$L$-curve method for evaluating the optimal parameter of a smoothing cubic spline

A universal device for filtering measuring noise is a smoothing cubic spline of defect 1. The magnitude of the filtering error (smoothing) is mostly determined by the value of the smoothing parameter. With the optimal smoothing parameter value, the smoothing error value takes a minimum value. In practice, there is no a priori information on the exact (not noisy) values of the signal, and it is impossible to calculate the value of the optimal smoothing parameter. In this regard, the algorithms used to solve practical problems for choosing the optimal parameter allow for estimating only acceptable smoothing errors, which sometimes significantly exceed the minimum values. The estimation of the optimal smoothing parameter with an unknown variance of the noise measurement of experimental data presents particular difficulty. The current article constructs and examines in detail an algorithm for estimating the optimal parameter based on the $L$-curve method. This method is used to select the regularization parameter in algorithms for solving incorrectly set tasks. Particular attention is paid to the estimation of the optimal parameter in conditions of correlated noise. Based on the results of these studies, the authors provide practical recommendations for the application of this selection algorithm in the practice of processing experimental data.