Estimation of small failure probabilities in high dimensions by subset simulation

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Abstract

A new simulation approach, called ‘subset simulation’, is proposed to compute small failure probabilities encountered in reliability analysis of engineering systems. The basic idea is to express the failure probability as a product of larger conditional failure probabilities by introducing intermediate failure events. With a proper choice of the conditional events, the conditional failure probabilities can be made sufficiently large so that they can be estimated by means of simulation with a small number of samples. The original problem of calculating a small failure probability, which is computationally demanding, is reduced to calculating a sequence of conditional probabilities, which can be readily and efficiently estimated by means of simulation. The conditional probabilities cannot be estimated efficiently by a standard Monte Carlo procedure, however, and so a Markov chain Monte Carlo simulation (MCS) technique based on the Metropolis algorithm is presented for their estimation. The proposed method is robust to the number of uncertain parameters and efficient in computing small probabilities. The efficiency of the method is demonstrated by calculating the first-excursion probabilities for a linear oscillator subjected to white noise excitation and for a five-story nonlinear hysteretic shear building under uncertain seismic excitation.

Introduction

The determination of the reliability of a system or component usually involves calculating its complement, the probability of failure:PF=P(θ∈F)=∫IF(θ)q(θ)dθwhere θ=[θ1,…,θn]∈Θ⊂Rn represents an uncertain state of the system with probability density function (PDF) q; F is the failure region within the parameter space Θ; IF is an indicator function: IF(θ)=1ifθ∈F and IF(θ)=0 otherwise. In practical applications, dependent random variables may often be generated by some transformation of independent random variables, and so it is assumed without much loss of generality that the components of θ are independent, that is, q(θ)=j=1nqj(θj), where for every j, qj is a one-dimensional PDF for θj.

Although PF is written as an n-fold integral over the parameter space Θ in Eq. (1), it cannot be efficiently evaluated by means of direct numerical integration if the dimension n is not small or the failure region has complicated geometry. Both of these conditions are likely to be encountered in real applications. Simulation methods then offer a feasible means to compute PF. Monte Carlo simulation (MCS) [1], [2] is well known to be robust to the type and dimension of the problem. Its main drawback, however, is that it is not suitable for finding small probabilities (e.g., PF≤10−3), because the number of samples, and hence the number of system analyses required to achieve a given accuracy, is proportional to 1/PF. Essentially, finding small probabilities requires information from rare samples corresponding to failure, and on average it would require many samples before a failure occurs.

Importance sampling techniques [1], [3], [4], [5] have been developed over the past few decades to shift the underlying distribution towards the failure region so as to gain information from rare events more efficiently. The success of the method relies on a prudent choice of the importance sampling density (ISD), which undoubtedly requires knowledge of the system in the failure region. When the dimension n of the uncertain parameter space is not too large and the failure region F is relatively simple to describe, many schemes for constructing the ISD, such as those based on design point(s) (e.g. [4], [6], [7], [8], [9], [10]) or adaptive pre-samples [11], [12], [13], [14], are found to be useful. The design point(s) or pre-samples are often obtained numerically by optimization or simulation where the integrand function IF(θ)q(θ) is directly used.

When the dimension n is large and the complexity of the problem increases, however, it may be difficult to gain sufficient knowledge to construct a good ISD [15]. Therefore, the development of efficient simulation methods for computing small failure probabilities in high-dimensional parameter spaces remains a challenging problem. Our objective has been to develop successful simulation methods that can adaptively generate samples, which explore the failure region efficiently, while at the same time retain the robustness of MCS to the dimension of the uncertain parameters and the complexity of the failure region.

A new simulation approach, called subset simulation, is presented to compute failure probabilities. It is robust to dimension size and efficient in computing small probabilities. In this approach, 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 problem of evaluating a small failure probability in the original probability space is thus replaced by a sequence of simulations of more frequent events in the conditional probability spaces. The conditional probabilities, however, cannot be evaluated efficiently by common techniques, and therefore a Markov chain MCS method based on the Metropolis algorithm [16] is used.

Section snippets

Basic idea of subset simulation

For convenience, we will use F to denote the failure event as well as its corresponding failure region in the uncertain parameter space. Given a failure event F, let F1F2⊃⋯⊃Fm=F be a decreasing sequence of failure events so that Fk=i=1kFi,k=1,…,m. For example, if failure of a system is defined as the exceedence of an uncertain demand D over a given capacity C, that is, F={D>C}, then a sequence of decreasing failure events can simply be defined as Fi={D>Ci}, where C1<C2<…<Cm=C. By definition

Markov chain MCS

Markov chain MCS, in particular, the Metropolis method [2], [16], is a powerful technique for simulating samples according to an arbitrary probability distribution. In this method, samples are simulated as the states of a Markov chain which, under the assumption of ergodicity, has the target distribution as its limiting stationary distribution. It has been recently applied to adaptive importance sampling for reliability analysis to construct an asymptotically optimal importance sampling density

Subset simulation procedure

Utilizing the modified Metropolis method, subset simulation proceeds as follows. First, we simulate samples by direct MCS to compute P̃1 from Eq. (3) as an estimate for P(F1). From these MCS samples, we can readily obtain some samples distributed as q(·|F1), simply as those which lie in F1. Starting from each of these samples, we can simulate Markov chain samples using the modified Metropolis method. These samples will also be distributed as q(·|F1). They can be used to estimate P(F2|F1) using

Choice of proposal PDFs {pj}

The proposal PDFs {pj} affect the deviation of the candidate state from the current state, and control the efficiency of the Markov chain samples in populating the failure region. Simulations show that the efficiency of the method is insensitive to the type of the proposal PDFs, and hence those which can be operated easily may be used. For example, the uniform PDF centered at the current sample with width 2lj is a good candidate for pj, and this is used in the examples in this work. The spread

Statistical properties of the estimators

In this section, we present results on the statistical properties of the estimators P̃i and P̃F. They are derived assuming that the Markov chain generated according to the modified Metropolis method is (theoretically) 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

Ergodicity of subset simulation procedure

The foregoing discussion assumes that the Markov chain generated according to the Metropolis method is ergodic, which guarantees that the conditional probability estimate based on the Markov chain samples from a single chain will tend to the corresponding theoretical conditional probability as N→∞. Theoretically, ergodicity can be always achieved by choosing a sufficiently large spread in the proposal PDFs {pj}. Practically, with a finite number of Markov chain samples, ergodicity often

Examples

The subset simulation methodology is applied to solving first-excursion failure probabilities for two examples. In these examples, the input W(t) is a Gaussian white noise process with spectral intensity S. The response of the system is computed at the discrete time instants {tk=(k−1)Δt:k=1,…n}, where the sampling interval is assumed to be Δt=0.02s and the duration of study is T=30s, so that the number of time instants is n=Tt+1=1501. The uncertain state vector θ then consists of the sequence

Conclusions

One of the major obstacles in applying simulation methods to estimating small failure probabilities is the need to simulate rare events. Subset simulation resolves this by breaking the problem into the estimation of a sequence of larger conditional probabilities. The Metropolis method has been modified to efficiently estimate the conditional probabilities by Markov chain simulation. Theoretical estimates for the coefficient of variation and results from numerical simulation demonstrate a

Acknowledgements

This paper is based upon work partly supported by the Pacific Earthquake Engineering Research Center under National Science Foundation Cooperative Agreement No. CMS-9701568. This support is gratefully acknowledged.

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