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：本文研究时间分数阶扩散方程中空间依赖源项的反演问题。为处理该问题的不适定性，我们将其转化为带有全变差正则化的最优控制问题。与包含$L^2$惩罚项的传统Tikhonov模型相比，引入全变差项在重建具有间断性或分段常数特性的解时展现出显著优势。通过全离散格式对控制问题进行近似，并在该框架下给出了收敛性结果。此外，基于等效鞍点重构形式，提出了线性化的原始-对偶迭代算法来求解离散控制模型，并通过若干数值实验验证了算法的有效性。