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

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.