A novel estimation method for failure-probability-based-sensitivity by conditional probability theorem

Abstract

By the average absolute difference between the unconditional failure probability and the conditional one on fixing an input at its realization, the failure-probability-based-sensitivity (FP-S) is defined to quantify the effect of the fixed input on the failure probability, which provides important information for reliability-based design optimization of the structure. Among the estimation methods for FP-S, the Bayes theorem-based methods are competitive, but the conditional probability density function (PDF) should be estimated in this type method. To alleviate the computational complexity of estimating conditional PDF, a novel FP-S estimation method is proposed by use of the conditional probability theorem. In the proposed method, the conditional failure probability on fixing the input at its realization is approximated by the conditional failure probability on fixing the input in a small interval, in which the conditional probability theorem of the random event can be used to transform FP-S as estimations of a series of probabilities, and they can be simultaneously completed by a numerical simulation for estimating the unconditional failure probability. For ensuring the precision of the approximation introduced by replacing the realization with the small interval, a selection strategy for the small interval is proposed. Comparing with the competitive Bayes theorem-based estimation for FP-S, the proposed method replaces the conditional PDF estimation with the conditional probability estimation, which greatly reduces the computational complexity and improves the accuracy of the FP-S estimation. By combining with the adaptive kriging surrogate model, the efficiency of the proposed method can be drastically improved, and the presented examples demonstrate the efficiency and accuracy of the proposed method.

Appendix

To construct a Kriging surrogate model, the output of the model Y = g(X) can be considered as a realization of stochastic process represented in Refs. (Jones et al. 1998; Bdour and Guiffaut 2016)

$$ {g}_K(X)={\beta}_0+Z(X) $$

(A1)

where β₀ is the mean value and Z(X) represents a stationary Gaussian process with zero mean and a covariance between two samples x⁽ⁱ⁾ and x^(j), the covariance is defined by.

$$ Cov\left[Z\left({x}^{(i)}\right),\kern0.5em Z\left({x}^{(j)}\right)\right]={\sigma}^2R\left({x}^{(i)},{x}^{(j)}\right) $$

(A2)

here, σ² is the process variance of Z(⋅), R is the spatial correlation function, which is only dependent on the spatial distance between x⁽ⁱ⁾ and x^(j). In this paper, the widely used Gaussian correlation is employed.

Therefore, for the arbitrary sample x, the Kriging model g_K(x) is a Gaussian random process, i.e., ${g}_K(x)\sim N\left({\mu}_{g_K}(x),{\sigma}_{g_K}^2(x)\right)$, where ${\mu}_{g_K}(x)$ and ${\sigma}_{g_K}^2(x)$ are the Kriging prediction mean and variance respectively. More details can be obtained in Refs. (Jones et al. 1998; Bdour and Guiffaut 2016).

In order to construct a convergent adaptive Kriging (AK) model for the output of model and to identify the input samples falling into the failure domain, a sample pool and an active learning strategy are required. Firstly, a sample pool of input variables is built asS = {x₁, x₂, ..., x_N}. Subsequently, N₀ initial samples are selected to establish the initial Kriging model. Then, an active learning function is employed to select new training sample with the most contribution to the prediction quality to refine the initial Kriging model. For the failure problem, a widely used learning function, named as U-learning function (Echard et al. 2011), is adopted and it is denoted as

$$ U(x)=\frac{\left|{\mu}_{g_K}(x)\right|}{\sigma_{g_K}(x)} $$

(A3)

The U-learning function implies a reliability index of misjudging the sign of g(x) by g_K(x), i.e.,

$$ \varPhi \left(-U(x)\right)=\varPhi \left(-\frac{\left|{\mu}_{g_K}(x)\right|}{\sigma_{g_K}(x)}\right) $$

(A4)

where Φ(⋅) denotes the calculative distribution function of a standard normal variable. Thus, the new training sample can be found in the sample pool by

$$ {x}_u=\arg\ \underset{x\in S}{\mathit{\min}}\left[U(x)\right] $$

(A5)

When U(x) = 2, it indicates that there is Φ(−2) = 0.0228 probability to misjudge the sign of g(x). Then the stop criterion of selecting the new training point in the sample pool S to update the current Kriging model is given as

$$ \underset{x\in S}{\mathit{\min}}\left[U(x)\right]\ge 2 $$

(A6)