Markus |
| contents | We analyze the Fréchet derivative $F$, that maps a perturbation in conductivity to the linearized change in boundary measurements governed by the conductivity equation. The domain is taken to be the unit ball $B \subset \mathbb{R}^d$ with $d \geq 2$, and we choose perturbations $η$ from the Hilbert space $L^2(B)$. Under the condition that the perturbations are rotationally symmetric, we show that the eigenfunctions of the linear approximation $F η$ correspond to the spherical harmonics. Furthermore, we establish an explicit formula for the associated eigenvalues and show that for perturbations from any bounded subset, the decay of these eigenvalues is uniform with respect to the degree of the spherical harmonics. The established structure of $F η$ enables us to show that the Fréchet derivative $F$ can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations. Both the extension to $L^2(B)$ perturbations and the approximability by finite-rank operators are favorable properties for further analysis of $F$ in numerical algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenstructure of the linearized electrical impedance tomography problem under radial perturbations Hirvensalo, Markus Analysis of PDEs 35B30, 35B35, 35Q60, 35R30 We analyze the Fréchet derivative $F$, that maps a perturbation in conductivity to the linearized change in boundary measurements governed by the conductivity equation. The domain is taken to be the unit ball $B \subset \mathbb{R}^d$ with $d \geq 2$, and we choose perturbations $η$ from the Hilbert space $L^2(B)$. Under the condition that the perturbations are rotationally symmetric, we show that the eigenfunctions of the linear approximation $F η$ correspond to the spherical harmonics. Furthermore, we establish an explicit formula for the associated eigenvalues and show that for perturbations from any bounded subset, the decay of these eigenvalues is uniform with respect to the degree of the spherical harmonics. The established structure of $F η$ enables us to show that the Fréchet derivative $F$ can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations. Both the extension to $L^2(B)$ perturbations and the approximability by finite-rank operators are favorable properties for further analysis of $F$ in numerical algorithms. |
| title | Eigenstructure of the linearized electrical impedance tomography problem under radial perturbations |
| topic | Analysis of PDEs 35B30, 35B35, 35Q60, 35R30 |
| url | https://arxiv.org/abs/2510.05966 |