  In this paper, we consider the inverse source problem for the time-fractional
diffusion equation, which has been known to be an ill-posed problem. To deal
with the ill-posedness of the problem, we propose to transform the problem into
a regularized problem with L^2 and total variational (TV) regularization terms.
Differing from the classical Tikhonov regularization with L^2 penalty terms,
the TV regularization is beneficial for reconstructing discontinuous or
piecewise constant solutions. The regularized problem is then approximated by a
fully discrete scheme. Our theoretical results include: estimate of the error
order between the discrete problem and the continuous direct problem; the
convergence rate of the discrete regularized solution to the target source
term; and the convergence of the regularized solution with respect to the noise
level. Then we propose an accelerated primal-dual iterative algorithm based on
an equivalent saddle-point reformulation of the discrete regularized model.
Finally, a series of numerical tests are carried out to demonstrate the
efficiency and accuracy of the algorithm.
