Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911397990891520 |
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| author | Machida, Manabu Notsu, Hirofumi Rabago, Julius Fergy Tiongson |
| author_facet | Machida, Manabu Notsu, Hirofumi Rabago, Julius Fergy Tiongson |
| contents | This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. We consider a Robin boundary condition, physically motivated in diffuse optical tomography to model partial reflection of light at tissue boundaries [Arr99, GFB83a]. This ensures well-posedness of the forward problem, while related inverse problems with Dirichlet or Neumann conditions have also been considered in previous studies [Mef21]. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Frechet derivative of a cost functional with respect to the unknown parameter, we also utilize its Eulerian derivative with respect to the unknown boundary interface for recovery. This non-conventional approach addresses parameter recovery when only a single boundary measurement can be obtained, providing a method for its reconstruction. Numerical experiments confirm the effectiveness of the proposed method, even for intricate and non-convex boundary interfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement Machida, Manabu Notsu, Hirofumi Rabago, Julius Fergy Tiongson Numerical Analysis Optimization and Control 49N45, 49Q10, 49K40, 65N21, 65N30 This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. We consider a Robin boundary condition, physically motivated in diffuse optical tomography to model partial reflection of light at tissue boundaries [Arr99, GFB83a]. This ensures well-posedness of the forward problem, while related inverse problems with Dirichlet or Neumann conditions have also been considered in previous studies [Mef21]. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Frechet derivative of a cost functional with respect to the unknown parameter, we also utilize its Eulerian derivative with respect to the unknown boundary interface for recovery. This non-conventional approach addresses parameter recovery when only a single boundary measurement can be obtained, providing a method for its reconstruction. Numerical experiments confirm the effectiveness of the proposed method, even for intricate and non-convex boundary interfaces. |
| title | Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement |
| topic | Numerical Analysis Optimization and Control 49N45, 49Q10, 49K40, 65N21, 65N30 |
| url | https://arxiv.org/abs/2503.14764 |