We consider the problem on constructing a stable approximate solution of an inverse problem formulated as a nonlinear irregular equation with a monotone operator. We suggest a two-stage method based on Lavrentiev’s regularization scheme and iterative approximation with the use of either modified Newton’s method or a regularized κ-process. We prove that the iterative processes converge and the iterations possess the Fejér property. We show that our method generates a regularization algorithm under a certain adjustment of control parameters. On the set of source-like representable solutions, we find an optimal-order error estimate for the algorithm.

中文翻译：

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单调算子的不适定问题的迭代过程

我们考虑在构造一个具有单调算子的非线性不规则方程的反问题的稳定近似解的问题。我们建议使用Lavrentiev正则化方案和使用修改的Newton方法或正则κ过程进行迭代逼近的两阶段方法。我们证明了迭代过程收敛并且迭代具有Fejér属性。我们证明了我们的方法在控制参数的一定调整下会生成一个正则化算法。在像源一样的可表示解的集合上，我们找到了算法的最佳阶误差估计。