The paper concerns with three iterative regularization methods for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the another is monotone and Lipschitz continuous, in a Hilbert space. We first describe how to incorporate regularization terms in the methods of forward-backward types, and then establish the strong convergence of the resulting methods. With several new stepsize rules considered, the methods can work with or without knowing previously the Lipschitz constant of cost operator. Unlike known hybrid methods, the strong convergence of the proposed methods comes from the regularization technique. Several applications to signal recovery problems and optimal control problems together with numerical experiments are also presented in this paper. Our numerical results have illustrated the fast convergence and computational effectiveness of the new methods over known hybrid methods.

本文涉及三种迭代正则化方法，用于解决两个算子之和的变分包含问题，一种在希尔伯特空间中是最大单调的，另一种是单调和Lipschitz连续的。我们首先描述如何将正则化项并入前向后向类型的方法中，然后建立所得方法的强收敛性。考虑到几个新的分步调整规则，这些方法可以在事先不知道成本算子的Lipschitz常数的情况下工作。与已知的混合方法不同，所提出方法的强收敛性来自于正则化技术。本文还介绍了信号恢复问题和最优控制问题的几种应用以及数值实验。