Recovering a Space-Dependent Coefficient in a Time Fractional Diffusion-Wave Equation via a Banach Space Regularization Scheme

This paper investigates a nonlinear inverse problem in a time fractional diffusion-wave equation, in which a spatially varying potential coefficient is recovered from noisy final-time measurements. A local uniqueness result for the inverse problem is first established in a finite-dimensional admissible space. To support the reconstruction, the Fréchet derivative of the forward map and its adjoint representation are derived to provide the gradient information required in the inversion procedure. A Banach space regularization scheme with a combined L1 and L2 penalty is then proposed to stabilize the nonlinear inverse problem, and the resulting nonsmooth minimization problem is solved by a locally linearized split Bregman iterative scheme. Numerical experiments in one and two spatial dimensions demonstrate the accuracy and stability of the proposed method for smooth, corner-type, localized, and discontinuous coefficient profiles.