tainty have been identified: deterministic model error and probability distribution parameter uncertainty. Thus, an uncertainty analysis will be carried out, as outlined in Section 2, to characterize the uncertainty in the predicted failure probability. As mentioned above, uncertainty associated with model error has been characterized based on validation studies as ε∼N(μ=0 , σ=0.1) mm. A Bayesian analysis using non-informative reference prior distributions is carried out to develop the uncertainty distributions for the probability distribution parameters of the model inputs. That is, the posterior distribution for the uncertain distribution parameters associated with each variable in Table 1 is computed, based on the available sample data. An automatically adapting Markov Chain Monte Carlo approach, based on the work of Haario et al (2001), is used to collect random samples of the distribution parameters from their posterior distributions. A variance decomposition of the epistemic variables, as discussed in Section 3, is used in order to identify what new information would allow for the greatest reduction in uncertainty. The structured Monte Carlo approach proposed by Saltelli et al (2004) is employed. The output statistic of interest is the failure probability, but here weuse γ=βin Eq. (1), where βis the reliability index, which for FORM is related to the failure probability by β=−Φ−1(p f ), (4) where Φ(·) is the standard normal inverse cumulative distribution function. The motivation for decomposition of the variance of the reliability index as opposed to the failure probability is that the distribution of the reliability index tends to be more symmetric (see, for example Der Kiureghian and Ditlevsen, 2009, Figure 1). This may improve the conditioning of the variance decomposition analysis, although admittedly further study and comparison is needed to better understand the difference. Choice of reliability method is an important consideration, both because it needs to be very efficient to be practical within the variance decomposition framework, and because stochastic methods may adversely impact the uncertainty quantification. For this work, the First Order Reliability Method (FORM; see, for example Haldar and Mahadevan, 2000) is utilized for the failure probability calculations.2 The method is advantageous here both because of its efficiency and because unlike a Monte Carlo approach, it will not introduce additional variance when used inside of variance decomposition. The FORM solution was verified for this problem using Monte Carlo simulation with 100,000 samples.3 The variance decomposition is performed using a base sample size of 50,000 (“N” in Saltelli et al, 2004), which necessitates a total of 350,000 evaluations of H(θ). Total computation time for the entire uncertainty analysis is on the order of six hours using a single-processor Linux machine. 2 Our FORM implementation is based on the nonlinear interior point solver from the Opt++ software package (Meza, 1994) 3 As mentioned in Section 2, the probabilistic method also introduces epistemic uncertainty to the result. With some ingenuity, it may also be possible to account for this uncertainty in the variance decomposition (similarly to the treatment of deterministic model error) in order to identify whether improvement of the probabilistic solution would be warranted as a means of reducing uncertainty. Variance Decomposition in the Presence of Epistemic and Aleatory Uncertainty 425
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