Estimation Of Small Failure Probabilities Research Articles

A multi-fidelity stochastic simulation scheme for estimation of small failure probabilities

Computing small failure probabilities is often of interest in the reliability analysis of engineering systems. However, this task can be computationally demanding since many evaluations of expensive high-fidelity models are often required. To address this, a multi-fidelity approach is proposed in this work within the setting of stratified sampling. The overall idea is to reduce the required number of high-fidelity model runs by integrating the information provided by different levels of model fidelity while maintaining accuracy in estimating the failure probabilities. More specifically, strata-wise multi-fidelity models are established based on Gaussian process models to efficiently predict the high-fidelity response and the system collapse from the low-fidelity response. Due to the reduced computational cost of the low-fidelity models, the multi-fidelity approach can achieve a significant speedup in estimating small failure probabilities associated with high-fidelity models. The effectiveness and efficiency of the proposed multi-fidelity stochastic simulation scheme are validated through an application to a two-story two-bay steel building under extreme winds.

Estimation of small failure probabilities using the Accelerated Weight Histogram method

Simulation of rare events, such as small failure probabilities, is a common problem in several scientific and engineering fields. This paper presents a novel Monte-Carlo-based simulation approach for this purpose, called the Accelerated Weight Histogram (AWH) method. The method was originally developed to solve challenging sampling problems in statistical and biological physics, but its algorithm has here been reformulated for estimation of rare event probabilities. The applicability of the method is investigated for a couple of simpler computational examples and for a more advanced practical case, which consists of a rock tunnel stability problem. To estimate the probability of failure of the latter in a realistic manner, a boolean indicator function was used to describe failure, based on a mechanical concept known as unbalanced force ratio. The investigated cases indicate that the AWH method performs well for both simpler limit states and more complex failure definitions.

Estimation of small failure probabilities by partially Bayesian active learning line sampling: Theory and algorithm

Line sampling (LS) has proved to be a highly promising advanced simulation technique for assessing small failure probabilities. Despite the great interest in practical engineering applications, many efforts from the research community have been devoted to improving the standard LS. This paper aims at offering some new insights into the LS method, leading to an innovative method, termed ‘partially Bayesian active learning line sampling’ (PBAL-LS). The problem of evaluating the failure probability integral in the LS method is treated as a Bayesian, rather than frequentist, inference problem, which allows to incorporate our prior knowledge and model the discretization error. The Gaussian process model is used as the prior distribution for the distance function, and the posterior mean, and an upper bound of the posterior variance of the failure probability are derived. Based on the posterior statistics of the failure probability, we also put forward a learning function and a stopping criterion, which enable us to use active learning. Besides, an efficient algorithm is also designed to implement the PBAL-LS method, with the ability to automatically adjust the important direction and efficiently process the lines. Five numerical examples are studied to demonstrate the performance of the proposed PBAL-LS method against several existing methods.

An efficient stratified sampling scheme for the simultaneous estimation of small failure probabilities in wind engineering applications

With the advent of performance-based engineering frameworks for extreme winds, greater efficiency in the estimation of failure probabilities associated with multiple limit states at ultimate load levels, including collapse, is required. To meet this demand, an optimal stratified sampling-based simulation scheme is proposed that shares several meritorious characteristics of standard Monte Carlo simulation (MCS), including the capability to deal with high-dimensional uncertain spaces and multiple limit state functions. The optimality of the scheme is governed by the allocation of MCS samples among the strata as well as the strata division itself. A constrained optimization problem is formulated to address efficient sample allocation which is an extension of the Neyman allocation. Details for the practical implementation of the scheme within wind engineering reliability assessment applications are discussed. It is observed that the proposed methodology can reduce the variance by several orders of magnitude compared to MCS, which is illustrated on a wind-excited steel structure for the limit states of member/system first yield and collapse. The applicability of Latin hypercube sampling within stratified sampling-based simulation and the resulting efficiency gains are also critically studied. The results show promise in the wider applicability of the proposed scheme in performance-based wind engineering.

