  In this paper, we consider an inverse space-dependent source problem for a
time-fractional diffusion equation. To deal with the ill-posedness of the
problem, we transform the problem into an optimal control problem with total
variational (TV) regularization. In contrast to the classical Tikhonov model
incorporating $L^2$ penalty terms, the inclusion of a TV term proves
advantageous in reconstructing solutions that exhibit discontinuities or
piecewise constancy. The control problem is approximated by a fully discrete
scheme, and convergence results are provided within this framework.
Furthermore, a lineraed primal-dual iterative algorithm is proposed to solve
the discrete control model based on an equivalent saddle-point reformulation,
and several numerical experiments are presented to demonstrate the efficiency
of the algorithm.
