Reconstruction of a potential parameter in time-fractional diffusion problems via a Kohn--Vogelius type functional: Theoretical aspects

Of concern is the problem of reconstructing a space-dependent potential from boundary observations in the Caputo time-fractional diffusion equation, utilizing a stable and robust recovery method. We develop an algorithm to minimize the Kohn-Vogelius (KV) cost function, which measures the difference between the solutions of two excitations. The inverse potential problem is recast into an optimization problem, where the objective is to minimize a Kohn-Vogelius-type functional within a set of admissible potentials. We establish the well-posedness of this optimization problem by proving the existence and uniqueness of a minimizer and demonstrating its stability with respect to perturbations in the boundary data. Furthermore, we analyze the Fr\'echet differentiability of the KV functional and prove the Lipschitz continuity of its gradient. These theoretical results enable the development of a convergent conjugate gradient algorithm for numerical reconstruction. The effectiveness and robustness of the proposed method are confirmed through several numerical examples in both one and two dimensions, including cases with noisy data.