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.

翻译：本文考虑时间分数阶扩散方程中与空间相关的反源问题。为解决该问题的不适定性，我们将其转化为带有总变差正则化的最优控制问题。与经典Tikhonov模型中包含$L^2$罚项的方法相比，引入总变差项在重构具有间断性或分段常数特性的解时展现出优势。该控制问题采用全离散格式逼近，并在该框架下给出了收敛性结果。进一步地，基于等价鞍点重构，提出一种线性化原始-对偶迭代算法求解离散控制模型，并通过数值实验验证了该算法的有效性。