Despite the enormous amount of attention that has been devoted to the problem, no procedure is available for its general solution, especially for interesting engineering problems where the system complexity is high and the probability of failure is low. It is effective for estimating small error probabilities because the computational effort grows much more slowly with lower error probabilities than standard Monte Carlo simulation.
Chapter 1 Introduction
Outline of this work
Two efficient simulation methods are developed in Chapters 4 and 5 to solve the first excursion problem. In Chapter 5, a method called subset simulation is developed to solve the first excursion problem in general, with no assumption about the structure and the modeling of excitation.
Problem definition
The symbol P(-) is reserved for the probability of a statement or the probability content of a region given in the argument. Where distribution is clearly implied in the discussion, the signature for distribution will be omitted.
Standard Monte Carlo simulation
In our context, the standard Monte Carlo simulation represents the most robust method for estimating the probability of failure. Given the efficiency and robustness of the simulation method, which requires more computational effort to calculate a given probability of failure, it is not worth continuing.
Chapter 2 Importance Sampling Simulation
- Optimal lSD and its implications
- Basic trade-off
- Variance of importance sampling estimator and relative entropy
- Importance sampling in high dimensions
For appropriate choice of ISD f, the significance sample estimator converges with probability 1 (Strong Law of Large Numbers) to Pp and is asymptotically normally distributed as N -+ oo (central limit theorem). The second source comes from the variability of the significance sampling coefficient given that the 9 generated by f lies in the F failure region.
2 IIO"W
Definition of applicability in high dimensions
Importance sampling is useful in high dimensions with an ISD selected from a class of PDFs Pxs(n) for the reliability problem 'R(qn, Fn), if. In the following, we will explore the conditions under which importance sampling is useful in high dimensions.
ISD with a single point
So suppose condition (1) is not true, that is, there are an infinite number of indices i E z+ for which. Since condition (1) holds as we have just shown, the second sum in the exponent of (2.68) consists of only a finite number of terms and is therefore finite.
ISD with multiple points
The next proposition shows that the variability of L(O) increases exponentially with the order of the Euclidean norm of the presamples, implying that importance sampling using ISD constructed from random presamples is not applicable in high dimensions. This of course excludes the computational effort of searching the design point and is only meaningful when the failure region can be well characterized at a design point so that the importance sampling estimate is unbiased.
Diagnosis for applicability in high dimensions
To obtain b.1s by simulation, we note that in the current example it is possible to generate conditional samples according to q(fJ\F), using (2.90). Then the variation of b.R and b.1s with dimension n is compared for different cases, as shown in Figures 2.10 to 2.13, where only the theoretical values are plotted.
Summary of this chapter
Also the variation of ll1s with n is similar to the variation of tlR; fl1s remains bounded as n increases when tlR does (Case 1), and 6.1s grows exponentially with n when tlR does (Cases 2 to 4). This suggests that, in this example, the boundedness of tlR as n -too may be a necessary and sufficient condition for the boundedness of 6.1s, although only the sufficiency part was proved in Section 2.4.3. The new results on the applicability of importance sampling in high dimensions should provide important guidelines for applications to simulation problems involving a large number of uncertain parameters, such as first excursion problems where a stochastic process is used for modeling the excitation, or reliability problems with uncertain structures with a large number of uncertain model parameters.
Markov Chain Monte Carlo Simulation
- Metropolis-Hastings algorithm
- MCMC estimator
- Proposal PDF
- MCMC and importance sampling
- High dimensional aspects of MCMC
Note that all samples of the Markov chain lie within the error range of F as enforced by step 2. The correlation between Markov chain samples is strongly affected by the PDF proposal, which we will discuss. In this case, all samples of the Markov chain will be distinct, independent, and identically distributed as the target PDF, equal to q(OIF).
According to the Metropolis-Hastings algorithm, assuming that the Markov chain is stationary, the 'rejection probability' is PR, i.e. the probability that.
Metropolis-Hastings algorithm
Modified MCMC
Similar problems can arise with the Metropolis-Hastings scheme, depending on the choice of proposal PDFs. Ina}, where there is a number of groups, some group (or partition) of indices {1,. 8n}· Without loss of generality, assume that grouping does not affect the index order.
Note from the algorithm that the distribution of the next sample Ok+l depends only on the current sample 0,., and thus the samples form a Markov chain.
Comparison of the modified and original scheme
This shows that the next sample of the Markov chain 9n+l will also be distributed as q(·IF), so that the latter is effectively a stationary distribution for the generated Markov chain. This result can be expected intuitively, since when it is large, it is unlikely that the candidate state is equal to the current state, since this requires all the components of the predecessor to { e(i) : j = 1,. The event that the next state is equal to the current state then almost corresponds to the event that the candidate state is rejected because it does not lie in F.
Regarding the selection of the proposal PDFs, the rules used in the original MCMC scheme can be applied to the modified scheme, except that in the latter the selection should be made group by group, rather than for the entire state 6.
