Simultaneous reliability and reliability-sensitivity analyses based on the information-reuse of sparse grid numerical integration

Abstract

In this paper, a new method is put forward for simultaneous reliability and reliability-sensitivity analyses based on the information-reuse of sparse grid numerical integration (SGNI). First, the reliability analysis is conducted on the basis of fractional exponential moments-based maximum entropy method (FEM-MEM), where the SGNI is employed for FEM assessments. The reliability index can be evaluated by integrating over the distribution derived by FEM-MEM. Then, the reliability-sensitivity analysis is carried out by reusing the output samples in previous reliability analysis and updating the corresponding weights, where no additional model evaluations are required. Then, the FEM-MEM is applied again to derive the conditional distribution and reliability index. By comparing the conditional and original reliability indexes, one can define the reliability-sensitivity index to identify the importance of each random variable to reliability. Since only one-round of model evaluations are necessary in the proposed method, the computational efficiency is highly desirable. Four numerical examples are investigated to check the effectiveness of the proposed method, where pertinent results obtained from Monte Carlo simulations (MCS) and Sobol’s index are compared. The results demonstrate the proposed method is accurate and efficient for simultaneous reliability and reliability-sensitivity analyses.

Appendix A: Sobol’s sensitivity index

Sensitivity analysis is used to analyze the contribution of parameters to the output value, which can be used to obtain the importance of the influence of parameters on the model. Sobol’s sensitivity analysis (Giap and Kosuke 2014) is a method based on variance to obtain various sensitivity indexes whose values vary from 0 to 1. The closer the index value is to 1, the greater the influence of this variable on the results. On the contrary, the closer the index value is to 0, the less the influence of this variable on the results. In this paper, we mainly consider the first-order Sobol’s index and the total Sobol’s index (Zhan and Zhang 2013; Gasanov et al. 2020; Dwight et al. 2016), which are widely applied in sensitivity analysis.

Consider a function as

$$\begin{aligned} Y = f\left( \mathbf{{X}} \right) \end{aligned}$$

(A.1)

where Y is the model output and ${\mathbf{{X}} = (X_1, X_2, X_3,\ldots,X_D)^{T}}$ is an D-dimensional random vector of input parameters. In the Sobol’s method, the total unconditional variance V(Y) can be broken down into the following parts

$$\begin{aligned} V\left( {{Y}} \right) = \sum _{i}^D{V_i}+\sum _{i}^D\sum _{j>i}^D{V_{ij}}+\cdots +V_{12...D}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} {V_i}= & {} V_{X_i}[{\mathbb {E}}_{X_{\sim {i}}}(Y|X_i)], \end{aligned}$$

(A.3)

$$\begin{aligned} {V_{ij}}= & {} V_{{X_i}{X_j}}[{\mathbb {E}}_{X_{\sim {ij}}}(Y|X_i,X_j)]-V_i-V_j,\nonumber \\&. . . \end{aligned}$$

(A.4)

$$\begin{aligned} {V_{12...D}}= & {} V_{{X_1}{X_2}...{X_D}}[{\mathbb {E}}_{X_{\sim {12...D}}}(Y|X_1,X_2,...,X_D)]. \end{aligned}$$

(A.5)

Therefore, the first-order sensitivity index ${S_i}$ which represents the influence of input variable ${X_i}$ alone can be expressed as

$$\begin{aligned} {S_i} = \frac{V_i}{V(Y)} = \dfrac{V_{X_i}[{\mathbb {E}}_{X_{\sim {i}}}(Y|X_i)]}{V(Y)}. \end{aligned}$$

(A.6)

Similarly, the second-order sensitivity index ${S_{ij}}$ representing the interaction between ${X_i}$ and ${X_j}$ can also be calculated such that

$$\begin{aligned} {S_{ij}} = \frac{V_{ij}}{V(Y)} = \dfrac{V_{{X_i}{X_j}}[{\mathbb {E}}_{X_{\sim {ij}}}(Y|X_i,X_j)]-V_i-V_j}{V(Y)}. \end{aligned}$$

(A.7)

The other high-order sensitivity indexes can be represented as

$$\begin{aligned} {S_{12...D}} = \frac{V_{12...D}}{V(Y)} = \dfrac{V_{{X_1}{X_2}...{X_D}}[{\mathbb {E}}_{X_{\sim {12...D}}}(Y|X_1,X_2,\ldots,X_D)]}{V(Y)}. \end{aligned}$$

(A.8)

The total sensitivity index ${S_{T_{i}}}$ is the sum of the effects of input variable ${X_i}$ and its interactions, which can be formulated as follows

$$\begin{aligned} \begin{aligned} {S_{T_{i}}}&= \dfrac{{\mathbb {E}}_{X_{\sim {i}}}[V_{X_i}(Y|X_{\sim {i}})]}{V(Y)}\\&= 1-\dfrac{V_{{X_i}{X_j}}[{\mathbb {E}}_{X_{\sim {ij}}}(Y|X_i,X_j)]-V_i-V_j}{V(Y)}\\&= S_i+\sum _{i\ne {j}}{S_{ij}}+\sum _{i\ne {j}\ne {k}}{S_{ijk}}+\cdots+S_{12...D}. \end{aligned} \end{aligned}$$

(A.9)

For example, when $D=4$,

$$\begin{aligned} {S_{T_1}} = S_1+S_{12}+S_{13}+S_{14}+S_{123}+S_{124}+S_{134}+S_{1234}. \end{aligned}$$

(A.10)