Computation of failure probability via hierarchical clustering

Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study

Passive systems, which completely depend on natural phenomena such as gravity, conduction and convection to accomplish the safety functions, are increasingly being used in new generation nuclear reactor designs. However, since the driving forces of passive systems are weak, they are more vulnerable to associated uncertainties and there may be a non-zero probability for the system to deviate from the intended behavior and leads to functional failure. Methods for quantification of the functional failure include Monte-Carlo simulation of the system uncertainties using a validated mechanistic code. Generally, the mechanistic codes used for complex system modeling are computationally expensive and Monte- Carlo simulation for estimating small failure probabilities requires more time and often become prohibitive. In this respect, recently, functional reliability methodologies including advanced simulation techniques such as subset simulation, Markov chain Monte-Carlo, importance sampling, and response conditioning method are reported in open literature. Unlike in the case of direct Monte Carlo simulation, for the probability estimates obtained using these advanced simulations, analytical formulas are not available to estimate standard error and confidence interval. In this paper, the estimation of standard error and confidence interval of functional reliability estimates using computationally efficient re-sampling methods based on bootstrap technique are described. Numerical application of these methods, to quantify the variability of functional reliability estimates, is also explained.

In the practical engineering, to solve the small failure probabilities with correlated high-dimensional variables, the subset simulation (SS) is combined together with the Monte Carlo and importance sampling (IS) method. The samples from the probability density functions (PDF) of the importance sampling are used to construct the intermediate failure events, by which the small failure probabilities are turned into a Markov chain (MC), which is a continuous product made of a series large failure probability or conditional failure probability (CFP) which is easily answered, on which the structural reliability can be efficiently simulated by directly obtaining the samples with correlated ones. Finally, the 3 planet carriers of 3 grade planetary reducers in shield tunneling machine(STM) are as examples to check the algorithm above, the results show that the SS of the IS with correlated variables can highly simulate failure probability.

The design of complex systems often requires reliability assessments involving a large number of uncertainties and low probability of failure estimations (in the order of 10−4). Estimating such rare event probabilities with crude Monte Carlo (CMC) is computationally intractable. Specific numerical methods to reduce the computational cost and the variance estimate have been developed such as importance sampling or subset simulation. However, these methods assume that the uncertainties are defined within the probability formalism. Regarding epistemic uncertainties, the interval formalism is particularly adapted when only their definition domain is known. In this paper, a method is derived to assess the reliability of a system with uncertainties described by both probability and interval frameworks. It allows one to determine the bounds of the failure probability and involves a sequential approach using subset simulation, kriging, and an optimization process. To reduce the simulation cost, a refinement strategy of the surrogate model is proposed taking into account the presence of both aleatory and epistemic uncertainties. The method is compared to existing approaches on an analytical example as well as on a launch vehicle fallout zone estimation problem.

A natural method for approximating out-of-sample predictive evaluation is leave-one-out cross-validation (LOOCV) --- we alternately hold out each case from a full data set and then train a Bayesian model using Markov chain Monte Carlo (MCMC) without the held-out; at last we evaluate the posterior predictive distribution of all cases with their actual observations. However, actual LOOCV is time-consuming. This paper introduces two methods, namely iIS and iWAIC, for approximating LOOCV with only Markov chain samples simulated from a posterior based on a \textit{full} data set. iIS and iWAIC aim at improving the approximations given by importance sampling (IS) and WAIC in Bayesian models with possibly correlated latent variables. In iIS and iWAIC, we first integrate the predictive density over the distribution of the latent variables associated with the held-out without reference to its observation, then apply IS and WAIC approximations to the integrated predictive density. We compare iIS and iWAIC with other approximation methods in three real data examples that respectively use mixture models, models with correlated spatial effects, and a random effect logistic model. Our empirical results show that iIS and iWAIC give substantially better approximates than non-integrated IS and WAIC and other methods.

<p>For strongly non-linear inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF). Particle methods are very well suited for parallelization and offer an alternative approach whereby the posterior PDF is approximated using the states and weights of a population of evolving particles. In addition, it provides reliable estimates of the evidence (marginal likelihood) that is needed for Bayesian model selection at essentially no cost. We consider adaptive sequential Monte Carlo (ASMC), which is an extension of annealed importance sampling (AIS). In these methods, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The main advantages of ASMC with respect to AIS are that it adaptively tunes the tempering between neighboring distributions and it performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field designed such that the posterior facies distribution includes two distinct modes with equal probability. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multi-point statistics (MPS) simulations. We focus here on the ability of ASMC to explore the posterior PDF and compare it with previously published results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget involving 24 particles for ASMC and 24 chains for PT, the ASMC implementation outperforms the results obtained by PT: the models fit the data better and the reference likelihood value is contained in the ASMC sampled likelihood range, while this is not the case for PT range. Moreover, we show that ASMC recovers both reference modes, while none of them is recovered by PT. However, with 24 particles there is one of the modes that has a higher weight than the other while the approximation is improved when moving to a larger number of particles. As a future development, we suggest that including fast surrogate modeling (e.g., polynomial chaos expansion) within ASMC for the MCMC steps used to evolve the particles in-between importance sampling steps would strongly reduce the computational cost while still ensuring results of similar quality as the importance sampling steps could still be performed using the regular more costly forward solver.</p>

