# On fractional asymptotical regularization of linear ill-posed problems in Hilbert spaces

## TL;DR

This paper introduces a fractional-order regularization method called FAR for solving linear ill-posed problems in Hilbert spaces, demonstrating accelerated convergence and numerical efficiency over existing methods.

## Contribution

It proposes a novel fractional asymptotical regularization method with theoretical acceleration properties and develops a new iterative scheme based on Adams-Moulton for practical implementation.

## Key findings

- FAR accelerates convergence compared to traditional methods.
- Numerical examples confirm the accuracy and efficiency of FAR.
- The method is applicable under certain smoothness conditions.

## Abstract

In this paper, we study a fractional-order variant of the asymptotical regularization method, called {\it Fractional Asymptotical Regularization (FAR)}, for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range $(1,2)$ yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.