Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations
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| Format: | Preprint |
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2022
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| author | Garde, Henrik Hyvönen, Nuutti |
| author_facet | Garde, Henrik Hyvönen, Nuutti |
| contents | We investigate a linearised Calderón problem in a two-dimensional bounded simply connected $C^{1,α}$ domain $Ω$. After extending the linearised problem for $L^2(Ω)$ perturbations, we orthogonally decompose $L^2(Ω) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(Ω)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(Ω)$ perturbation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_10164 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations Garde, Henrik Hyvönen, Nuutti Analysis of PDEs Numerical Analysis 35R30, 35R25 We investigate a linearised Calderón problem in a two-dimensional bounded simply connected $C^{1,α}$ domain $Ω$. After extending the linearised problem for $L^2(Ω)$ perturbations, we orthogonally decompose $L^2(Ω) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic-type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general $L^2(Ω)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(Ω)$ perturbation. |
| title | Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations |
| topic | Analysis of PDEs Numerical Analysis 35R30, 35R25 |
| url | https://arxiv.org/abs/2204.10164 |