We present a new technique for statistical static timing analysis (SSTA) based on Markov chain Monte Carlo (MCMC), that allows fast and accurate estimation of the right-hand tail of the delay distribution. A ¿naive¿ MCMC approach is inadequate for SSTA. Several modifications and enhancements, presented in this paper, enable application of MCMC to SSTA. Moreover, such an approach overcomes inherent limitations of techniques such as importance sampling and Quasi-Monte Carlo. Our results on open source designs, with an independent delay variation model, demonstrate that our technique can obtain more than an order of magnitude improvement in computation time over simple Monte Carlo, given an estimation accuracy target at a point in the tail. Our approach works by providing a large number of samples in the region of interest. Open problems include extension of algorithm applicability to a broader class of synthesis conditions, and handling of correlated delay variations. In a broader context, this work aims to show that MCMC and associated techniques can be useful in rare event analyses related to circuits, particularly for high-dimensional problems.

In statically indeterminate timber structures, the stiffness of the members and the semi-rigid behaviour of joints influence the force distribution in the structure, affecting its performance and reliability. With the ambition to provide a reliable basis for the increasing interest in building more complex and statically indeterminate timber structures, this paper aims at identifying the impact of connections on the reliability of such structures by performing a study of a simple statically indeterminate system, i.e. a simple beam with rotational springs at its supports. Using UQlab, a Matlab-based toolbox, a parametric study varying the stiffness and the level of ductility of connections is performed. Different methods such as Monte Carlo Simulations, Subset, Importance Sampling and Adaptive Kriging with Subset simulation are applied and compared in terms of efficiency and computational time. Advanced modelling strategies such as importance sampling are effective means to reduce computational costs especially in complex models. The sensitivity analyses on input parameters reveal that the joint stiffness and ductility are amongst the most important input variables. Moreover, the results showed that an increment in reliability is gained increasing the level of ductility of joints of intermediate and high stiffness.

Accurate prediction of complex phenomena can be greatly enhanced through the use of data and observations to update simulations. The ability to create these data-driven simulations is limited by error and uncertainty in both the data and the simulation. The stochastic engine project addressed this problem through the development and application of a family of Markov Chain Monte Carlo methods utilizing importance sampling driven by forward simulators to minimize time spent search very large state spaces. The stochastic engine rapidly chooses among a very large number of hypothesized states and selects those that are consistent (within error) with all the information at hand. Predicted measurements from the simulator are used to estimate the likelihood of actual measurements, which in turn reduces the uncertainty in the original sample space via a conditional probability method called Bayesian inferencing. This highly efficient, staged Metropolis-type search algorithm allows us to address extremely complex problems and opens the door to solving many data-driven, nonlinear, multidimensional problems. A key challenge has been developing representation methods that integrate the local details of real data with the global physics of the simulations, enabling supercomputers to efficiently solve the problem. Development focused on large-scale problems, and on examining the mathematical robustness of the approach in diverse applications. Multiple data types were combined with large-scale simulations to evaluate systems with {approx}{sup 10}20,000 possible states (detecting underground leaks at the Hanford waste tanks). The probable uses of chemical process facilities were assessed using an evidence-tree representation and in-process updating. Other applications included contaminant flow paths at the Savannah River Site, locating structural flaws in buildings, improving models for seismic travel times systems used to monitor nuclear proliferation, characterizing the source of indistinct atmospheric plumes, and improving flash radiography. In the course of developing these applications, we also developed new methods to cluster and analyze the results of the state-space searches, as well as a number of algorithms to improve the search speed and efficiency. Our generalized solution contributes both a means to make more informed predictions of the behavior of very complex systems, and to improve those predictions as events unfold, using new data in real time.

