Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation

TL;DR: In this article, a set simulation approach is proposed to compute small failure probabilities encountered in reliability analysis of engineering systems, which can be expressed as a product of larger conditional failure probabilities by introducing intermediate failure events.

About: This article is published in Probabilistic Engineering Mechanics.The article was published on 2001-10-01 and is currently open access. It has received 1890 citations till now. The article focuses on the topics: Subset simulation & Law of total probability.

Summary

1. Introduction

• A new simulation approach, called subset simulation, is presented to compute failure probabilities.
• The failure probability is expressed as a product of conditional probabilities of some chosen intermediate failure events, the evaluation of which only requires simulation of more frequent events.
• The conditional probabilities, however, cannot be evaluated ef®ciently by common techniques, and therefore a Markov chain MCS method based on the Metropolis algorithm [16] is used.

2. Basic idea of subset simulation

• N} are independent and identically distributed (i.i.d.) samples simulated according to PDF q.
• In general, the task of ef®ciently simulating conditional samples is not trivial.

3.1. Modi®ed Metropolis algorithm

• Thus, Markov chain simulation accelerates the ef®ciency of exploring the failure region.
• The higher acceptance rate, however, is gained at the expense of introducing dependence between successive samples, which inevitably reduces the ef®ciency of the conditional failure probability estimators, as shown later.

4. Subset simulation procedure

• The proposal PDFs {p p j } affect the deviation of the candidate state from the current state, and control the ef®ciency of the Markov chain samples in populating the failure region.
• Simulations show that the ef®ciency of the method is insensitive to the type of the proposal PDFs, and hence those which can be operated easily may be used.
• The spread of the proposal PDFs affects the size of the region covered by the Markov chain samples, and consequently it controls the ef®ciency of the method.
• Small spread tends to increase the dependence between successive samples due to their proximity, slowing down convergence of the estimator and it may also cause ergodicity problems (see later).
• Excessively large spread, however, may reduce the acceptance rate, increasing the number of repeated Markov chain samples and so slowing down convergence.

6. Statistical properties of the estimators

• They are derived assuming that the Markov chain generated according to the modi®ed Metropolis method is ergodic, that is, its stationary distribution is unique and independent of the initial state of the chain.
• A discussion on ergodicity will follow after this section.
• It is assumed in this section that the intermediate failure events are chosen a priori.
• In the case where the intermediate threshold levels are chosen dependent on the conditional samples and hence vary in different simulation runs, as is the case in the examples presented in this paper, the derived results should hold approximately, provided such variation is not signi®cant.
• Nevertheless, this approximate analysis is justi®ed since the objective is to have an assessment of the quality of the probability estimate based on information available in one simulation run.

7. Ergodicity of subset simulation procedure

• The foregoing argument suggests that the subset simulation procedure is likely to produce an ergodic estimator for failure probability, nevertheless it offers no guarantee for practical ergodicity.
• Whether ergodicity problems become an issue depends on the particular application and the choice of the proposal PDFs.
• Importance sampling using design point(s) implicitly assumes that the main contribution of failure probability comes from the neighborhood of the known design points and there are no other design points of signi®cant contribution.
• Thus, in situations such as when the failure region is highly concave or there are other unknown design points, the importance-sampling estimator is biased and has an ergodicity problem.
• In view of this, one should appreciate the ergodic property of standard MCS, since it is a totally global procedure in the sense that it does not accumulate information about the failure region developed from local states only.

8. Examples

• The subset simulation methodology is applied to solving ®rst-excursion failure probabilities for two examples.
• The two examples are described ®rst, followed by a discussion of the simulation results.

8.3. Discussion of results

• Note that, from a theoretical point of view, Example 2 is much more complex than Example 1, since it involves the ®rst excursion failure of a vector-valued response process, the system is nonlinear hysteretic, and the excitation is nonwhite and nonstationary.
• Thus, it is expected that subset simulation can be applied ef®ciently to ®rst excursion problems for a wide range of dynamical systems.

9. Conclusions

• Numerical simulations show that the above phenomenon occurs in more general situations; for example, when the components u j are not all identically distributed.
• For the modi®ed Metropolis algorithm, note that the next state is equal to the current state either when the candidate state generated in Step 1 of the algorithm is equal to the current state or when the candidate state does not lie in F i and hence is rejected in Step 2.
• The event that the next state is equal to the current state then nearly corresponds to the event where a candidate state is rejected for not lying in F i .
• When the dimension n is large, P R can be expected not to increase systematically with n, and hence the modi®ed Metropolis algorithm is applicable even when the dimension is large.