A scaling fractional asymptotical regularization method for linear inverse problems

In this paper, we propose a Scaling Fractional Asymptotical Regularization (S-FAR) method for solving linear ill-posed operator equations in Hilbert spaces, inspired by the work of (2019 Fract. Calc. Appl. Anal. 22(3) 699-721). Our method is incorporated into the general framework of linear regularization and demonstrates that, under both Hölder and logarithmic source conditions, the S-FAR with fractional orders in the range (1, 2] offers accelerated convergence compared to comparable order optimal regularization methods. Additionally, we introduce a de-biasing strategy that significantly outperforms previous approaches, alongside a thresholding technique for achieving sparse solutions, which greatly enhances the accuracy of approximations. A variety of numerical examples, including one- and two-dimensional model problems, are provided to validate the accuracy and acceleration benefits of the FAR method.