This work is motivated by the need to estimate the probability of rare events in engineering systems with random inputs. We introduce a multilevel estimator which is based on and generalizes the idea of subset simulation. The novel estimator employs a hierarchy of approximations to the system response computed with different resolutions. This leads to reduced computational costs compared to subset simulation. We study the statistical properties and implementation details of the proposed estimator. Markov chain Monte Carlo (MCMC) runs are required within the estimator, and we demonstrate that the nestedness of the associated multilevel failure domains enables a perfect MCMC simulation without burn-in. We show that nestedness follows from certain simple one-dimensional failure domains. In high dimensions we propose a modification of the multilevel estimator which uses level-dependent stochastic input dimensions. We report on numerical experiments in one- and two-dimensional physical space; in particular, we...

Although Markov chain Monte Carlo methods have been widely used in many disciplines, exact eigen analysis for such generated chains has been rare. In this paper, a special Metropolis-Hastings algorithm, Metropolized independent sampling, proposed first in Hastings (1970), is studied in full detail. The eigenvalues and eigenvectors of the corresponding Markov chain, as well as a sharp bound for the total variation distance between the nth updated distribution and the target distribution, are provided. Furthermore, the relationship between this scheme, rejection sampling, and importance sampling are studied with emphasis on their relative efficiencies. It is shown that Metropolized independent sampling is superior to rejection sampling in two respects: asymptotic efficiency and ease of computation.

Structural reliability is an engineering method that focuses on the calculation and prediction of the probability of failure for a structural system. Structural reliability is used by many pipeline operators to assess the probability of failure, and these probability of failure estimates are then used to manage safety, environmental, and operational risks for threats such as corrosion, cracking, and third-party damage. In the structural reliability approach, the probability of failure is obtained from a multi-dimensional integral, for which the solution is commonly estimated numerically using Direct Monte Carlo (DMC) simulation. DMC is straight-forward and robust but requires a significant amount of computational effort to estimate small probabilities, which are typical of threats to pipeline integrity. Subset Simulation is an approach that improves on the efficiency issues of DMC by representing the rare event probability as the product of a number of more frequent events, which are each estimated separately. Previous work published at IPC in 2020 by Bandstra and Fraser [1] showed very close agreement between DMC and Subset Simulation for a single defect reference case with a single limit state. This study extends that initial work by applying Subset Simulation to various failure models for a variety of pipeline defects. Subset Simulation is applied to the CSA corrosion model, PRCI MAT-8 crack model, and the EPRG dent failure model, and the performance is evaluated by comparing the results to DMC. For each of these comparisons, simulations are performed across a large-scale grid of validation cases that consider a range of pipeline and defect sizes. An approach that utilizes Subset Simulation to handle multiple limit states is also presented and applied for relevant failure models and the results are evaluated against DMC.

Markov chain Monte Carlo (MCMC) algorithms have become powerful tools for Bayesian inference. However, they do not scale well to large-data problems. Divide-and-conquer strategies, which split the data into batches and, for each batch, run independent MCMC algorithms targeting the corresponding subposterior, can spread the computational burden across a number of separate computer cores. The challenge with such strategies is in recombining the subposteriors to approximate the full posterior. By creating a Gaussian-process approximation for each log-subposterior density we create a tractable approximation for the full posterior. This approximation is exploited through three methodologies: firstly a Hamiltonian Monte Carlo algorithm targeting the expectation of the posterior density provides a sample from an approximation to the posterior; secondly, evaluating the true posterior at the sampled points leads to an importance sampler that, asymptotically, targets the true posterior expectations; finally, an alternative importance sampler uses the full Gaussian-process distribution of the approximation to the log-posterior density to re-weight any initial sample and provide both an estimate of the posterior expectation and a measure of the uncertainty in it.

For strongly non-linear and high-dimensional inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF) given a realistic computational budget and are generally poorly amenable to parallelization. Particle methods approximate the posterior PDF using the states and weights of a population of evolving particles and they are very well suited to parallelization. We focus on adaptive sequential Monte Carlo (ASMC), an extension of annealed importance sampling (AIS). In AIS and ASMC, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The AIS and ASMC algorithms also provide estimates of the evidence (marginal likelihood) as needed for Bayesian model selection, at basically no additional cost. ASMC performs better than AIS as it adaptively tunes the tempering schedule and performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field defined such that the posterior facies distribution includes two distinct modes. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multiple-point statistics (MPS) simulations. We examine how ASMC explores the posterior PDF and compare with results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget, ASMC outperforms PT as the ASMC-derived models fit the data better and recover the reference likelihood. Moreover, we show that ASMC partly retrieves both posterior modes, while none of them is recovered by PT. Lastly, we demonstrate how the power posteriors obtained by ASMC can be used to assess the influence of the assumed data errors on the posterior means and variances, as well as on the evidence. We suggest that ASMC can advantageously replace MCMC for solving many challenging inverse problems arising in the field of water resources.

Group Importance Sampling for particle filtering and MCMC