Simultaneous recovery of a corroded boundary and admittance using the Kohn-Vogelius method
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| Format: | Preprint |
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2025
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| author | Essahraoui, Moustapha Cherrat, Elmehdi Afraites, Lekbir Rabago, Julius Fergy Tiongson |
| author_facet | Essahraoui, Moustapha Cherrat, Elmehdi Afraites, Lekbir Rabago, Julius Fergy Tiongson |
| contents | We address the problem of identifying an unknown portion $Γ$ of the boundary of a $d$-dimensional ($d \in \{1, 2\}$) domain $Ω$ and its associated Robin admittance coefficient, using two sets of boundary Cauchy data $(f, g)$--representing boundary temperature and heat flux--measured on the accessible portion $Σ$ of the boundary. Identifiability results \cite{Bacchelli2009,PaganiPierotti2009} indicate that a single measurement on $Σ$ is insufficient to uniquely determine both $Γ$ and $α$, but two independent inputs yielding distinct solutions ensure the uniqueness of the pair $Γ$ and $α$. In this paper, we propose a cost function based on the energy-gap of two auxiliary problems. We derive the variational derivatives of this objective functional with respect to both the Robin boundary $Γ$ and the admittance coefficient $α$. These derivatives are utilized to develop a nonlinear gradient-based iterative scheme for the simultaneous numerical reconstruction of $Γ$ and $α$. Numerical experiments are presented to demonstrate the effectiveness and practicality of the proposed method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous recovery of a corroded boundary and admittance using the Kohn-Vogelius method Essahraoui, Moustapha Cherrat, Elmehdi Afraites, Lekbir Rabago, Julius Fergy Tiongson Numerical Analysis Optimization and Control 49Q10, 35R25, 35R30, 49Q12 We address the problem of identifying an unknown portion $Γ$ of the boundary of a $d$-dimensional ($d \in \{1, 2\}$) domain $Ω$ and its associated Robin admittance coefficient, using two sets of boundary Cauchy data $(f, g)$--representing boundary temperature and heat flux--measured on the accessible portion $Σ$ of the boundary. Identifiability results \cite{Bacchelli2009,PaganiPierotti2009} indicate that a single measurement on $Σ$ is insufficient to uniquely determine both $Γ$ and $α$, but two independent inputs yielding distinct solutions ensure the uniqueness of the pair $Γ$ and $α$. In this paper, we propose a cost function based on the energy-gap of two auxiliary problems. We derive the variational derivatives of this objective functional with respect to both the Robin boundary $Γ$ and the admittance coefficient $α$. These derivatives are utilized to develop a nonlinear gradient-based iterative scheme for the simultaneous numerical reconstruction of $Γ$ and $α$. Numerical experiments are presented to demonstrate the effectiveness and practicality of the proposed method. |
| title | Simultaneous recovery of a corroded boundary and admittance using the Kohn-Vogelius method |
| topic | Numerical Analysis Optimization and Control 49Q10, 35R25, 35R30, 49Q12 |
| url | https://arxiv.org/abs/2506.17938 |