Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912031063408640 |
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| author | Bisch, Joanna Hirvensalo, Markus Hyvönen, Nuutti |
| author_facet | Bisch, Joanna Hirvensalo, Markus Hyvönen, Nuutti |
| contents | This work considers the Fréchet derivative of the idealized forward map of two-dimensional electrical impedance tomography, i.e., the linear operator that maps a perturbation of the coefficient in the conductivity equation over a bounded two-dimensional domain to the linear approximation of the corresponding change in the Neumann-to-Dirichlet boundary map. It is proved that the Fréchet derivative is bounded from the space of square-integrable conductivity perturbations to the space of Hilbert--Schmidt operators on the mean-free $L^2$ functions on the domain boundary, if the background conductivity coefficient is constant and the considered simply-connected domain has a $C^{1,α}$ boundary. This result provides a theoretical framework for analyzing linearization-based one-step reconstruction algorithms of electrical impedance tomography in an infinite-dimensional setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10671 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators Bisch, Joanna Hirvensalo, Markus Hyvönen, Nuutti Analysis of PDEs 35B30, 35B35, 35Q60, 35R30 This work considers the Fréchet derivative of the idealized forward map of two-dimensional electrical impedance tomography, i.e., the linear operator that maps a perturbation of the coefficient in the conductivity equation over a bounded two-dimensional domain to the linear approximation of the corresponding change in the Neumann-to-Dirichlet boundary map. It is proved that the Fréchet derivative is bounded from the space of square-integrable conductivity perturbations to the space of Hilbert--Schmidt operators on the mean-free $L^2$ functions on the domain boundary, if the background conductivity coefficient is constant and the considered simply-connected domain has a $C^{1,α}$ boundary. This result provides a theoretical framework for analyzing linearization-based one-step reconstruction algorithms of electrical impedance tomography in an infinite-dimensional setting. |
| title | Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators |
| topic | Analysis of PDEs 35B30, 35B35, 35Q60, 35R30 |
| url | https://arxiv.org/abs/2409.10671 |