Reconstruction and stability analysis of potential appearing in time‐fractional subdiffusion

Abstract

In this work, we establish a local stability estimate for a problem involving the reconstruction of a domain in the time‐fractional diffusion equation with an order of : from the knowledge of a Neumann additional data on a part of the boundary. This problem is motivated by several applications in anomalous diffusion phenomena. The stability estimate is obtained through a straightforward approach utilizing shape optimization techniques. To reconstruct the domain , we formulate the inverse problem as a shape optimization one by minimizing a least‐squares cost function. Using the topological derivative method, we perform a noniterative algorithm for numerically recovering the location and shape of the unknown domain. Finally, we demonstrate the effectiveness of our approach in reconstructing multiple subdomains of varying shapes and sizes, considering noisy data, through numerical experiments.