# Numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation

## TL;DR

This paper develops a numerical method using Tikhonov regularization and iterative thresholding to reconstruct the spatial component of a source term in a time-fractional diffusion equation, demonstrating stability, convergence, and efficiency.

## Contribution

It introduces a stabilized nonlinear minimization approach with a novel convergence proof for the discrete scheme in reconstructing the source term.

## Key findings

- The method is stable and convergent under the proposed scheme.
- Numerical experiments confirm the efficiency and accuracy of the iterative thresholding algorithm.
- The approach effectively reconstructs the spatial component in fractional diffusion problems.

## Abstract

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 $H^1$-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.