Probabilistic Engineering Mechanics 16 +2001) 263±277 www.elsevier.com/locate/probengmech

Estimation of small failure probabilities in high dimensions by subset simulation

Siu-Kui Au, James L. Beck*

Division of Engineering and Applied Science, California Institute of Technology, Mail Code 104-44, Pasadena, CA 91125, USA

Abstract A newsimulation 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 suf®ciently 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 ef®ciently estimated by means of simulation. The conditional probabilities cannot be estimated ef®ciently 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 ef®cient in computing small probabilities. The ef®ciency of the method is demonstrated by calculating the ®rst-excursion probabilities for a linear oscillator subjected to white noise excitation and for a ®ve-story nonlinear hysteretic shear building under uncertain seismic excitation. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Markov chain Monte Carlo method; Monte Carlo simulation; Reliability; First excursion probability; First passage problem; Metropolis algorithm

1. Introduction Both of these conditions are likely to be encountered in real applications. Simulation methods then offer a feasible

The determination of the reliability of a system or compo- means to compute PF. Monte Carlo simulation +MCS) nent usually involves calculating its complement, the prob- [1,2] is well known to be robust to the type and dimension ability of failure: of the problem. Its main drawback, however, is that it is not 23 Z suitable for ®nding small probabilities +e.g., PF # 10 †; PF ˆ P u [ F†ˆ IF u†q u†du 1† because the number of samples, and hence the number of system analyses required to achieve a given accuracy, is

n proportional to 1/PF. Essentially, ®nding small probabilities where u ˆ‰u1; ¼; unŠ [ Q , R represents an uncertain state of the system with probability density function requires information from rare samples corresponding to +PDF) q; F is the failure region within the parameter failure, and on average it would require many samples space Q; I is an indicator function: I u†ˆ1ifu [ F before a failure occurs. F F Importance sampling techniques [1,3±5] have been and IF u†ˆ0 otherwise. In practical applications, depen- dent random variables may often be generated by some developed over the past fewdecades to shift the underlying transformation of independent random variables, and so it distribution towards the failure region so as to gain informa- is assumed without much loss of generality that the compo- tion from rare events more ef®ciently. The success of the Qn method relies on a prudent choice of the importance nents of u are independent, that is, q u†ˆ jˆ1 qj uj†; where for every j, q is a one-dimensional PDF for u . sampling density +ISD), which undoubtedly requires knowl- j j edge of the system in the failure region. When the dimen- Although PF is written as an n-fold integral over the para- meter space Q in Eq. +1), it cannot be ef®ciently evaluated sion n of the uncertain parameter space is not too large and by means of direct numerical integration if the dimension n the failure region F is relatively simple to describe, many is not small or the failure region has complicated geometry. schemes for constructing the ISD, such as those based on design point+s) +e.g. [4,6,7±10]) or adaptive pre-samples [11±14], are found to be useful. The design point+s) or * Corresponding author. Tel.: 11-626-395-4132; fax: 11-626-568-2719. E-mail address: [email protected] +J.L. Beck). pre-samples are often obtained numerically by optimization

0266-8920/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-8920+01)00019-4 264 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 or simulation where the integrand function IF+u)q+u)is sequence of conditional probabilities {P Fi11uFi† : i ˆ directly used. 1; ¼; m 2 1} and P+F1). The idea of subset simulation is When the dimension n is large and the complexity of the to estimate the failure probability PF by estimating these problem increases, however, it may be dif®cult to gain suf®- quantities. When the probability of failure is estimated by cient knowledge to construct a good ISD [15]. Therefore, means of simulation, the dif®culty often increases with the development of ef®cient simulation methods for decreasing failure probability. Basically, the smaller the computing small failure probabilities in high-dimensional PF, the more rare the failure event is, and the more the parameter spaces remains a challenging problem. Our number of samples required to realize failure events for objective has been to develop successful simulation meth- computing PF. Observe that, although PF is small, by choos- ods that can adaptively generate samples, which explore the ing the intermediate failure events {Fi : i ˆ 1; ¼; m 2 1} failure region ef®ciently, while at the same time retain the appropriately, the conditional probabilities involved in Eq. robustness of MCS to the dimension of the uncertain para- +2) can be made suf®ciently large so that they can be eval- meters and the complexity of the failure region. uated ef®ciently by simulation procedures. For example, A newsimulation approach, called subset simulation, is suppose P F1†; P Fi11uFi† , 0:1; i ˆ 1; 2; 3; then PF , presented to compute failure probabilities. It is robust to 1024 which is too small for ef®cient estimation by MCS. dimension size and ef®cient in computing small probabil- However, the conditional probabilities, which are of the ities. In this approach, the failure probability is expressed as order of 0.1, may be evaluated ef®ciently by simulation a product of conditional probabilities of some chosen inter- because the failure events are more frequent. mediate failure events, the evaluation of which only requires The problem of simulating rare events in the original simulation of more frequent events. The problem of evalu- probability space is thus replaced by a sequence of simula- ating a small failure probability in the original probability tions of more frequent events in the conditional probability space is thus replaced by a sequence of simulations of more spaces. frequent events in the conditional probability spaces. The To compute PF based on Eq. +2), one needs to compute conditional probabilities, however, cannot be evaluated ef®- the probabilities P F1†; {P Fi11uFi† : i ˆ 1; ¼; m 2 1}: ciently by common techniques, and therefore a Markov P+F1) can be readily estimated by MCS: chain MCS method based on the Metropolis algorithm 1 XN [16] is used. P F † < P~ ˆ I u † 3† 1 1 N F1 k kˆ1

