واریانس تجزیه و تحلیل حساسیت جانبی برای پارامترهای توزیع و محاسبه آن

The output variance is an important measure for the performance of a structural system, and it is always influenced by the distribution parameters of inputs. In order to identify the influential distribution parameters and make it clear that how those distribution parameters influence the output variance, this work presents the derivative based variance sensitivity decomposition according to Sobol′s variance decomposition, and proposes the derivative based main and total sensitivity indices. By transforming the derivatives of various orders variance contributions into the form of expectation via kernel function, the proposed main and total sensitivity indices can be seen as the “by-product” of Sobol′s variance based sensitivity analysis without any additional output evaluation. Since Sobol′s variance based sensitivity indices have been computed efficiently by the sparse grid integration method, this work also employs the sparse grid integration method to compute the derivative based main and total sensitivity indices. Several examples are used to demonstrate the rationality of the proposed sensitivity indices and the accuracy of the applied method.

Development of probabilistic sensitivity analysis is frequently considered as an essential component of a probabilistic analysis, and it is often critical towards understanding the physical mechanisms and modifying the design to mitigate and manage risk [1]. Traditional sensitivity analysis (SA) can be classified into two groups: local SA and global SA [2]. Local SA usually investigates how small variation of a distribution parameter around a reference point changes the value of the output. One classical local SA is the derivative based SA by defining the derivative of probabilistic statistics quantity with respect to the distribution parameters of inputs [2]. The main drawback of the derivative based SA is that it depends on the choice of the nominal point, but it is generally mathematically simple and straightforward [3]. Global SA studies how the uncertainty in the output of a computational model can be decomposed according to the input sources of uncertainty [4]. Contrary to the local SA, global SA explores the whole range of uncertainty of the model inputs by letting them vary simultaneously [5]. At present, a number of global sensitivity indices have been suggested, e.g. Helton and Saltelli [6] and [7] proposed the nonparametric sensitivity indices (input–output correlation), Sobol, Iman and Saltelli [7], [8] and [9] proposed the variance based sensitivity indices, Chun, Liu and Borgonovo [10] and [11] proposed moment independent sensitivity indices. In this work, we mainly investigate the variance based sensitivity indices which have been applied to design under uncertainty problems and are capable of identifying the contributions of any random variable. However, the variance based sensitivity indices are far more computationally demanding [12]. It is noticed that in the classical variance based SA, the influences of distribution parameters are not involved. If the variation of a distribution parameter can lead to a considerable change to the variance, the computational results of the variance based SA will be vulnerable and less reliable. Thus, it is significant to identify how the distribution parameters influence the variance. To do this, this work employs the derivative based SA by defining the derivatives of the variance with respect to the distribution parameters. According to the Sobol′s variance decomposition theory [8], the influence of the distribution parameters on the variance can be transmitted by various orders variance contributions. Thus, the influence of the distribution parameter on the various orders variance contribution is investigated. Based on that, this work presents the derivative based main and total sensitivity indices, which can be used to identify the influence of the distribution parameter on the main and total variance contribution. By employing the kernel function to simplify the derivatives of the variance contribution with respect to the distribution parameters, the proposed main and total sensitivity indices can be computed easily and can be seen as the “by-product” of the classical variance based SA without additionally computational cost. Since a large number of methods have been used to compute the Sobol′s variance based sensitivity indices, such as the quasi-Monte Carlo method [12], FAST method [13], the meta-model based method [14], the sparse grid integration method [15] etc, and the sparse grid integration (SGI) method has been proved to be a high efficiency one in reference [16], this work would employ the SGI method to compute the proposed main and total indices. The remainder of this work is organized as follows: Section 2 gives a brief review of the variance based SA. Section 3 gives the decomposition of the derivative based variance sensitivity and proposes the corresponding main and the total sensitivity indices. In Section4, the kernel function is used to simplify the derivatives of the variance contributions with respect to the distribution parameters and some discussions are given. The spares grid integration method is employed in Section 5 to compute the proposed main and total sensitivity indices. In Section 6, two numerical examples are first given to validate the rationality of the proposed sensitivity indices and the accuracy of the SGI method. Then a simple cantilever beam with explicit response and a ten-bar structure with implicit response are analyzed. Finally, some conclusions are drawn in Section 7.

This work investigates the influence of the distribution parameters on the output variance. By employing the derivative based sensitivity analysis, the influential parameters can be identified. For those influential parameters, the derivative based variance sensitivity is decomposed according to Sobol's variance decomposition theory, thus how the influential distribution parameters influence the variance can be made clear. It is noticed that if there is no interaction effect in a model (additive model), the influence of one input's distribution parameters is only transmitted by the corresponding first-order variance contribution of this input, while if there is interaction effect in a model, the influence of one input's distribution parameters may be transmitted by the various orders variance contributions of other input. Thus, by collecting the information and improving the understanding of those most influential parameters can make the output variance stable. Meanwhile, the results of the classical variance based sensitivity analysis would also stable. It needs to be emphasized that the proposed sensitivity indices is defined by normalizing the derivative of the variance contributions by that of the output variance. Compared with the way of computing the derivatives of the Sobol's sensitivity indices, the proposed sensitivity indices can reflect of the transition of the influence of the distribution parameters clearly and they are much more comprehensible. For the proposed DBMI and DBTI can be simplified by the kernel function, they can be seen as the “by-product” of the classical variance based sensitivity analysis without additionally computational cost. In order to compute the proposed sensitivity indices efficiently, the sparse grid integration method is employed. The computational results of several examples demonstrate that the spares grid integration method is high efficiency with enough accuracy. The advantage of this method for copying the high dimensionality problems is also validated in this work.