A shape-optimization approach for inverse diffusion problems by single boundary measurement

This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Fr\'{e}chet derivative of a cost functional with respect to the unknown parameter, we also utilize its shape derivative with respect to the unknown boundary interface for recovery. This non-conventional approach addresses the problem of parameter recovery from a single measurement, which represents the key innovation of this work. Various numerical experiments, conducted in both two and three spatial dimensions, demonstrate the effectiveness of the proposed method, even for complex, non-convex boundary interfaces.