Identifying source term in the subdiffusion equation with L 2-TV regularization *

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.