Approximation and uniqueness results for the nonlocal diffuse optical tomography problem
| Title: | Approximation and uniqueness results for the nonlocal diffuse optical tomography problem |
|---|---|
| Authors: | Lin, Yi-Hsuan, Zimmermann, Philipp |
| Publication Year: | 2024 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Analysis of PDEs, 35R30, 26A33, 35J10, 35J70 |
| More Details: | We investigate the inverse problem of recovering the diffusion and absorption coefficients $(\sigma,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( \sigma \nabla))^s u+q u =0 \text{ in }\Omega$ from the nonlocal Dirichlet-to-Neumann (DN) map $\Lambda^s_{\sigma,q}$. 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}( \sigma \nabla v)=0 \text{ in }\Omega$ can be approximated in $H^1(\Omega)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $\Lambda_\sigma$ related to conductivity equation can be approximated by the nonlocal DN map $\Lambda_{\sigma,q}^s$. - Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partial\Omega$ provided $\sigma$ 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 $\Omega$, 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. Comment: 37 pages |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2406.06226 |
| Accession Number: | edsarx.2406.06226 |
| Database: | arXiv |
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| Items | – Name: Title Label: Title Group: Ti Data: Approximation and uniqueness results for the nonlocal diffuse optical tomography problem – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lin%2C+Yi-Hsuan%22">Lin, Yi-Hsuan</searchLink><br /><searchLink fieldCode="AR" term="%22Zimmermann%2C+Philipp%22">Zimmermann, Philipp</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2024 – Name: Subset Label: Collection Group: HoldingsInfo Data: Mathematics – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics+-+Analysis+of+PDEs%22">Mathematics - Analysis of PDEs</searchLink><br /><searchLink fieldCode="DE" term="%2235R30%2C+26A33%2C+35J10%2C+35J70%22">35R30, 26A33, 35J10, 35J70</searchLink> – Name: Abstract Label: Description Group: Ab Data: We investigate the inverse problem of recovering the diffusion and absorption coefficients $(\sigma,q)$ in the nonlocal diffuse optical tomography equation $(-\text{div}( \sigma \nabla))^s u+q u =0 \text{ in }\Omega$ from the nonlocal Dirichlet-to-Neumann (DN) map $\Lambda^s_{\sigma,q}$. 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}( \sigma \nabla v)=0 \text{ in }\Omega$ can be approximated in $H^1(\Omega)$ by solutions to the nonlocal diffuse optical tomography equation and the DN map $\Lambda_\sigma$ related to conductivity equation can be approximated by the nonlocal DN map $\Lambda_{\sigma,q}^s$. - Local uniqueness: We prove that the absorption coefficient $q$ can be determined in a neighborhood $\mathcal{N}$ of the boundary $\partial\Omega$ provided $\sigma$ 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 $\Omega$, 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.<br />Comment: 37 pages – Name: TypeDocument Label: Document Type Group: TypDoc Data: Working Paper – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="http://arxiv.org/abs/2406.06226" linkWindow="_blank">http://arxiv.org/abs/2406.06226</link> – Name: AN Label: Accession Number Group: ID Data: edsarx.2406.06226 |
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| RecordInfo | BibRecord: BibEntity: Subjects: – SubjectFull: Mathematics - Analysis of PDEs Type: general – SubjectFull: 35R30, 26A33, 35J10, 35J70 Type: general Titles: – TitleFull: Approximation and uniqueness results for the nonlocal diffuse optical tomography problem Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lin, Yi-Hsuan – PersonEntity: Name: NameFull: Zimmermann, Philipp IsPartOfRelationships: – BibEntity: Dates: – D: 10 M: 06 Type: published Y: 2024 |
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