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.

翻译：在本文中,我们考虑了时间折射扩散方程式的反源问题,众所周知,时间折射扩散方程式是一个不恰当的问题。为了解决这个问题的弊端,我们提议将问题转换成L ⁇ 2和完全变异(TV)的正规化条件的常规化问题。与古典的Tikhonov正规化和L ⁇ 2的罚款条件不同,电视正规化有助于重建不连续或片断的常数解决方案。然后,常规化的问题被完全独立的方案所近似。我们的理论结果包括:对离散问题和连续的直接问题之间的误差顺序的估计;离散常规化解决办法对目标源术语的趋同率;以及常规化解决办法对噪音水平的趋同。然后,我们提议在对离散的常规化模式进行同等的马鞍点重新组合的基础上加速初线迭代算法。最后,进行了一系列数字测试,以证明算法的效率和准确性。