Inspired by certain regularization techniques for linear inverse problems, in this work we investigate the convergence properties of the Levenberg-Marquardt method using singular scaling matrices. Under a completeness condition, we show that the method is well-defined and establish its local quadratic convergence under an error bound assumption. We also prove that the search directions are gradient-related allowing us to show that limit points of the sequence generated by a line-search version of the method are stationary for the sum-of-squares function. The usefulness of the method is illustrated with some examples of parameter identification in heat conduction problems for which specific singular scaling matrices can be used to improve the quality of approximate solutions.

翻译：受线性逆问题中某些正则化技术的启发，本文研究了使用奇异尺度矩阵的Levenberg-Marquardt方法的收敛性质。在完备性条件下，我们证明了该方法的适定性，并在误差界假设下建立了其局部二次收敛性。我们还证明了搜索方向与梯度相关，从而表明该方法通过线性搜索生成的序列的极限点是平方和函数的驻点。通过热传导问题中参数识别的若干示例，验证了该方法的实用性，其中特定的奇异尺度矩阵可用于提高近似解的质量。