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

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.