In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

中文翻译：

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用于大规模非线性不适定问题的非精确非平稳 Tikhonov 程序

在这项工作中，我们考虑具有非零残差的大规模不适定非线性最小二乘问题的稳定数值解。我们提出了一种具有不精确步长计算的非平稳 Tikhonov 方法，专为大规模问题而设计。在每次迭代时，该方法都需要求解椭圆信任域子问题来计算步长。该任务采用 Lanczos 方法来执行，通过该方法计算近似解。选择信任区域半径以确保生成的 Tikhonov 正则化参数满足模型上的规定条件，这被证明可以确保该方法的正则化特性。所提出的方法在参数识别问题和图像配准问题上进行了测试，