On the Robust Estimation of Small Failure Probabilities for Strong Nonlinear Models

Abstract Structural reliability methods are nowadays a cornerstone for the design of robustly performing structures, thanks to advancements in modeling and simulation tools. Monte Carlo-based simulation tools have been shown to provide the necessary accuracy and flexibility. While standard Monte Carlo estimation of the probability of failure is not hindered in its applicability by approximations or limiting assumptions, it becomes computationally unfeasible when small failure probability needs to be estimated, especially when the underlying numerical model evaluation is time consuming. In this case, variance reduction techniques are commonly employed, allowing for the estimation of small failure probabilities with a reduced number of samples and model calls. As a competing approach to variance reduction techniques, surrogate models can be used to substitute the computationally expensive model and performance function with an easy to evaluate numerical function calibrated through a supervised learning procedure. Both these tools provide accurate results for structural application. However, particular care should be taken into account when the reliability problems deal with high-dimensional or strongly nonlinear structural performances since the accuracy of the estimate is largely dependent on choices made during the surrogate modeling process. In this work, we compare the performance of the most recent state-of-the-art advance Monte Carlo techniques and surrogate models when applied to strongly nonlinear performance functions. This will provide the analysts with an insight to the issues that could arise in these challenging problems and help to decide with confidence on which tool to select in order to achieve accurate estimation of the failure probabilities within feasible times with their available computational capabilities.

Estimation of small failure probabilities based on thermodynamic integration and parallel tempering

Abstract Estimating small failure probabilities of systems is one of the most challenging tasks in the reliability analysis of engineering systems. Classic Monte Carlo simulation estimates the expected value of a step function (indicator function for failure), which is computationally demanding. In this paper, a new simulation method based on thermodynamic integration and parallel tempering (TIPT) is proposed to estimate small failure probabilities. Through thermodynamic integration, we convert the problem into estimating expected values for a series of smooth functions, which results in a series of much simpler problems. To solve these simpler problems in parallel, we adopt a Markov chain Monte Carlo (MCMC) method called parallel tempering. Thus, each sub-problem has each own chain, and the chains communicate to assist each other. For high-dimensional problems (i.e. with a large number of uncertain variables), we use the preconditioned Crank-Nicolson version of MCMC in parallel tempering to increase efficiency. Three widely used analytic examples are used to test the efficiency of TIPT. Finally, TIPT is applied to estimate the failure probability in a benchmark problem for water distribution systems.

A Bayesian Monte Carlo-based algorithm for the estimation of small failure probabilities of systems affected by uncertainties

The estimation of system failure probabilities in presence of uncertainties may be a difficult task when the values involved are very small, so that sampling-based Monte Carlo methods may become computationally impractical, especially if the computer codes used to model the system response require large computational efforts, both in terms of time and memory. In this work, we propose to exploit the Bayesian Monte Carlo (BMC) approach to the estimation of definite integrals for developing a new, efficient algorithm for estimating small failure probabilities. The Bayesian framework allows an effective use of all the information available, i.e. the computer code evaluations and the input uncertainty distributions, and, at the same time, the analytical formulation of the Bayesian estimator guarantees the construction of a computationally lean algorithm. The proposed method is first satisfactorily tested with reference to an analytic, two-dimensional case study of literature, offering satisfactory results; then, it is applied to a realistic case study of a natural convection-based cooling system of a gas-cooled fast reactor, operating under a post-loss-of-coolant accident (LOCA), showing performances comparable to those of other efficient alternative methods of literature.

Assessing small failure probabilities by AK–SS: An active learning method combining Kriging and Subset Simulation

With complex performance functions and time-demanding computation of structural responses, the estimation of small failure probabilities is a challenging problem in engineering. Although Subset Simulation (SS) is a powerful tool for small probabilities, the computation amount is still large for time-consuming numerical procedures. Metamodelling is an important approach to increase the computational efficiency for engineering problems, however, a larger set of sample points is required for higher accuracy. This is a time-consuming task when the performance function needs to be numerically evaluated. To address this issue, AK–SS: an active learning method combining Kriging model and SS is proposed in this paper. The efficiency of this new method relies upon the advantages of SS in evaluating small failure probabilities and the Kriging model with active learning and updating characteristic for approximating the true performance function. The proposed method is applied to several benchmark functions in the literature, and to the reliability analysis of a shield tunnel, which requires finite element analysis. The results demonstrated that as compared to the other approaches in literature, AK–SS can provide accurate solutions more efficiently, making it a promising approach for structural reliability analyses involving small failure probabilities, high-dimensional performance functions, and time-consuming simulation codes in practical engineering.

