Stability analysis of an inverse coefficients problem in a system of partial differential equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916726378070016 |
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| author | Meftahi, Houcine Nssibi, Chayma |
| author_facet | Meftahi, Houcine Nssibi, Chayma |
| contents | In this study, we address the inverse problem of recovering the Lamé parameters ($λ, μ$) and the density $ρ$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lamé parameters ($λ, μ$) are know and we look for the inverse problem of recovering the density $ρ$. In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $λ$, $μ$ and $ρ$ simultameousely.
We establish Lipschitz stability estimate, provided that the parameters $λ$, $μ$ and $ρ$ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability analysis of an inverse coefficients problem in a system of partial differential equations Meftahi, Houcine Nssibi, Chayma Optimization and Control In this study, we address the inverse problem of recovering the Lamé parameters ($λ, μ$) and the density $ρ$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lamé parameters ($λ, μ$) are know and we look for the inverse problem of recovering the density $ρ$. In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $λ$, $μ$ and $ρ$ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $λ$, $μ$ and $ρ$ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials. |
| title | Stability analysis of an inverse coefficients problem in a system of partial differential equations |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2505.05116 |