# Smoothing $\mathcal{L}^2$ gradients in iterative regularization

## TL;DR

This paper introduces a new iterative regularization method that smooths the gradient to improve robustness and effectiveness in solving large-scale inverse problems with noisy data, demonstrated through image deblurring and tomography applications.

## Contribution

It proposes a novel iterative regularization technique using smoothed descent directions, enhancing stability and performance over traditional gradient-based methods.

## Key findings

- More robust recovery in high-noise scenarios
- Improved results compared to Tikhonov, TV, and CGLS methods
- Effective in large-scale inverse problems

## Abstract

Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, which are a necessity due to the ill-posedness of inverse problems. Tikhonov-type regularization methods are very popular in this regard. However, its direct implementation for large-scale linear or non-linear problems is a non-trivial task. In such scenarios, iterative regularization methods usually serve as a better alternative. In this paper we propose a new iterative regularization method which uses descent directions, different from the usual gradient direction, that enable a more smoother and effective recovery than the later. This is achieved by transforming the original noisy gradient, via a smoothing operator, to a smoother gradient, which is more robust to the noise present in the data. It is also shown that this technique is very beneficial when dealing with data having large noise level. To illustrate the computational efficiency of this method we apply it to numerically solve some classical integral inverse problems, including image deblurring and tomography problems, and compare the results with certain standard regularization methods, such as Tikhonov, TV, CGLS, etc.