In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method together with an extrapolation strategy. The general formulation of iteratively regularized Landweber iteration method in Banach spaces excludes the use of certain functions such as total variation like penalty functionals, $L^1$ functions etc. The novel scheme presented in this paper allows to use such non-smooth penalty terms that can be helpful in practical applications involving the reconstruction of several important features of solutions such as piecewise constancy and sparsity. We carefully discuss the choices for important parameters, such as combination parameters and step sizes involved in the design of the method. Additionally, we discuss an example to validate our assumptions.

翻译：在本文中,我们引入了一种新的双点梯度方法,以解决巴纳赫空间中存在的问题,并研究其趋同分析。该方法基于众所周知的迭代正规化土地网迭代法以及外推法。巴纳赫空间迭代正规化土地网迭代法的一般提法排除了某些功能的使用,如刑罚功能等全变、1美元功能等。本文件介绍的新办法允许使用非移动的处罚术语,这些术语有助于实际应用,包括重建若干重要解决方案的特征,如节纸凝固和宽度。我们仔细讨论了重要参数的选择,如在设计方法时涉及的组合参数和步骤大小。此外,我们讨论了验证我们假设的范例。