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

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 Fréchet 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.