  Recently, inverse problems have attracted more and more attention in
computational mathematics and become increasingly important in engineering
applications. After the discretization, many of inverse problems are reduced to
linear systems. Due to the typical ill-posedness of inverse problems, the
reduced linear systems are often ill-posed, especially when their scales are
large. This brings great computational difficulty. Particularly, a small
perturbation in the right side of an ill-posed linear system may cause a
dramatical change in the solution. Therefore, regularization methods should be
adopted for stable solutions. In this paper, a new class of accelerated
iterative regularization methods is applied to solve this kind of large-scale
ill-posed linear systems. An iterative scheme becomes a regularization method
only when the iteration is early terminated. And a Morozov's discrepancy
principle is applied for the stop criterion. Compared with the conventional
Landweber iteration, the new methods have acceleration effect, and can be
compared to the well-known accelerated v-method and Nesterov method. From the
numerical results, it is observed that using appropriate discretization
schemes, the proposed methods even have better behavior when comparing with
v-method and Nesterov method.