2. Basic idea of subset simulation where {uk : k ˆ 1; ¼; N} are independent and identically distributed +i.i.d.) samples simulated according to PDF q.It For convenience, we will use F to denote the failure event is natural to compute the conditional failure probabilities as well as its corresponding failure region in the uncertain based on an estimator similar to Eq. +3), which necessitates parameter space. Given a failure event F, let F1 . F2 . the simulation of samples according to the conditional ¼ distribution of u given that it lies in F , that is, q uuF †ˆ . Fm ˆ F Tbe a decreasing sequence of failure events i i so that F ˆ k F ; k ˆ 1; ¼; m: For example, if failure q u†I u†=P F †: Although one can followa `direct' MCS k iˆ1 i Fi i of a system is de®ned as the exceedence of an uncertain approach and obtain such samples as those simulated from q demand D over a given capacity C, that is, F ˆ {D . C}; which lie in the failure region Fi, it is not ef®cient to do so then a sequence of decreasing failure events can simply be since on average it takes 1/P+Fi) samples before one such de®ned as Fi ˆ {D . Ci}; where C1 , C2 , ¼ , Cm ˆ sample occurs. In general, the task of ef®ciently simulating C: By de®nition of conditional probability, we have conditional samples is not trivial. Fortunately, this can be ! achieved by a Markov chain MCS method based on the \m Metropolis algorithm that is presented in the Section 3. PF ˆ P Fm†ˆP Fi 2† iˆ1 ! ! 3. Markov chain MCS m\2 1 m\2 1 ˆ PFu F P F m i i Markov chain MCS, in particular, the Metropolis method iˆ1 iˆ1 [2,16], is a powerful technique for simulating samples ! ÀÁm\2 1 according to an arbitrary probability distribution. In this ˆ PFmuFm21 P Fi ˆ ¼ method, samples are simulated as the states of a Markov iˆ1 chain which, under the assumption of ergodicity, has the target distribution as its limiting stationary distribution. It mY2 1 has been recently applied to adaptive importance sampling ˆ P Fi† P Fi11uFi† for reliability analysis to construct an asymptotically opti- iˆ1 mal importance sampling density [14]. The signi®cance of Eq. +2) expresses the failure probability as a product of a the Metropolis algorithm to the current application is that if S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 265 we can simulate a sample having the conditional distri- expressed as a product of the component transition PDFs: bution q ´uF †; then we can use the method to simulate i Yn as the next state of the Markov chain a newsample p uk11uuk†ˆ pj uk11 j†uuk j†† 4† which will also be distributed as q ´uFi†: +Even if the jˆ1 current sample is not distributed as q ´uFi†; the limiting distribution property guarantees that the distribution of where pj is the transition PDF for the jth component of uk. simulated samples will tend to q ´uF † as the number of For uk11 j† ± uk j†; i ) Markov steps increases.) These Markov chain samples, qj uk11 j†† which are dependent in general, can be used for statis- p u j†uu j†† ˆ pp u j†uu j††min 1; 5† j k11 k j k11 k q u j†† tical averaging as if they were i.i.d., although with some j k reduction in ef®ciency. p Using Eq. +5), together with the symmetry property of pj We ®rst describe the method to simulate from the current and the identity min{1; a=b}b ˆ min{1; b=a}a for any posi- sample the next Markov chain sample so that the stationary tive numbers a,b, it is straightforward to show that pj satis- distribution of the Markov chain is equal to q uuFi†ˆ ®es the `reversibility' condition with respect to q : Qn j q u†IF u†=P Fi†ˆ‰ jˆ1 qj uj†ŠIF u†=P Fi†; where u ˆ i i p u j†uu j††q u j†† ˆ p u j†uu j††q u j†† 6† ‰u1; ¼; unŠ: The algorithm presented here is a new j k11 k j k j k k11 j k11 modi®ed version of the original Metropolis algorithm Note that the equality in Eq. +6) is trivial when u j†ˆ since it has been found that the latter has dif®culties k11 u j†: Combining Eqs. +4) and +6) and the fact that all the when simulating random vectors with many independent k states lie in F , the transition PDF for the whole state u also components. The reader is referred to the Appendix for i k satis®es the following reversibility condition with respect to the description of the original Metropolis algorithm q ´uF † : and the problems encountered in high dimensional i simulation. p uk11uuk†q ukuFi†ˆp ukuuk11†q uk11uFi† 7†

Thus, if the current sample uk is distributed as q ´uFi†; 3.1. Modi®ed Metropolis algorithm then Z p For every j ˆ 1; ¼; n; let pj juu†, called the `proposal p uk11†ˆ p uk11uuk†q ukuFi†duk 8† PDF', be a one-dimensional PDF for j centered at u with p p the symmetry property pj juu†ˆpj uuj†: Generate a Z sequence of samples {u ; u ; ¼} from a given sample u 1 2 1 ˆ p ukuuk11†q uk11uFi†duku; by 7† by computing uk11 from uk ˆ‰uk 1†; ¼; uk n†Š; k ˆ 1; 2; ¼; as follows: Z ˆ q uk11uFi† p ukuuk11†duk 1. Generate a `candidate' state u~ : For each component j ˆ p 1; ¼; n; simulate jj from pj ´uuk j††: Compute the ratio ~ ˆ q uk11uFi† rj ˆ qj jj†=qj uk j††: Set u j†ˆjj with probability min{1,r } and set u~ j†ˆu j† with the remaining prob- R j k since p ukuuk11†duk ˆ 1: This shows that the next Markov ability 1 2 min{1; rj}: chain sample u will also be distributed as q ´uF †; and so ~ ~ ~ k11 i 2. Accept/reject u: Check the location of u: If u [ Fi; the latter is indeed the stationary distribution for the gener- ~ accept it as the next sample, i.e. uk11 ˆ u; otherwise ated Markov chain. reject it and take the current sample as the next sample, Step 1 can be viewed as a `local' random walk in the i.e. u ˆ u : k11 k neighborhood of the current state uk, while Step 2 ensures that the next sample always lie in Fi so as to produce the In brief, we ®rst generate a candidate state u~ +dependent correct conditioning in the samples. Thus, Step 2 is in prin- ~ on the current state) and then take either u or uk as the next ciple similar to the direct MCS approach in that both are sample uk11 according to whether the candidate state lies in based on accepting samples that lie in Fi. However, the Fi or not. `acceptance' rate for the modi®ed Metropolis algorithm We nowshowthat the next sample willbe distributed as should be considerably higher than for direct MCS because ~ q ´uFi† if the current sample is, and hence q ´uFi† is the the candidate state u is simulated in the neighborhood of the p stationary distribution of the Markov chain. Since all the current state uk [ Fi if the proposal PDFs pj are suf®ciently Markov chain samples lie in Fi, it is suf®cient to consider local and so uk should have a high probability of lying in Fi. the transition between the states in Fi, which is governed by Thus, Markov chain simulation accelerates the ef®ciency of Step 1. According to Step 1, the transition of the individual exploring the failure region. The higher acceptance rate, components of uk are independent, so the transition PDF of however, is gained at the expense of introducing depen- the Markov chain between any two states in Fi can be dence between successive samples, which inevitably 266 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277

