Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination
Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination
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DOI：
10.1016/j.jsv.2007.03.040
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发表时间：
2007-07
影响因子：
4.7
通讯作者：
H. Choi;A. Thite;D. Thompson
中科院分区：
文献类型：
--
作者：
H. Choi;A. Thite;D. Thompson
在进行结构传声的传递路径分析时，需要结构传声路径内激励点处和/或连接处的作用力。这些力可以通过逆技术获得，但所用的测量数据将包含一些未知误差。因此，由于这些测量数据的病态矩阵的求逆，重建的力可能包括很大的误差。在这项研究中，Tikhonov正则化被用来改善矩阵求逆的条件化。有几种方法可以选择最佳正则化参数。本文的目的是比较普通交叉验证法和广义交叉验证法以及L曲线准则的性能。对于测量数据中的不同噪声水平，进行了模拟，表示在矩形平板上的测量。此外，还通过改变板的形状、力的位置和测量数据中包含的噪声水平来研究结论的稳健性。研究发现，L曲线方法的性能优于OCV或GCV，特别是在操作响应中的高噪声水平下，但当这些噪声水平较低时，表现得不那么好。因此，人们发现它不太容易产生大的重建误差，但在低噪声存在的情况下，它往往会过度规则化解决方案，导致低估力量。在实际操作中，操作响应的测量可能会受到噪声污染的影响，这表明L曲线法在实际情况下可能是最合适的方法。然而，在确定最合适的正则化技术之前，获得信号中可能的噪声的良好估计是重要的。一般认为，当矩阵条件数较高时，普通交叉验证法比广义交叉验证法具有更好的性能。由于条件数越高，正则化的必要性越大，因此发现普通交叉验证法总体上比广义交叉验证法给出更可靠的结果。
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.