SESC: A new subset simulation method for rare-events estimation
Introduction
During the past decades, the structural reliability analysis has been developed as a rational tool for the analysis and design of engineering systems in the presence of uncertainties [1]. It not only estimates the safety level of the structural systems, but can also be used for rare event probability estimations in other scientific fields [2]. This makes the application range of structural reliability theory broader than what it was originally meant for.
As the most accurate and robust reliability method, the crude Monte Carlo Simulation (MCS) generates random samples based on the statistical parameters of the random variables and estimates the failure probability (Pf) of systems based on the law of large numbers [3]. However, since its computational costs are heavy, it is not the most capable approach for time-consuming cases (e.g. finite element-based problems) and rare event probability estimations [4]. Hence, many variance reduction and approximation methods such as the first- and second-order reliability methods (FORM and SORM) [5], [6], [7], importance sampling (IS) [8], directional simulation (DS) [9], line sampling [10], dimension reduction-based [11], entropy-based [12], subset simulation (SubSim) [13], [14], asymptotic sampling [15] and so on have been proposed to solve such problems. Efficient Meta model-based approaches have also been used to solve reliability problems [16], [17], but since they are generally derived based on some simplifications and assumptions, their generality is limited compared to the MCS.
Among the mentioned approaches, SubSim is more capable (compared to other reliability methods) because it uses a suitable idea for rare event probability estimations. By introducing intermediate failure events, it expresses the small Pf as a product of larger conditional probabilities and solves problems with fewer function calls compared to the crude MCS [13]. SubSim has been suitably developed in the last two decades, especially for solving system-level reliability problems [18], [19]. However, its main weakness (discussed in the next section) has not been rectified.
Section snippets
Problem statement
On the one hand, the reliability analysis of a system with small failure probability and complex performance function is the main challenge in the reliability theory, and on the other, modern societies tend to design/construct engineering systems with high safety and small failure probability. Hence, it is essential to handle such problems with qualified and reliable methods.
SubSim is the mainstream method for estimating rare events and small failure probabilities [13]. It transmits samples
Control variates method
When the crude MCS is not used in the reliability analysis because of the assumptions made in other reliability methods, the estimated probability may be inaccurate and erroneous; hence, Pf can be reformulated as follows:where Ω={<=0} represents the failure domain, is the inexact probability obtained by a reliability approach, and ξ is the estimation error. To solve Eq. (2), CV may be used to estimate the probability error through small sample size simulations [21], [22],
SESC
The occurrence probability of a rare event may be increased by a times increasing the standard deviation of random variables (σ) (Fig. 4). Hence, a fast, but imprecise problem estimation, may be possible by . For a problem with D normal random variables, may be presented as . Accordingly, the CV refines the obtained probability as follows:
Once the imprecise probability is approximated, the importance failure region of the
Comparison of the SESC and SubSim
The main differences between the proposed SESC and conventional SubSim are in: A) basic idea behind both methods and B) performance.
A) Basic ideas behind the methods: While the SubSim formulation comes from the Bayes theorem that of the SESC is derived from the CV technique.
Using the Bayes theorem, SubSim presents a small failure probability as the product of the “conditional probabilities” as follows:where . In this method, the
Simulation examples
This section will examine the capabilities of the proposed approach by first investigating its accuracy/efficiency for solving high dimensional and complex problems with non-normal and small failure probability, then solving the SubSim counterexamples and, finally, examining its application for solving complex real-world problems. The default parameter of the partial failure probability in the SubSim is set at 0.2 with 1000 samples.
Conclusions
The original contribution of this study is presenting the SESC (sequential space conversion) as a new simulation method that may address the drawbacks of the conventional SubSim method; the latter is originated form the Bayes theorem, but the proposed approach is based on the CV (control variates) technique.
The mathematical framework of this method relies on the space conversion technique that turns a probability integral with two PDFs (probability density functions) and one integral domain
CRediT authorship contribution statement
Mohsen Rashki: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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