We construct and study a class of numerically implementable iteratively regularized Gauss–Newton type methods for approximate solution of irregular nonlinear operator equations in Hilbert space. The methods include a general finite-dimensional approximation for equations under consideration and cover the projection, collocation and quadrature discretization schemes. Using an a posteriori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. We also investigate projected versions of the processes which take into account a priori information about a convex compact containing the solution. An iteratively regularized quadrature process is applied to an inverse 2D problem of gravimetry.

中文翻译：

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具有后验停止的有限维迭代正则化过程，用于求解不规则非线性算子方程

我们构建并研究了一类数值可实现的迭代正则化高斯-牛顿型方法，用于希尔伯特空间中不规则非线性算子方程的近似解。这些方法包括对所考虑方程的一般有限维近似，并涵盖投影、搭配和正交离散化方案。使用迭代过程的后验停止规则和解决方案的标准源条件，我们建立了方法生成的近似值的准确度估计。我们还研究了过程的投影版本，这些版本考虑了包含解决方案的凸紧凑的先验信息。将一个迭代正则化的求积过程应用于重力的反二维问题。