Sequential exploration-exploitation with dynamic trade-off for efficient reliability analysis of complex engineered systems

Abstract

A new sequential sampling method, named sequential exploration-exploitation with dynamic trade-off (SEEDT), is proposed for reliability analysis of complex engineered systems involving high dimensionality and a wide range of reliability levels. The proposed SEEDT method is built based on the ideas of two previously developed sequential Kriging reliability methods, namely efficient global reliability analysis (EGRA) and maximum confidence enhancement (MCE) methods. It employs Kriging-based sequential sampling to build a surrogate model (i.e., Kriging model) that approximates the performance function of an engineered system, and performs Monte Carlo simulation on the surrogate model for reliability analysis. A new acquisition function, referred to as expected utility (EU), is developed to sequentially locate a computationally efficient set of sample points for constructing the Kriging model. The SEEDT method possesses three technical contributions: (i) defining a new utility function with several desirable properties that facilitates the joint consideration of exploration and exploitation over the course of sequential sampling; (ii) introducing a new exploration-exploitation trade-off coefficient that dynamically weighs exploration and exploitation to achieve a fine balance between these two activities; and (iii) developing a new convergence criterion based on the uncertainty in the prediction of the limit-state function (LSF). The effectiveness of the proposed method in reliability analysis is evaluated with several mathematical and practical examples. Results from these examples suggest that, given a certain number of sample points, the SEEDT method is capable of achieving better accuracy in predicting the LSF than the existing sequential sampling methods.

Appendix A Derivation of analytical formula of expected utility function

This appendix details the derivation of the analytical formula of the expected utility in (18). Replacing the utility function in (13) with the proposed one in (16) yields the following expected utility function:

$$ \mathrm{EU}\left(\mathbf{x}\right)={\int}_{-\infty}^{+\infty } u\left(\widehat{G}\right){f}_{\widehat{G}}\left(\widehat{G}\right) d\widehat{G}={\int}_{-\infty}^{+\infty}\omega \cdot {e}^{-\frac{{\widehat{G}}^2}{2\sigma {\hbox{'}}^2}}\cdot \phi \left(\widehat{G};{\mu}_{\widehat{G}},{\sigma}_{\widehat{G}}\right)\; d\widehat{G} $$

(28)

By rewriting u(G) as a normal PDF function and a constant, (27) can be rewritten as:

$$ \mathrm{EU}\left(\mathbf{x}\right)={\int}_{-\infty}^{+\infty}\omega \cdot \sqrt{2\pi}{\sigma}^{\hbox{'}}\phi \left(\widehat{G};0,{\sigma}^{\hbox{'}}\right)\cdot \phi \left(\widehat{G};{\mu}_{\widehat{G}},{\sigma}_{\widehat{G}}\right) d\widehat{G} $$

(29)

It is known that the product of two normal PDFs is proportional to a normal PDF. By using this rule, (27) can be further simplified as:

$$ \mathrm{EU}\left(\mathbf{x}\right)={\int}_{-\infty}^{+\infty}\omega \cdot \sqrt{2\pi}{\sigma}^{\hbox{'}}\phi \left({\mu}_{\hat{G}};0,\sqrt{\sigma {\hbox{'}}^2+{\sigma}_{\hat{G}}^2}\right)\cdot \phi \left(\widehat{G};{\mu}_0,{\sigma}_0\right) d\widehat{G} $$

(30)

where μ ₀ and σ ₀ are two constant parameters. Since $\phi \left({\mu}_{\widehat{G}};0,\sqrt{\sigma {\prime}^2+{\sigma}_{\widehat{G}}^2}\right)$ is not a function of $\widehat{G}$, this term, along with ω and $\sqrt{2\pi}{\sigma}^{\prime }$, can be extracted from the integral, which then yields the following:

$$ \mathrm{EU}=\omega \cdot \sqrt{2\pi}{\sigma}^{\hbox{'}}\phi \left({\mu}_{\hat{G}};0,\sqrt{\sigma {\hbox{'}}^2+{\sigma}_{\hat{G}}^2}\right){\int}_{-\infty}^{+\infty}\phi \left(\widehat{G};{\mu}_0,{\sigma}_0\right) d\widehat{G} $$

(31)

The integration of any standard normal PDF in the range of [−∞, +∞] equals to one. Therefore, the integral in (30) simply equals one. By simplifying $\phi \left({\mu}_{\widehat{G}},0,\sqrt{\sigma {\prime}^2+{\sigma}_{\widehat{G}}^2}\right)$, (30) can be rewritten as:

$$ \mathrm{EU}=\omega \cdot \sqrt{2\pi}{\sigma}^{\hbox{'}}\frac{1}{\sqrt{2\pi \left(\sigma {\hbox{'}}^2+{\sigma}_{\hat{G}}^2\right)}}{e}^{-\frac{\mu_{\hat{G}}^2}{2\sqrt{\sigma {\hbox{'}}^2+{\sigma}_{\hat{G}}^2}}} $$

(32)

Then, the expected utility function can be expressed as:

$$ \mathrm{EU}=\omega \cdot \frac{\alpha_t}{1+{\alpha_t}^2}{e}^{-\frac{{\mu_{\hat{G}}}^2}{2{\sigma}_{\hat{G}}\sqrt{1+{\alpha_t}^2}}} $$

(33)

This simplified form of the expected utility function is also shown in (18) and is used as the acquisition function in this study.