Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination

Abstract

In performing transfer path analysis of structure-borne sound transmission, the operational forces at the excitation points and/or at the connections within the structure-borne paths are required. These forces can be obtained by using inverse techniques but the measured data used will contain some unknown errors. Therefore the reconstructed forces may include large errors due to the inversion of an ill-conditioned matrix of these measured data. In this study, Tikhonov regularization is used in order to improve the conditioning of the matrix inversion. Several methods are available to select the optimal regularization parameter. The purpose of this paper is to compare the performance of the ordinary and generalized cross validation methods and the L-curve criterion. Simulations are carried out, representing measurements on a rectangular plate, for different noise levels in measured data. Also, the robustness of the conclusions is investigated by varying the shape of the plates, the force positions, and the noise levels included in the measured data. The L-curve method is found to perform better than OCV or GCV, particularly for high noise levels in the operational responses, but less well when these noise levels are low. It is therefore found to be less susceptible to producing large reconstruction errors but it tends to over-regularize the solution in the presence of low noise, leading to under-estimates of the forces. In practice, measurements of operational responses may be susceptible to noise contamination which suggests that the L-curve method is likely to be the most appropriate method in practical situations. Nevertheless, it is important to obtain good estimates of the likely noise in the signals before determining the most appropriate regularization technique. Ordinary cross validation method is generally found to have a better performance than generalized cross validation method if the matrix condition numbers are high. Since the need for regularization is greater with high condition numbers, it is consequently found that the ordinary cross validation method gives more reliable results overall than the generalized cross validation method.