In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H¹-norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm.

中文翻译：

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时间分数扩散方程源项中空间分量的数值重构

在本文中，我们关注对时间分数阶扩散方程的源项中的空间分量进行数值重构的分析。此病态问题是由适当选择的Tikhonov正则化通过一个稳定的非线性最小化系统来解决。证明了优化系统的存在性和稳定性。非线性优化问题可以通过一个完全离散的方案来近似，其收敛性是根据本研究证明的新结果证明的：H ¹离散前向系统的解的-范数是一致有界的。提出了一种迭代阈值算法来解决离散最小化问题，并通过数值实验证明了算法的有效性和准确性。