p reduces the ef®ciency of the conditional failure probability down convergence. The optimal choice of the spread of pj is estimators, as shown later. therefore a trade-off between acceptance rate and correla- p tion due to proximity. Roughly speaking, the spread of pj may be chosen as some fraction of the standard deviation of 4. Subset simulation procedure the corresponding component uj as speci®ed by the PDF q+u), although the optimal choice depends on the particular Utilizing the modi®ed Metropolis method, subset simula- type of problem. tion proceeds as follows. First, we simulate samples by ~ direct MCS to compute P1 from Eq. +3) as an estimate for P+F1). From these MCS samples, we can readily obtain 5.2. Choice of intermediate failure events some samples distributed as q ´uF1†; simply as those which lie in F1. Starting from each of these samples, we can simu- The choice of the intermediate failure events {Fi} plays a late Markov chain samples using the modi®ed Metropolis key role in the subset simulation procedure. Two issues are method. These samples will also be distributed as q ´uF1†: basic to the choice of the intermediate failure events. The ~ They can be used to estimate P F2uF1† using an estimator P2 ®rst is the parameterization of the target failure event F similar to Eq. +3). Observe that the Markov chain samples which allows the generation of intermediate failure events which lie in F2 are distributed as q ´uF2† and thus they by varying the value of the de®ned parameter. The second is provide `seeds' for simulating more samples according to the choice of the speci®c sequence of values of the de®ned q ´uF2† to estimate P F3uF2†: Repeating this process, we can parameter, which affects the values of the conditional prob- compute the conditional probabilities of the higher condi- abilities {P Fi11uFi†} and hence the ef®ciency of the subset tional levels until the failure event of interest, F + ˆ Fm), has simulation procedure. been reached. At the ith conditional level, 1 # i # m 2 1; Many failure events encountered in engineering applica- i† tions can be de®ned using a combination of union and inter- let {uk : k ˆ 1; ¼; N} be the Markov chain samples with distribution q ´uFi†; possibly coming from different chains section of some component failure events. In particular, generated by different seeds. Then consider a failure event F of the following form:

XN L ~ 1 i† [L \j P Fi11uFi† < Pi11 ˆ IF u † 9† N i11 k F ˆ {Djk u† . Cjk u†} 11† kˆ1 jˆ1 kˆ1 Finally, combining Eqs. +2), +3) and +9), the failure prob- ability estimator is where Djk+u) and Cjk+u) may be viewed as the demand and capacity variables of the +j,k) component of the system. The Ym ~ ~ failure event F in Eq. +11) can be considered as the failure of PF ˆ Pi 10† iˆ1 a system with L sub-systems connected in series, where the jth sub-system consists of Lj components connected in paral- lel. 5. Implementation issues In order to apply subset simulation to compute the failure probability PF, it is desirable to parameterize F with a single p parameter so that the sequence of intermediate failure 5.1. Choice of proposal PDFs {pj } events {Fi : i ˆ 1; ¼; m 2 1} can be generated by varying p the parameter. This can be accomplished as follows. For the The proposal PDFs {pj } affect the deviation of the candi- date state from the current state, and control the ef®ciency of failure event in Eq. +11), de®ne the `critical demand-to- the Markov chain samples in populating the failure region. capacity ratio' +CDCR) Y as: Simulations showthat the ef®ciency of the method is insen- D u† sitive to the type of the proposal PDFs, and hence those Y u†ˆ max min jk 12† which can be operated easily may be used. For example, jˆ1;¼;L kˆ1;¼;Lj Cjk u† the uniform PDF centered at the current sample with width p Then it can be easily veri®ed that 2lj is a good candidate for pj ; and this is used in the exam- ples in this work. The spread of the proposal PDFs affects F ˆ {Y u† . 1} 13† the size of the region covered by the Markov chain samples, and consequently it controls the ef®ciency of the method. and so the sequence of intermediate failure events can be Small spread tends to increase the dependence between generated as: successive samples due to their proximity, slowing down convergence of the estimator and it may also cause Fi ˆ {Y u† . yi} 14† ergodicity problems +see later). Excessively large spread, ¼ however, may reduce the acceptance rate, increasing the where 0 , y1 , , ym ˆ 1 is a sequence of +normalized) number of repeated Markov chain samples and so slowing intermediate threshold values. S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 267

Similarly, consider a failure F of the form: abilities are equal to a ®xed value p0 [ 0; 1†: This is accomplished by choosing the intermediate threshold level \L [Lj yi i ˆ 1; ¼; m 2 1† as the 1 2 p0†Nth largest value +i.e. an F ˆ {Djk u† . Cjk u†} 15† i21† jˆ1 kˆ1 order statistic) among the CDCRs {Y uk † : k ˆ 1; ¼; N} i21† where the uk are the Markov chain samples generated at which can be considered as the failure of a system with L the +i 2 1)th conditional level for i ˆ 2; ¼; m 2 1; and the sub-systems connected in parallel with the jth sub-system 0† uk are the samples from the initial MCS. Here, p0 is consisting of L components connected in series. One can j assumed to be chosen so that p0N and hence 1 2 p0†N are easily verify that the de®nition positive integers, although this is not strictly necessary. This choice of the intermediate threshold levels implies that they Djk u† Y u†ˆ min max 16† are dependent on the conditional samples and will vary in jˆ1;¼;L kˆ1;¼;L C u† j jk different simulation runs. In the examples presented in this satis®es Eq. +13) and hence the sequence of failure events paper, the value p0 is chosen to be 0.1, which is found to can again be generated based on Eq. +14). yield good ef®ciency. The foregoing discussion can be generalized to failure events consisting of multiple stacks of union and intersec- 6. Statistical properties of the estimators tion. Essentially, Y is de®ned using `max' and `min' in the same order corresponding to each occurrence of union + < ) In this section, we present results on the statistical proper- and intersection + > )inF, respectively. As another exam- ~ ~ ties of the estimators Pi and PF: They are derived assuming ple, consider the ®rst excursion failure of the interstory drift that the Markov chain generated according to the modi®ed in any one story of a ns-story building beyond a given Metropolis method is +theoretically) ergodic, that is, its threshold level b. Let the interstory drift response {Xj t; u† : stationary distribution is unique and independent of the j ˆ 1; ¼; n } be computed at the n time instants t ; ¼; t s t 1 nt initial state of the chain. A discussion on ergodicity will within the duration of interest. Then: followafter this section. It is assumed in this section that

