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arXiv · 2503.14764

Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement

Manabu Machida·Hirofumi Notsu·Julius Fergy Tiongson Rabago
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Abstract

This paper proposes a unified shape and coefficient optimization approach for inverse problems governed by diffusion equations. The associated forward problem is considered with a Robin boundary condition, physically motivated in diffuse optical tomography to model partial reflection of light at tissue boundaries. The main objective is the recovery of the piecewise-defined absorption coefficient together with its underlying interface from a single boundary measurement. To this end, a shape-based reconstruction approach is formulated in which the interface is introduced as a geometric unknown governing the piecewise structure of the absorption coefficient. While classical approaches rely on the Fr\'{e}chet derivative with respect to spatially varying parameters, the Eulerian derivative with respect to the interface is additionally exploited. This leads to a unified framework for the simultaneous recovery of the coefficient and the geometry under the single-measurement setting. Numerical experiments demonstrate the effectiveness of the proposed method, even for complex and non-convex interfaces.

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Manabu Machida, Hirofumi Notsu, Julius Fergy Tiongson Rabago. 2025-03-18. Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement. https://doi.org/10.3934/ipi.2026049

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