  In this article, we are concerned with the analysis on the numerical
reconstruction of the spatial component in the source term of a time-fractional
diffusion equation. This ill-posed problem is solved through a stabilized
nonlinear minimization system by an appropriately selected Tikhonov
regularization. The existence and the stability of the optimization system are
demonstrated. The nonlinear optimization problem is approximated by a fully
discrete scheme, whose convergence is established under a novel result verified
in this study that the $H^1$-norm of the solution to the discrete forward
system is uniformly bounded. The iterative thresholding algorithm is proposed
to solve the discrete minimization, and several numerical experiments are
presented to show the efficiency and the accuracy of the algorithm.
