In these notes we propose and analyze an inertial type method for obtaining stable approximate solutions to nonlinear ill-posed operator equations. The method is based on the Levenberg-Marquardt (LM) iteration. The main obtained results are: monotonicity and convergence for exact data, stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a parameter identification problem in elliptic PDEs, ii) a parameter identification problem in machine learning; the computational efficiency of the proposed method is compared with canonical implementations of the LM method.

翻译：本文提出并分析了一种惯性型方法，用于获得非线性不适定算子方程的稳定近似解。该方法基于Levenberg-Marquardt（LM）迭代。获得的主要结果包括：精确数据下的单调性与收敛性，噪声数据下的稳定性与半收敛性。在数值实验方面，我们考察了：i) 椭圆型偏微分方程中的参数辨识问题；ii) 机器学习中的参数辨识问题；并将所提方法的计算效率与经典LM方法的实现进行了比较。