  Inverse problems arise in a wide spectrum of applications in fields ranging
from engineering to scientific computation. Connected with the rise of interest
in inverse problems is the development and analysis of regularization methods,
such as Tikhonov-type regularization methods or iterative regularization
methods, which are a necessity in most of the inverse problems. In the last few
decades, regularization methods motivating sparsity has been the focus of
research, due to the high dimensionalty of the real-life data, and
$\mathcal{L}^1$-regularization methods (such as LASSO or FISTA) has been in its
center (due to their computational simplicity). In this paper we propose a new
(semi-) iterative regularization method which is not only simpler than the
mentioned algorithms but also yields better results, in terms of accuracy and
sparsity of the recovered solution. Furthermore, we also present a very
effective and practical stopping criterion to choose an appropriate
regularization parameter (here, it's iteration index) so as to recover a
regularized (sparse) solution. To illustrate the computational efficiency of
this algorithm we apply it to numerically solve the image deblurring problem
and compare our results with certain standard regularization methods, like
total variation, FISTA, LSQR etc.
