In this paper, we propose and analyze a novel Kaczmarz type method for solving inverse problems which can be written as systems of nonlinear operator equations in Banach spaces. The proposed method is formulated by combining homotopy perturbation iteration and Kaczmarz approach with uniformly convex penalty terms. The penalty term is allowed to be non-smooth, including the $L^1$ and the total variation like penalty functionals, to reconstruct special features of solutions such as sparsity and piecewise constancy. To accelerate the iteration, we introduce a sophisticated rule to determine the step sizes per iteration. Under certain conditions, we present the convergence result of the proposed method in the exact data case. When the data is given approximately, together with a suitable stopping rule, we establish the stability and regularization properties of the method. Finally, some numerical experiments on parameter identification in partial differential equations by boundary as well as interior measurements are provided to validate the effectiveness of the proposed method.

在本文中，我们提出并分析了一种新颖的Kaczmarz型求解逆问题的方法，该方法可以写为Banach空间中的非线性算子方程组。该方法是将同伦摄动迭代和Kaczmarz方法与一致凸惩罚项相结合而提出的。处罚条款可以是不光滑的，包括${\mathrm{大号}}^{\mathrm{1个}}$以及诸如惩罚函数之类的总变化量，以重建解决方案的特殊功能，例如稀疏性和分段恒定性。为了加快迭代速度，我们引入了一个复杂的规则来确定每次迭代的步长。在一定条件下，我们给出了在精确数据情况下该方法的收敛结果。当大约给出数据时，再加上合适的停止规则，我们便建立了该方法的稳定性和正则化性质。最后，通过边界和内部测量，对偏微分方程参数识别进行了一些数值实验，以验证该方法的有效性。