In this paper, we propose an inexact Newton regularization combined with two-point gradient methods for nonlinear ill-posed problems. The basic idea of the proposed method is to linearize the equation around each outer iteration and subsequently apply a so-called two-point gradient method in the inner loop to accelerate the iterative process. Under suitable assumptions, we show that the iteration sequence generated by the proposed algorithm converges to a solution of the related problem in the noiseless situation. Furthermore, the stability and regularization properties of the proposed algorithm are analyzed in the noise-data case. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.

在本文中，我们针对非线性不适定问题提出了一种不精确的牛顿正则化与两点梯度法相结合的方法。所提出方法的基本思想是围绕每次外部迭代线性化方程，然后在内环中应用所谓的两点梯度法以加快迭代过程。在适当的假设下，我们证明了该算法产生的迭代序列收敛于无噪声情况下相关问题的解决方案。此外，在噪声数据的情况下，分析了该算法的稳定性和正则化性质。提供了几个数值示例，以验证理论结果并证明所提出方法的效率。