[ns [nt the intermediate failure events are chosen a priori. In the F ˆ {uXj tk; u†u . b} ˆ {Y u† . 1} 17† case where the intermediate threshold levels are chosen jˆ1 kˆ1 dependent on the conditional samples and hence vary in where different simulation runs, as is the case in the examples presented in this paper, the derived results should hold uX t †u approximately, provided such variation is not signi®cant. Y u†ˆ min max j k 18† jˆ1;¼;ns kˆ1;¼;nt b Nevertheless, this approximate analysis is justi®ed since the objective is to have an assessment of the quality of the The choice of the sequence of intermediate threshold probability estimate based on information available in one values {y ; ¼; y } appearing in the parameterization of 1 m simulation run. intermediate failure events affects the values of the condi- tional probabilities and hence the ef®ciency of the subset ~ 6.1. MCS estimator P1 simulation procedure. If the sequence increases slowly, then ~ the conditional probabilities will be large, and so their esti- As well-known, the MCS estimator P1 in Eq. +3) mation requires less samples N. A slowsequence, however, computed using the i.i.d. samples {u1; ¼; uN } converges requires more simulation levels m to reach the target failure almost surely to P1 +Strong Lawof Large Numbers), is event, increasing the total number of samples NT ˆ mN in unbiased, consistent, and Normally distributed as N ! 1 the whole procedure. Conversely, if the sequence increases +Central Limit Theorem). The coef®cient of variation ~ too rapidly that the conditional failure events become rare, it +c.o.v.) of P1; d 1, de®ned as the ratio of the standard devia- ~ will require more samples N to obtain an accurate estimate tion to the mean of P1; is given by: of the conditional failure probabilities in each simulation s level, which again increases the total number of samples. 1 2 P1 d1 ˆ 19† It can thus be seen that the choice of the intermediate thresh- P1N old values is a trade-off between the number of samples required in each simulation level and the number of simula- ~ 6.2. Conditional probability estimator Pi 2 # i # m† tion levels required to reach the target failure event. The choice of the intermediate threshold values {yi : i ˆ Since the Markov chains generated at each conditional 1; ¼; m 2 1} deserves a detailed study which is left for level are started with samples +selected from the previous future work. One strategy is to choose the yi a priori, but simulation level) distributed as the corresponding target then it is dif®cult to control values of the conditional prob- conditional PDF, the Markov chain samples used for abilities P FiuFi21†: Therefore, in this work, the yi are computing the conditional probability estimators based on chosen `adaptively' so that the estimated conditional prob- Eq. +9) are identically distributed as the target conditional 268 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 PDF. Using this result and taking expectation on both side Substituting Eq. +23) into +20) yields: ~ of Eq. +9) shows that the conditional estimators Pi i ˆ "# 1 N=NXc 2 1 kN 2; ¼; m† are unbiased. On the other hand, the Markov E‰P~ 2 P Š2 ˆ R 0† 1 2 1 2 c R k† 24† i i N i N i chain samples are dependent. In spite of this dependence, kˆ1 all the estimators P~ ; still have the usual convergence prop- i "# erties of estimators using independent samples [17]. For R 0† N=NXc 2 1 kN ~ ˆ i 1 1 2 1 2 c r k† example, Pi converges almost surely to P FiuFi21† +Strong N N i Lawof Large Numbers), is consistent, and Normally distrib- kˆ1 uted as N ! 1 +Central Limit Theorem). where An expression for the c.o.v. for P~ is next derived. At the i r k†ˆR k†=R 0† 25† i 2 1†th conditional level, suppose that the number of i i i Markov chains +each with a possibly different starting is the correlation coef®cient at lag k of the stationary i† i† point) is Nc, and N=Nc samples have been simulated from sequence {Ijk : k ˆ 1; ¼; N=Nc}: Finally, since Ijk is a i† each of these chains, so that the total number of Markov Bernoulli random variable, Ri 0†ˆVar‰Ijk ŠˆPi 1 2 Pi†; ~ chain samples is N. Although the samples generated by and so the variance of Pi is given by different chains are in general dependent because the Pi 1 2 Pi† seeds for each chain may be dependent, it is assumed s 2 ˆ E‰P~ 2 P Š2 ˆ ‰1 1 g Š 26† i i i N i for simplicity in analysis that they are uncorrelated 0 where through the indicator function IF ´†; i.e. E‰IF u†IF u †Š 2 2 0 i i i i† P F uF † ˆ 0ifu and u are from different chains. Let u N=N 2 1  i i21 jk Xc kN be the kth sample in the jth Markov chain at simulation level g ˆ 2 1 2 c r k† 27† i N i i. For simplicity in notation, let I i† ˆ I u i21†† and P ˆ kˆ1 jk Fi jk i P F uF †; i ˆ 2; ¼; m: Then: ~ i i21 The c.o.v. d i of Pi is thus given by: 2 32 s N N=N 1 Xc Xc 1 2 Pi ~ 2 4 i† 5 d ˆ 1 1 g † 28† E‰Pi 2 PiŠ ˆ E Ijk 2 Pi† i i N PiN jˆ1 kˆ1 The covariance sequence {R k† : i ˆ 0; ¼; N=N 2 1} "#2 i c 1 XNc NX=Nc can be estimated using the Markov chain samples {u i21† : ˆ E I i† 2 P † 20† jk 2 jk i j ˆ 1; ¼; N ; k ˆ 1; ¼; N=N } at the +i 2 1)th conditional N jˆ1 kˆ1 c c level by: For the jth chain, 0 1 1 XNc N=NXc 2 k "# ~ @ i† i† A ~2 2 Ri k† < Ri k†ˆ Ijl Ij;l1k 2 Pi 29† NX=Nc NX=Nc N 2 kN i† i† i† c jˆ1 lˆ1 E Ijk 2 Pi† ˆ E Ijk 2 Pi† Ijl 2 Pi† kˆ1 k;lˆ1 from which the correlation sequence {ri k† : k ˆ 1; ¼; N=N 2 1} and hence the correlation factor g in Eq. NX=Nc c i ˆ R k 2 1† 21† +27) can also be estimated. Consequently, the c.o.v. d i for i ~ k;lˆ1 the conditional probability estimator Pi can be estimated by Eq. +28), where P is approximated by P~ using Eq. +9). where i i The factor 1 2 Pi†=PiN in Eq. +28) is the familiar one for the square of the c.o.v. of the MCS estimators with N inde- R k†ˆE I i† 2 P † I i† 2 P †ˆE‰I i†I i† Š 2 P2 22† i j1 i j;11k i j1 j;11k i ~ pendent samples. The c.o.v. of Pi can thus be considered as is the covariance between I i† and I i† ; for any l ˆ the one in MCS with an effective number of independent jl j;l1k samples N= 1 1 g †: The ef®ciency of the estimator using 1; ¼; N=Nc; and it is independent of l due to stationarity. It i is also independent of the chain index j since all chains are dependent samples of a Markov chain gi . 0† is therefore probabilistically equivalent. Evaluating the sum in Eq. +21) reduced compared to the case when the samples are inde- with respect to k 2 l; pendent gi ˆ 0†; and smaller values of g i imply higher ef®ciency. The value of g i depends on the choice of the "#2  NX=Nc N=NXc 2 1 spread of the proposal PDFs +Section 5.1). i† N N E Ijk 2 Pi† ˆ Ri 0† 1 2 2 k Ri k† Nc Nc ~ kˆ1 kˆ1 6.3. Failure probability estimator PF 23† ~ ~ Since P1 ! P F1† and Pi ! P FiuQFi21† 2 # i # m† ~ m 2 1 "# almost surely as N ! 1; PF ! P F1† iˆ1 P Fi11uFi†ˆ N N=NXc 2 1 kN ˆ R 0† 1 2 1 2 c R k† PF almost surely also. Due to the correlation between the N i N i ~ ~ c kˆ1 conditional estimators {Pi}; PF is biased for every N, but it S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 269 is asymptotically unbiased. This correlation is due to the Proof. From Eq. +31) ~ fact that the samples used for computing Pi which lie in "#2 32 ~ P~ 2 P 2 Xm X Ym Fi are used to start the Markov chains to compute Pi11: F F 4 ¼ 5 E ˆ E diZi 1 didjZiZj 1 1 diZi The results concerning the mean and c.o.v. of the failure PF ~ iˆ1 i.j iˆ1 probability estimator PF are summarized in the following two propositions. 