We investigate a class of finite dimensional iteratively regularized Gauss–Newton methods for solving nonlinear irregular operator equations in a Hilbert space. The developed technique allows to investigate in a uniform style various discretization methods such as projection, quadrature and collocation schemes and to take into account available restrictions on the solution. We propose an a posteriori stopping rule for the iterative process and establish an accuracy estimate for obtained approximation. The regularized Gauss–Newton method combined with the quadrature discretization and the a posteriori iteration stopping is applied to a model ionospheric radiotomography problem. The problem is reduced to a nonlinear integral equation describing the phase shift of a sounding radio signal in dependence of the free electron concentration in the ionospheric plasma. We establish the unique solvability of the inverse problem in the class of analytic functions.

我们研究了一类有限维迭代正则化高斯-牛顿方法，用于求解希尔伯特空间中的非线性不规则算子方程。所开发的技术允许以统一的方式研究各种离散化方法，例如投影、正交和搭配方案，并考虑到解决方案的可用限制。我们为迭代过程提出了一个后验停止规则，并为获得的近似值建立了一个准确度估计。正则化 Gauss-Newton 方法结合正交离散化和后验迭代停止应用于模型电离层放射断层摄影问题。该问题被简化为一个非线性积分方程，该方程描述了探测无线电信号的相移，该相移取决于电离层等离子体中的自由电子浓度。我们在解析函数类中建立了逆问题的唯一可解性。