Summary of this chapter
Since the factors in the first product are always less than 1, the first term tends to vanish as increases. Consequently, when the number of groups after and thus the dimension n is large, it can be expected that PR does not systematically increase with n, and thus the modified Metropolis-Hastings algorithm is applicable even when the dimension is large. The choice of the proposal PDF for applications to solve the first excursion problem will be discussed in Chapter 5.
Linear Systems and Importance Sampling using Elementary Events
- Discrete-time linear systems
- Analysis of the failure region
- Elementary failure region
- Interaction of elementary failure regions
- Development of importance sampling density
- Proposed ISD
- Properties of proposed lSD and failure probability estimator
- Summary of proposed importance sampling procedure
- Generalization to non-causal systems
- Illustrative examples
- Example 1: SDOF oscillator
- Example 2: Seismic response of moment-resisting steel frame
- Summary of this chapter
The latter implies that the contribution of all parts of the failure region will be accounted for in the estimator. Consider the failure probability that the peak interstory drift ratio across all stories of the structure exceeds a threshold level b. It is seen in Figure 4.8 that the variation of the response standard deviation ai(t) with t approximately follows that of the envelope function e(t).
The results are shown in Table 4.6, showing that the variation of the failure probability estimates among independent runs is indeed small.
Chapter 5 General Systems and Subset Simulation Method
- Basic idea of subset simulation
- Subset simulation procedure
- Choice of intermediate failure events
- Choice of proposal PDF
- Choice of proposal PDF for first excursion problems
- Statistical properties of the estimators
- MCS estimator P 1
- Conditional probability estimator Pi (2 Sis m)
- Failure probability estimator Pp
- Ergodicity of subset simulation procedure
This choice of intermediate thresholds means that they depend on conditional patterns and volition. The value of 'Yi depends on the choice of spread of PDF templates. Markov state to the next, a candidate state is generated near the current state according to PDF suggestions.
Whether ergodicity issues become an issue depends on the specific application and the choice of proposal PDFs.
5. 7 Summary of this chapter
Applications to Probabilistic Assessment of Seismic Performance
- Lifetime reliability
- Stochastic ground motion model
The main goal is to demonstrate that subset simulation can be efficiently applied to calculate the failure probability of a structure given that an earthquake of uncertain magnitude and location occurs in a region of interest around where the structure is located. This information can be used to estimate the lifetime reliability, or equivalently, the lifetime failure probability of a structure in an uncertain seismic environment. The lifetime failure probability of a structure is closely related to the failure probability of a structure when an earthquake occurs.
In particular, assuming that the occurrence of earthquakes follows a Poisson process and the failure event of the structure in different earthquake events is independent with the same failure probability, then the lifetime failure probability P(Fhre) is given by
Radiation Spectrum
To generate a series of time histories for the ground acceleration for a given moment magnitude M and epicentral distance r, a discrete-time white noise series is {Wj = J27r I 6.t Zj: j = 1,. The synthetic ground motion a(t; Z, M, r) generated by the A-S model is a function of the additive excitation parameters Z = [Z1,. The term 1/R is the geometric dispersion factor, where R = ./h2 + r2 is the radial distance from the earthquake source to the site, r is the epicentral distance (in km), and h is the nominal fault depth ( km ).
On the other hand, in Figure 6.2, the spectral amplitude decreases at all frequencies as the epicentral distance r increases, and there is no significant dependence of the frequency content on r.
Envelope function
It should be noted that both M and r have roughly a multiplicative effect on the synthetic ground acceleration a(t; Z,M,r) and thus on the structural response.
Distribution of stochastic excitation model parameters
Illustrative examples
- Example 1: Linear SDOF oscillator
As the simulation level increases, the spectrum develops a peak near 1 Hz, which is the natural frequency of the structure. The inter-story drift ratios of the columns at different floors corresponding to these ground excitations are shown in Figure 6.28. The interstory drift ratio of the left column (which is essentially the same as that of the right column) corresponding to these ground excitations is shown in Figure 6.44.
As the simulation level increases, the spectrum develops a peak near 2 Hz, which is close to the natural frequency of the structure (2.6 Hz).
Summary of this chapter
Nevertheless, in this case the spectral peak is less distinct and does not occur at the natural frequency of the small amplitude structure. This is possibly due to the softening behavior of the structure at large vibration amplitudes, resulting in an apparently smaller 'resonance frequency' adapted by the Markov chain samples from the additive excitation. In view of this, the results of the failure analysis either provide channels for calibrating the stochastic excitation models or must otherwise be interpreted carefully.
Nevertheless, assuming that the quality of stochastic ground motion models will improve, subset simulation provides an effective tool for error probability estimation and error analysis.
Chapter 7 Conclusion
- Conclusions
- Future work
- l Neighborhood of points in the failure region
- Proximity of neighboring design points
- Overshooting of design point response
- A reciprocal relationship of design point responses
- Simulation formula for Zf
Conclusions from error analysis should be interpreted taking into account the inherent assumptions and limitations of the probabilistic models used. Petrov-Galerkin finite element solution for the first passage probability and moments of the first passage time of a randomly accelerated free particle. By intuition, we can expect that the maximum value of the response y( r; w;) corresponding to the excitation w; occurs at time t, since w; it is.
The proof then follows from the symmetry of the above expression with respect to s and t.
