Approximation and uniqueness results for the nonlocal diffuse optical tomography problem
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arXiv
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| Format: | Preprint |
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2024
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| author | Lin, Yi-Hsuan Zimmermann, Philipp |
| author_facet | Lin, Yi-Hsuan Zimmermann, Philipp |
| contents | We investigate the inverse problem of recovering the diffusion and absorption coefficients $(σ,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( σ\nabla))^s u+q u =0 \text{ in }Ω$ from the nonlocal Dirichlet-to-Neumann (DN) map $Λ^s_{σ,q}$. The purpose of this article is to establish the following approximation and uniqueness results.
- Approximation: We show that solutions to the conductivity equation $ \text{div}( σ\nabla v)=0 \text{ in }Ω$ can be approximated in $H^1(Ω)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $Λ_σ$ related to conductivity equation can be approximated by the nonlocal DN map $Λ_{σ,q}^s$.
- Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partialΩ$ provided $σ$ is already known in $\mathcal{N}$.
- Global uniqueness: Under the same assumptions as for the local uniqueness result, and if one of the potentials vanishes in $Ω$, then one can turn with the help of \ref{item 1 abstract} the local determination into a global uniqueness result.
It is worth mentioning that the approximation result relies on the Caffarelli--Silvestre type extension technique and the geometric form of the Hahn--Banach theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation and uniqueness results for the nonlocal diffuse optical tomography problem Lin, Yi-Hsuan Zimmermann, Philipp Analysis of PDEs 35R30, 26A33, 35J10, 35J70 We investigate the inverse problem of recovering the diffusion and absorption coefficients $(σ,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( σ\nabla))^s u+q u =0 \text{ in }Ω$ from the nonlocal Dirichlet-to-Neumann (DN) map $Λ^s_{σ,q}$. The purpose of this article is to establish the following approximation and uniqueness results. - Approximation: We show that solutions to the conductivity equation $ \text{div}( σ\nabla v)=0 \text{ in }Ω$ can be approximated in $H^1(Ω)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $Λ_σ$ related to conductivity equation can be approximated by the nonlocal DN map $Λ_{σ,q}^s$. - Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partialΩ$ provided $σ$ is already known in $\mathcal{N}$. - Global uniqueness: Under the same assumptions as for the local uniqueness result, and if one of the potentials vanishes in $Ω$, then one can turn with the help of \ref{item 1 abstract} the local determination into a global uniqueness result. It is worth mentioning that the approximation result relies on the Caffarelli--Silvestre type extension technique and the geometric form of the Hahn--Banach theorem. |
| title | Approximation and uniqueness results for the nonlocal diffuse optical tomography problem |
| topic | Analysis of PDEs 35R30, 26A33, 35J10, 35J70 |
| url | https://arxiv.org/abs/2406.06226 |