Solving time-fractional inverse source problems using topological gradient methods

This paper addresses an inverse source problem related to a fractional diffusion equation, focussing on the identification of the source support from boundary measurements. We employ the Kohn-Vogelius formulation alongside the topological gradient method. The unknown support is characterized as the solution to an optimization problem that minimizes a Kohn-Vogelius-type functional. In the theoretical framework, we derive a topological asymptotic expansion that describes how the shape function varies with small geometric perturbations of the source term. Furthermore, we compute the Riesz representation of the source term operator, which is instrumental in formulating the expression for the topological gradient. On the numerical side, we introduce a straightforward and efficient method for solving the inverse source problem. The derived topological gradient is utilized to detect and reconstruct the unknown support, and the efficacy of the proposed algorithm is demonstrated through various numerical results.