In the spirit of the method in Margotti and Rieder (An inexact Newton regularization in Banach spaces based on the non-stationary iterated Tikhonov method. J Inverse Ill-Posed Probl. 2015;23:373–392), we propose an inexact Newton regularization which is based on the non-stationary iterated Tikhonov method for non-smooth solutions of nonlinear inverse problems in Banach spaces. The method consists of an inner iteration and an outer iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method. The inner iteration is performed by the non-stationary iterated Tikhonov method. The remarkable point is that the penalty terms can be general convex functions. Under certain assumptions on nonlinear operators, the convergence analysis of our method is given by making use of tools from convex analysis. Furthermore, numerical simulations are provided to support the theoretical results and test the performance of the method.

本着 Margotti 和 Rieder 中方法的精神（基于非平稳迭代 Tikhonov 方法的 Banach 空间中的不精确牛顿正则化。J Inverse Ill-Posed Probl. 2015;23:373-392），我们提出了不精确的牛顿正则化基于非平稳迭代 Tikhonov 方法求解 Banach 空间中非线性逆问题的非光滑解。该方法由内迭代和外迭代组成。外部迭代由差异原理终止，由不精确的牛顿正则化方法组成。内迭代由非平稳迭代的Tikhonov方法进行。值得注意的是，惩罚项可以是一般的凸函数。在非线性算子的某些假设下，我们的方法的收敛性分析是利用凸分析工具给出的。此外，还提供了数值模拟来支持理论结果并测试该方法的性能。