34† Xm Proposition 1. P~ is biased for every N. The fractional bias F ˆ didjE‰ZiZjŠ 1 o 1=N† is bounded above by: i;jˆ1 "# ~ X PF 2 PF Xm uE u # didj 1 o 1=N†ˆO 1=N† 30† P # didj 1 o 1=N†ˆO 1=N† F i.j i;jˆ1 ~ p PF is thus asymptotically unbiased and the bias is O+1/N). since E‰ZiZjŠ # 1 and di ˆ O 1= N†: Note that the c.o.v. d in Eq. +33) is de®ned through the ~ Proof. De®ne Zi ˆ Pi 2 Pi†=si; then it is clear that expected deviation about the target failure probability P 2 ~ F E‰ZiŠˆ0 and E‰Zi Šˆ1; and Pi ˆ Pi 1 siZi: ~ instead of E‰PFŠ so that the effects of bias are accounted P~ 2 P Ym for. The upper bound corresponds to the case when the F F ˆ P~ =P 2 1 31† conditional probability estimators {P~ } are fully correlated. P i i i F iˆ1 ~ The actual c.o.v. depends on the correlation between the Pis: ~ If all the Pis were uncorrelated, then Ym Xm ˆ 1 1 diZi† 2 1 2 2 iˆ1 d ˆ di 35† iˆ1 Xm X X Ym Although the P~ s are generally correlated, simulations ¼ i P ˆ diZi 1 didjZiZj 1 didjdkZiZjZk 1 1 diZi 2 m 2 showthat d may be well approximated by iˆ1 di : This iˆ1 i.j i.j.k iˆ1 will be illustrated in Section 8. To get an idea of the number of samples required to Taking expectation and using E‰ZiŠˆ0; ~ achieve a given accuracy in PF; consider the case "# 2 P~ 2 P X X where P F1†ˆP Fi11uFi†ˆp0: Assume di ˆ 1 1 g† 1 2 E F F ˆ d d E‰Z Z Š 1 d d d E‰Z Z Z Š 1 ¼ p †=p N to be the same for all levels. Using Eqs. +28) and P i j i j i j k i j k 0 0 F i.j i.j.k +34), and noting that the number of simulation levels to !"# achieve the target failure probability is m ˆ logP =logp ; Ym Ym F 0 1 d E Z we conclude that to achieve a given c.o.v. of d in the esti- i i ~ iˆ1 iˆ1 mate PF; the total number of samples required is roughly 32† 1 1 g† 1 2 p † N < mN ˆ ulogP ur 0 36† T F r 2 If {Zi} are uncorrelated, E‰ZiZjŠ, E[Zi Zj Zk],¼, are all p0ulogp0u d zero, and hence P~ will be unbiased. In general, however, F where r # 3 depends on the actual correlation of the P~ s: {Z } are correlated, so P~ is biased for every N. On the other i i pF Thus, for a ®xed p and d, N / ulogP ur: Compared with hand, since d is O 1= N† from Eq. +28), the ®rst term in Eq. 0 T F i MCS, where N / 1=P ; this implies a substantial improve- +32) is O+1/N) while the remaining sums of higher products T F ment in ef®ciency when estimating small probabilities. of the d i's are o+1/N). Taking absolute value on both sides of Eq. +32), using the Cauchy±Schwarz inequality to obtain p A uE‰Z Z Šu # 2 2 ˆ 1 and applying it to the R.H.S. i j E‰Zi ŠE‰Zj Š of Eq. +32) proves the required proposition. 7. Ergodicity of subset simulation procedure A The foregoing discussion assumes that the Markov chain ~ Proposition 2. The c.o.v. d of PF is bounded above by: generated according to the Metropolis method is ergodic, which guarantees that the conditional probability estimate "#2 P~ 2 P Xm based on the Markov chain samples from a single chain will d2 ˆ E F F # d d 1 o 1=N†ˆO 1=N† 33† P i j tend to the corresponding theoretical conditional probability F i;jˆ1 as N ! 1: Theoretically, ergodicity can be always achieved ~ PFpis thus a consistent estimator and its c.o.v. d is by choosing a suf®ciently large spread in the proposal PDFs p O 1= N†: {pj }: Practically, with a ®nite number of Markov chain 270 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 samples, ergodicity often becomes an issue of whether for as the Markov chain samples in these regions propagate theMarkovchainsamplesusedtoestimatethecondi- through higher conditional levels. This argument holds, of tional probabilities populate suf®ciently well the impor- course, as long as unimportant failure regions +i.e. those tant regions of the failure domain, where the main with negligible contribution) at lower conditional levels contributions to the failure probability come from. Intui- remain unimportant at higher conditional levels, since tively, if there are some parts of the failure region of otherwise there may not be enough seeds +if any) at lower signi®cant probability content that are not well visited conditional levels to develop more Markov chain samples to by the Markov chain samples, the contribution of the account for their importance at higher levels. In conclusion, probability mass from such regions will not be re¯ected even though a Markov chain started from a single state may in the estimator, and the conditional probability estimate have ergodicity problems with disconnected failure regions, will be signi®cantly biased. using Markov chain samples from multiple chains, as in the For a Markov chain started at a single point, ergodi- proposed method, can be expected to achieve practical ergo- city problems may arise due to the existence of discon- dicity in spite of the existence of disconnected failure nected failure regions that are separated by safe regions regions. whose size is large compared to the spread of the The foregoing argument suggests that the subset p proposal PDFs {pj }: To see this, recall that to go simulation procedure is likely to produce an ergodic from the current Markov state to the next, a candidate estimator for failure probability, nevertheless it offers state is generated in the neighborhood of the current no guarantee for practical ergodicity. Whether ergodi- state according to the proposal PDFs. To transit from city problems become an issue depends on the particular one failure region to another, the candidate state has to application and the choice of the proposal PDFs. It is lie in the second failure region, but this is unlikely to worth-noting that similar issues related to ergodicity are happen if the spread of the proposal PDF is small expected to arise in any simulation method which tries compared to the safe region between the two failure to conclude `global' information from some known regions. +Any candidate state that falls in the safe local information, assuming implicitly that the known region is rejected by the Metropolis algorithm.) So, local information dominates the problem. For example, the safe region prohibits transition of the Markov importance sampling using design point+s) implicitly chain between disconnected failure regions, resulting assumes that the main contribution of failure probability in ergodicity problems. The situation of disconnected comes from the neighborhood of the known design failure regions may arise, for example, in failure points and there are no other design points of signi®cant problems for systems with components connected in contribution. Thus, in situations such as when the fail- series. ure region is highly concave or there are other unknown By using the samples from multiple Markov chains design points, the importance-sampling estimator is biased with different initial states obtained from previous and has an ergodicity problem. In viewof this, one should conditional levels, as in the present methodology, ergo- appreciate the ergodic property of standard MCS, since it is dicity problems due to disconnected failure regions may a totally global procedure in the sense that it does not accu- be resolved. To see this, ®rst note that the subset simu- mulate information about the failure region developed from lation procedure begins with MCS, which produces local states only. independent samples distributed in the whole parameter space. The MCS samples which lie in F1 populate the different regions of F1 according to the relative impor- 8. Examples tance +probability content) of the regions. We can expect these samples populate F1 suf®ciently well, The subset simulation methodology is applied to solving ~ since otherwise the c.o.v. of P1 will be large and subse- ®rst-excursion failure probabilities for two examples. In ~ quently the c.o.v. of PF will not be acceptable as the these examples, the input W+t) is a Gaussian white noise errors accumulate through subsequent conditional simu- process with spectral intensity S. The response of the system lation levels. For the next conditional level, we simulate is computed at the discrete time instants {tk ˆ k 2 1†Dt : multiple Markov chains, starting at the conditional MCS k ˆ 1; ¼n}; where the sampling interval is assumed to be samples distributed among different regions in F1.This Dt ˆ 0:02 s and the duration of study is T ˆ 30 s; so that the allows us to have seeds in the different regions of F1, which number of time instants is n ˆ T=Dt 1 1 ˆ 1501: The could possibly be disconnected. The Markov chain initiated uncertain state vector u then consists of the sequence of in each disconnected region will populate it at least locally, i.i.d. standard Gaussian random variables which generate and so the combined Markov chains correctly account for thep white noise input at the discrete time instants, {W tk†ˆ the contributions from the regions to the conditional prob- 2pS=Dtuk : k ˆ 1; ¼; n}; and so the number of uncertain ability estimate. parameters involved in the problem is n ˆ 1501: The inter-