An improved adaptive kriging-based importance technique for sampling multiple failure regions of low probability

The estimation of system failure probabilities may be a difficult task when the values involved are very small, so that sampling-based Monte Carlo methods may become computationally impractical, especially if the computer codes used to model the system response require large computational efforts, both in terms of time and memory. This paper proposes a modification of an algorithm proposed in literature for the efficient estimation of small failure probabilities, which combines FORM to an adaptive kriging-based importance sampling strategy (AK-IS). The modification allows overcoming an important limitation of the original AK-IS in that it provides the algorithm with the flexibility to deal with multiple failure regions characterized by complex, non-linear limit states. The modified algorithm is shown to offer satisfactory results with reference to four case studies of literature, outperforming in general several other alternative methods of literature.

Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions

Estimation of small failure probabilities is one of the most important and challenging computational problems in reliability engineering. The failure probability is usually given by an integral over a high-dimensional uncertain parameter space that is difficult to evaluate numerically. This paper focuses on enhancements to Subset Simulation (SS), proposed by Au and Beck, which provides an efficient algorithm based on MCMC (Markov chain Monte Carlo) simulation for computing small failure probabilities for general high-dimensional reliability problems. First, we analyze the Modified Metropolis algorithm (MMA), an MCMC technique, which is used in SS for sampling from high-dimensional conditional distributions. The efficiency and accuracy of SS directly depends on the ergodic properties of the Markov chains generated by MMA, which control how fast the chain explores the parameter space. We present some observations on the optimal scaling of MMA for efficient exploration, and develop an optimal scaling strategy for this algorithm when it is employed within SS. Next, we provide a theoretical basis for the optimal value of the conditional failure probability p 0, an important parameter one has to choose when using SS. We demonstrate that choosing any p 0 ∈ [0.1, 0.3] will give similar efficiency as the optimal value of p 0. Finally, a Bayesian post-processor SS+ for the original SS method is developed where the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples from SS are viewed as informative data relevant to the system’s reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the information in the sampled data. This PDF quantifies the uncertainty in the value of the failure probability and it may be further used in risk analyses to incorporate this uncertainty. To demonstrate SS+, we consider its application to two different reliability problems: a linear reliability problem and reliability analysis of an elasto-plastic structure subjected to strong seismic ground motion. The relationship between the original SS and SS+ is also discussed.

Geometric insight into the challenges of solving high-dimensional reliability problems

In this paper we adopt a geometric perspective to highlight the challenges associated with solving high-dimensional reliability problems. Adopting a geometric point of view we highlight and explain a range of results concerning the performance of several well-known reliability methods. We start by investigating geometric properties of the N -dimensional Gaussian space and the distribution of samples in such a space or in a subspace corresponding to a failure domain. Next, we discuss Importance Sampling (IS) in high dimensions. We provide a geometric understanding as to why IS generally does not work in high dimensions [Au SK, Beck JL. Importance sampling in high dimensions. Structural Safety 2003;25(2):139–63]. We furthermore challenge the significance of “design point” when dealing with strongly nonlinear problems. We conclude by showing that for the general high-dimensional nonlinear reliability problems the selection of an appropriate fixed IS density is practically impossible. Next, we discuss the simulation of samples using Markov Chain Monte Carlo (MCMC) methods. Firstly, we provide a geometric explanation as to why the standard Metropolis–Hastings (MH) algorithm does “not work” in high-dimensions. We then explain why the modified Metropolis–Hastings (MMH) algorithm introduced by Au and Beck [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 2001;16(4):263–77] overcomes this problem. A study of the correlation of samples obtained using MMH as a function of different parameters follows. Such study leads to recommendations for fine-tuning the MMH algorithm. Finally, the MMH algorithm is compared with the MCMC algorithm proposed by Katafygiotis and Cheung [Katafygiotis LS, Cheung SH. Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study, Structural Safety 2006 (in print)] in terms of the correlation of samples they generate.

Application of line sampling simulation method to reliability benchmark problems

A procedure denoted as Line Sampling (LS) has been developed for estimating the reliability of static and dynamical systems. The efficiency and accuracy of the method is shown by application to the subset of the entire spectrum of the posed benchmark problems [Schuëller GI, Pradlwarter HJ, Koutsourelakis PS. Benchmark study on reliability estimation in higher dimensions of structural systems. In URL: http://www.uibk.ac.at/mechanik/Publications/benchmark.html. Institute of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, 2004], i.e. in particular linear systems with random properties. The notion of design point excitation for non-linear systems is discussed and its use extended for reliability estimations of conservative non-linear MDOF systems considering critical conditional excitation. For solving the hysteretic MDOF system with uncertain structural parameters subjected to general Gaussian excitation, however, the general applicable subset procedure [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 2001;16:263–277] has been used combined with Importance Sampling.

Estimation of small failure probabilities in high dimensions by subset simulation

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.