In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg–Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional electrical impedance tomography problem. The performance of our strategy is compared with standard implementations of the Levenberg–Marquardt method (using a priori choice of the multipliers).

中文翻译：

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Levenberg-Marquardt方法中选择拉格朗日乘数的范围松弛标准

在本文中，我们提出了一种新的策略，用于选择Levenberg-Marquardt方法中的拉格朗日乘子来解决由希尔伯特空间之间作用的非线性算子建模的不适定问题。针对该方法建立了收敛性分析结果，包括迭代误差的单调性，残差的几何衰减，精确数据的收敛性，噪声数据的稳定性和半收敛性。给出了椭圆形参数识别二维电阻抗层析成像问题的数值实验。我们将策略的执行效果与Levenberg-Marquardt方法的标准实现（使用乘数的先验选择）进行了比较。