In this manner, the contribution from disconnected failure mediate thresholds {bi} are chosen `adaptively' +see Section regions in higher conditional levels can also be accounted 5.2) so that the estimates for the conditional probabilities S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 271

Fig. 1. Failure probability estimate for Example 1.

p {P FiuFi21†} are equal to p0 ˆ 0:1: The proposal PDFs {pj } 8.2. Example 2 Five-story nonlinear hysteretic shear are chosen to be uniform PDFs with width 2, which are building found to give good ef®ciency. The two examples are described ®rst, followed by a discussion of the simulation Consider a ®ve-story shear building with hysteretic results. behavior under earthquake motion modeled by a nonsta- tionary stochastic process. The ¯oor masses are 3 8.1. Example 1 SDOF linear oscillator 45:4 £ 10 kg for all stories. The linear interstory stiff- ness for the ®rst to ®fth stories are 41:1 £ 106; 38:5 £ Consider a single degree-of-freedom +DOF) oscillator 106; 33:4 £ 106; 25:6 £ 106 and 15:2 £ 106 N/m, respec- with natural frequency v ˆ 7:85 rad=s +1.25 Hz) and damp- tively. Each story, of height 2.7 m, is modeled by a ing ratio z ˆ 2% subjected to white noise excitation: Bouc±Wen hysteretic element [18] with parameters a ˆ b ˆ 0:5; h ˆ 1 and has strength equal to 12 £ 1023 of the corresponding interstory stiffness. The small-ampli- X t† 1 2zvX_ t† 1 v2X t†ˆW t† 37† tude natural frequency of the structure is 1.25 Hz. The system is assumed to start from rest, that is, X 0†ˆ0 Rayleigh viscous damping is also assumed so that the and X_ 0†ˆ0: The spectral intensity for the white noise is ®rst two modes have 5% of critical damping at small assumed to be S ˆ 1: response amplitudes. Failure is de®ned as the displacement response exceeding The structure is subjected to base excitation a t† modeled a threshold level b within the ®rst 30 s, that is, by Clough±Penzien ®ltered white noise W+t) modulated by an envelope function e+t): [n F ˆ {uX tk†u . b} ˆ { max uX tk†u . b} 38† 2 2 kˆ1;¼;n a t† 1 2zs2vs2a_ t† 1 vs2a t†ˆ2zs1vs1a_1 t† 1 vs1a1 t† 40† kˆ1

The intermediate failure events {Fi} can then be chosen as: 2 a1 t† 1 2zs1vs1a_1 t† 1 vs1a1 t†ˆe t†W t† 41†

Fi ˆ { max uX tk†u . bi} 39† kˆ1;¼;n where vs1 ˆ 15:7 rad=s +2.5 Hz) and vs2 ˆ 1:57 rad=s +0.25 Hz) are the dominant and lower-cutoff frequency of where b1 , ¼ , bm ˆ b are the intermediate thresholds. the spectrum, respectively; zs1 ˆ 0:6 and zs2 ˆ 0:8 are the 272 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277

Fig. 2. Failure probability estimate for Example 2.

damping parameters associated with the dominant and exp[2+t 2 24)2/2] starting from t ˆ 24 s: The spectral lower-cutoff frequency, respectively. The envelope function intensity for the Gaussian white noise W+t) in Eq. +41) is e+t) is assumed to vary quadratically as +t/4)2 for the ®rst 4 s, assumed to be S ˆ 2:5 £ 1023m2=s3: The nonlinear response then settle at unity for 20 s, and ®nally decay as of the structure is computed using the Newmark constant

