An efficient method by nesting adaptive Kriging into Importance Sampling for failure-probability-based global sensitivity analysis

Abstract

Failure-probability-based global sensitivity (FP-GS) analysis can measure the effect of the input uncertainty on the failure probability. The state-of-the-art for estimating the FP-GS are less efficient for the rare failure event and the implicit performance function case. Thus, an adaptive Kriging nested Importance Sampling (AK-IS) method is proposed in this work to efficiently estimate the FP-GS. For eliminating the dimensionality dependence in the calculation, an equivalent form of the FP-GS transformed by the Bayes’ formula is employed by the proposed method. Then the AK model is nested into IS for recognizing the failure samples. After all the failure samples are correctly identified from the IS sample pool, the failure samples are transformed into those subjected to the original conditional probability density function (PDF) on the failure domain by the Metropolis–Hastings algorithm, on which the conditional PDF of the input on the failure domain can be estimated for the FP-GS finally. The proposed method highly improves the efficiency of estimating the FP-GS comparing with the state-of-the-art, which is illustrated by the results of several examples in this paper.

Appendices

Appendix A: The variation coefficient of the estimates

The variation coefficient of $\hat{P}\left\{ F \right\}$ is estimated by the following equation:

$$ {\text{COV}}\left( {\hat{P}\left\{ F \right\}} \right) = \frac{{\sqrt {{\text{var}} \left( {P\left\{ F \right\}} \right)} }}{{E\left( {P\left\{ F \right\}} \right)}} = \sqrt {\frac{{1 - P\left\{ F \right\}}}{{\left( {N - 1} \right)P\left\{ F \right\}}}} . $$

The variation coefficient of $\hat{P}\left\{ {F\left| {x_{ji} } \right.} \right\}$ is estimated by the following equation:

$$ {\text{COV}}\left( {\hat{P}\left\{ {F\left| {x_{ji} } \right.} \right\}} \right) = \frac{{\sqrt {{\text{var}} \left( {P\left\{ {F\left| {x_{ji} } \right.} \right\}} \right)} }}{{E\left( {P\left\{ {F\left| {x_{ji} } \right.} \right\}} \right)}} = \sqrt {\frac{{1 - P\left\{ {F\left| {x_{ji} } \right.} \right\}}}{{\left( {N_{2} - 1} \right)P\left\{ {F\left| {x_{ji} } \right.} \right\}}}} . $$

It is difficult to derive the variation coefficient of $\hat{\xi }_{i}$ and no derivation process about the variation coefficient of $\hat{\xi }_{i}$ has been found in the existing literature. Generally, the variation coefficient of $\hat{\xi }_{i}$ can be obtained by repeated calculation process. Assumed that $\hat{\xi }_{i}$ is calculated for $N_{C}$ times by $N_{1}$ samples, which are randomly generated for each time. The variation coefficient of $\hat{\xi }_{i}$ can be obtained by the following equation:

$$ {\text{COV}}\left( {\hat{\xi }_{i} } \right) = \frac{{\sqrt {{\text{var}} \left[ {\hat{\xi }_{i}^{\left( q \right)} } \right]} }}{{E\left[ {\hat{\xi }_{i}^{\left( q \right)} } \right]}}{ = }\frac{{\sqrt {E\left[ {\left( {\hat{\xi }_{i}^{\left( q \right)} } \right)^{2} } \right]{ - }E^{2} \left[ {\hat{\xi }_{i}^{\left( q \right)} } \right]} }}{{E\left[ {\hat{\xi }_{i}^{\left( q \right)} } \right]}}{ = }\frac{{\sqrt {\frac{1}{{N_{C} }}\sum\limits_{q = 1}^{{N_{C} }} {\left( {\hat{\xi }_{i}^{\left( q \right)} } \right)^{2} } { - }\left[ {\frac{1}{{N_{C} }}\sum\limits_{q = 1}^{{N_{C} }} {\hat{\xi }_{i}^{\left( q \right)} } } \right]^{2} } }}{{\frac{1}{{N_{C} }}\sum\limits_{q = 1}^{{N_{C} }} {\hat{\xi }_{i}^{\left( q \right)} } }}, $$

where $\hat{\xi }_{i}^{\left( q \right)}$ is the estimated value of $\hat{\xi }_{i}$ calculated for the $q$-th time, and $q = 1,2, \ldots ,N_{C}$.

_{Appendix B: Determination for} $N$ _, $N_{1}$ _, $N_{2}$ _and $M$

Taking $N$ as example, to meet the requirement for the variation coefficients of $\hat{P}\left\{ F \right\}$, the number $N$ is determined by the following process:

• (1) Generate $N$ random samples of the inputs and construct the sample pool ${\mathbf{S}}^{f}$, $N{ = }10000$ is advised at the first time to execute step (1);

• (2) Estimate the failure probability;

• (3) Calculate the variation coefficient of $\hat{P}\left\{ F \right\}$, i.e., $COV\left( {\hat{P}\left\{ F \right\}} \right)$, if $COV\left( {\hat{P}\left\{ F \right\}} \right) < 5\%$, turn to step (5), else turn to step (4);

• (4) Set $N{ = }N + 5000$ and turn to step (1);

• (5) Record the results of $\hat{P}\left\{ F \right\}$, $COV\left( {\hat{P}\left\{ F \right\}} \right)$.

The processes for determining $N_{1}$, $N_{2}$ and $M$ are similar to that for $N$ and no longer illustrated here.