Fig. 3. Sample average of failure probability estimates from 50 simulation runs for Example 1. S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 273

Fig. 4. Sample average of failure probability estimates from 50 simulation runs for Example 2.

acceleration scheme for a duration of 30 s, which is suf®- Failure is de®ned as the exceedence of the interstory cient to capture the whole earthquake response history. P±D drift of any one of the stories above a given threshold effects have been taken into account in computing the level b within the ®rst 30 s. Let Xj+tk) denote the interstory nonlinear response. drift response for the jth story at time tk, j ˆ 1; ¼; 5;

Fig. 5. Coef®cient of variation of failure probabilty estimate for Example 1. 274 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277

Fig. 6. Coef®cient of variation of failure probability estimate for Example 2.

k ˆ 1; ¼; n: Then the failure event of interest is independent simulation runs is computed. The results are compared with those obtained by MCS in Figs. 3 and 4 [5 [n for Examples 1 and 2, respectively. It is seen in both ®gures F ˆ {uXj tk†u . b} ˆ { max max uXj tk†u . b} jˆ1;¼;5 kˆ1;¼;n jˆ1 kˆ1 that the average of the failure probability estimates almost 42† coincide with the MCS results, except in the low probability region where the error in the MCS estimates is signi®cant. The intermediate failure events {Fi} can then be chosen These results showthat the bias due to the correlation as: between conditional probability estimates at different levels is negligible, and the failure probability estimate obtained Fi ˆ { max max uXj tk†u . bi} 43† jˆ1;¼;5 kˆ1;¼;n by subset simulation is practically unbiased. where {b } are the intermediate interstory drift thresholds. To investigate the actual variability of the estimates, as i well as the validity of the upper bound on the c.o.v. in Eq. 8.3. Discussion of results +33), the sample c.o.v. of the failure probability estimates over 50 sets of independent subset simulation runs are Figs. 1 and 2 showthe failure probability estimates from plotted with circles in Figs. 5 and 6 for Example 1 and 2, three independent simulation runs for Examples 1 and 2, respectively. In these ®gures, the dashed line shows the respectively. In these ®gures, the failure probability esti- upper bound computed using Eq. +33) averaged over the mates corresponding to the intermediate thresholds are 50 simulation runs. The averaged estimates of the c.o.v. shown with circles, which are computed using a total based on Eq. +35), which assumes the conditional probabil- number of samples equal to 500, 1000, 1500 and 2000, ity estimators are uncorrelated, are also plotted with a solid respectively. That is, N ˆ 500 samples have been used to line in the ®gure. In both ®gures, the sample c.o.v. +circles) estimate each of the conditional probabilities. Note that a lie close to the one assuming uncorrelated conditional prob- single subset simulation run yields failure probability esti- ability estimates +solid line), showing that there is only a mates for all threshold levels up to the largest one consid- small reduction in ef®ciency due to the correlation. The ered, not just for the bi. For comparison, the results using smaller the failure probability and hence higher the simula- MCS with 100,000 samples are also shown in the ®gures. It tion level, the more the sample c.o.v. is greater than the one is seen in both examples that the results by subset simulation assuming uncorrelated conditional probability estimates, and MCS agree well. since correlation effects accumulate with increasing simula- To investigate the bias of the failure probability esti- tion levels. To compare the ef®ciency of subset simulation mates, the average of failure probability estimates over 50 with MCS, the c.o.v. of a MCS estimate at a particular S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 275 failure probability level using the same total number of Acknowledgements samples as in subset simulation +that is, NT ˆ 21 22 23 24 500; 1000; 1500; 2000 for PF ˆ 10 ; 10 ; 10 ; 10 ; This paper is based upon work partly supported by the respectively) is plotted with squares in Figs. 5 and 6 for Paci®c Earthquake Engineering Research Center under Examples 1 and 2, respectively. At a given failure probabil- National Science Foundation Cooperative Agreement No. ity level PF, the c.o.v. of failure probability estimates CMS-9701568. This support is gratefully acknowledged. computed by subset simulation and MCS using the same total number of samples +and hence approximately the same computational effort) can be compared. In Figs. 5 Appendix A and 6, the c.o.v. achieved by subset simulation and MCS This appendix describes the original Metropolis algo- are approximately the same in the large probability region rithm and explains why it is not applicable to simulating +e.g., P , 0:1). In particular, the values of c.o.v. for subset F random vectors with independent components in high simulation +circle) and MCS +square) coincide at P ˆ 0:1; F dimensions. since according to the subset simulation procedure with p ˆ 0:1 this probability estimate is computed based on an 0 A.1. Metropolis algorithm initial MCS. While the c.o.v. for subset simulation grows approximately in a linear fashion with the logorithm of Let the proposal PDF, pp ju u†; be a chosen n-dimen- decreasing failure probabilities, the c.o.v. for MCS grows sional joint PDF for j centered at u with the symmetry exponentially. Thus, subset simulation becomes more and property p p juu†ˆp p uuj†: Generate a sequence of samples more ef®cient compared with MCS as the target probability {u1,u2,¼} from a given sample u1 by computing uk11 from of failure gets smaller. uk; k ˆ 1; 2; ¼; as follows: Note that, from a theoretical point of view, Example 2 is much more complex than Example 1, since it involves the 1. Generate a `candidate' state u~: Simulate j according to ®rst excursion failure of a vector-valued response process, p ~ p ju uk†: Compute the ratio r ˆ q j†=q uk†: Set u ˆ j the system is nonlinear hysteretic, and the excitation is non- ~ with probability min{1,r} and set u ˆ uk with the white and nonstationary. In spite of this difference in remaining probability 1 2 min{1,r}. complexity, however, the trend of the c.o.v. with decreasing ~ ~ ~ 2. Accept/reject u: Check the location of u.Ifu [ Fi; failure probability in both examples is similar, and so is the ~ accept it as the next sample, i.e. uk11 ˆ u; otherwise ef®ciency of subset simulation. Thus, it is expected that reject it and take the current sample as the next sample, subset simulation can be applied ef®ciently to ®rst excursion i.e. uk11 ˆ uk: problems for a wide range of dynamical systems. Essentially, the original Metropolis algorithm differs from the modi®ed Metropolis algorithm in the way the 9. Conclusions candidate state is generated in Step 1. Note that the Markov chain samples generated according to the Metropolis algo- One of the major obstacles in applying simulation rithm are not all distinct. In particular, there is a non-zero methods to estimating small failure probabilities is the probability, PR, that the next state in the Markov chain will need to simulate rare events. Subset simulation resolves be equal to the current state. A large PR implies that the this by breaking the problem into the estimation of a current and the next state will be highly correlated. This sequence of larger conditional probabilities. The Metro- generally increases the statistical variability of the estimator ~ polis method has been modi®ed to ef®ciently estimate Pi and reduces the ef®ciency of the simulation process. It is the conditional probabilities by Markov chain simula- thus essential that PR be generally small among the samples tion. Theoretical estimates for the coef®cient of varia- in order to achieve satisfactory ef®ciency. tion and results from numerical simulation demonstrate It has been found that when the uncertain parameters u a substantial improvement in ef®ciency over standard are independent and the dimension n is large, PR is close to MCS. 1, rendering the original Metropolis algorithm almost inap- It should be noted that the choice of the proposal plicable. To demonstrate this, consider the case when Qn PDFs is crucial to the success of the subset simulation q u†ˆ jˆ1 q uj†; where q+uj) is the one-dimensional PDF methodology. This choice deserves a detailed study for the component uj, being equal for all components j.Asa p Qn p when the method is applied to different types of common choice, assume that p juu†ˆ jˆ1 p jjuuj†; p problems. Also, better choices of intermediate threshold where p jjuuj† denotes the one-dimensional PDF for jj p levels may be developed to further improve the ef®- centered at uj with the symmetric property p jjuuj†ˆ p Qn ciency of subset simulation. Future research may also p ujujj†: Let r ˆ jˆ1 q jj†=q uj† and let u be uniformly focus on using the conditional samples generated in the distributed on [0,1], independent of everything else. subset simulation procedure for system analysis after According to the Metropolis algorithm, the event {r , 1} failure occurs. and an independent failed Bernoulli trial +with probability 276 S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277

1 2 r† imply that the next state will be equal to the current qj jj†=qj uj†; j ˆ 1; ¼; n and let {uj : j ˆ 1; ¼; n} be inde- state. Thus, pendent and uniformly distributed on [0,1], independent of everything else. For the modi®ed Metropolis algorithm, P $ P r , 1; u . r† A1† R note that the next state is equal to the current state either when the candidate state generated in Step 1 of the algo- ˆ P u . rur , 1†P r , 1† rithm is equal to the current state or when the candidate state

does not lie in Fi and hence is rejected in Step 2. So, ˆ E‰1 2 rur , 1ŠP r , 1† 08 9 1 <\n = ˆ {1 2 E‰rur , 1Š}P r , 1† P ˆ P@ r , 1; u . r < u~ Ó F A A7† R : j j j; i jˆ1 To assess the quantities appearing in Eq. +A1), ®rst note 0 1 that, since { jj; uj† : j ˆ 1; ¼; n} are independently and p \n identically distributed as p jjuuj†q uj†; we have # P@ r , 1; u . r A 1 P u~ Ó F †  j j j i log r† 1 Xn q j † q j† jˆ1 ˆ log j ! E log A2† n n q u † q u† jˆ1 j Yn ˆ P r , 1; u . r † 1 P u~ Ó F † where the convergence is in an almost sure sense due to the j j j i jˆ1 Strong Lawof Large Numbers; j and u in Eq. +A2) are p jointly distributed as p juu†q u†: Since exp+´) is a convex Yn ~ function, Jensen's inequality gives: ˆ P uj . rjurj , 1†P rj , 1† 1 P u Ó Fi†  jˆ1 q j† q j† q j† exp E log # E exp log ˆ E A3† q u† q u† q u† Yn ~ ˆ {1 2 E‰rjurj , 1Š}P rj , 1† 1 P u Ó Fi† Further, jˆ1  q j† ZZ q j† E ˆ p p juu†q u†dudj A4† Since the factors in the ®rst product are always less than q u† q u† one, the ®rst term will often tend to zero as n ! 1: In this ~ Z Z case, combining Eq. +A7) with the fact that PR $ P u Ó p p p ~ ˆ q j† p uuj†dudj since P juu†ˆp uuj† Fi†; it can be concluded that PR ! P u Ó Fi† as n ! 1. This result can be expected intuitively, since when n is Z Z large, it is unlikely that the candidate state is equal to the ˆ q j†dj since p p uuj†du ˆ 1 current state, as this requires all the n `component candidate states' {jj : j ˆ 1; ¼; n} be rejected in Step 1 of the algo- rithm. The event that the next state is equal to the current ˆ 1 state then nearly corresponds to the event where a candidate

Combining Eqs. +A3) and +A4) gives: state is rejected for not lying in Fi. Consequently, when the  q j† dimension n is large, PR can be expected not to increase E log ˆ 2C , 0 A5† systematically with n, and hence the modi®ed Metropolis q u† algorithm is applicable even when the dimension is large. for some ®xed positive constant C. Thus, Eqs. +A2) and +A5) imply, almost surely as n ! 1; References r ! exp 2nC† A6† and hence r is exponentially small when the dimension n of [1] Rubinstein RY. Simulation and the Monte-Carlo method. NewYork: Wiley, 1981. the uncertain parameter space is large. It follows that [2] Fishman GS. Monte Carlo: concepts, algorithms, and applications. E‰r‰r , 1Š!0; P r , 1†!1 and so PR ! 1asn ! 1: NewYork: Springer, 1996. Thus, it is unlikely that the Metropolis chain will transit to [3] Hammersley JM, Handscomb DC. Monte-Carlo methods. London: any other newdistinct state as the dimension n is large, in Methuen, 1964. which case nearly all the Markov chain samples will be [4] SchueÈller GI, Stix R. A critical appraisal of methods to determine failure probabilities. Struct Safety 1987;4:293±309. equal. [5] Shinozuka M. Basic analysis of structural safety. J Struct Engng, Although it has not been demonstrated analytically, ASCE 1983;109:721±40. numerical simulations showthat the above phenomenon [6] Melchers RE. Importance sampling in structural systems. Struct occurs in more general situations; for example, when the Safety 1989;6:3±10. components u are not all identically distributed. In contrast, [7] Hohenbichler M, Rackwitz R. Improvement of second-order reliabil- j ity estimates by importance sampling. J Engng Mech, ASCE the modi®ed Metropolis algorithm presented in Section 3 is 1988;114+12):2195±8. applicable in high dimensions. To see this, let rj ˆ [8] Papadimitriou C, Beck JL, Katafygiotis LS. Asymptotic expansions S.-K. Au, J.L. Beck / Probabilistic Engineering Mechanics 16 62001) 263±277 277

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