We here consider the subset simulation method which approaches a failure event using a decreasing sequence of nested intermediate failure events. The method resembles importance sampling, which actively explores a probability space by conditioning the next evaluation on the previous evaluations using a Markov chain Monte Carlo (MCMC) algorithm. A Markov chain typically requires many steps to estimate the target distribution, which is impractical with expensive numerical models. Therefore, we propose to approximate each step of a Markov chain locally with Gaussian process (GP) regression. Benchmark examples of reliability analysis show that local approximations significantly improve overall efficiency of subset simulation. They reduce the number of expensive limit-state evaluations by over $80\%$. However, GP regression becomes computationally impractical with increasing dimension. Therefore, to make our use of a GP feasible, we employ the partial least squares (PLS) regression, a gradient-free reduction method, locally to explore and utilize a low-dimensional subspace within a Markov chain. Numerical experiments illustrate a significant computational gain with maintained sufficient accuracy.
Surrogate-Guided Adaptive Importance Sampling for Failure Probability Estimation
Aerospace Engineering and the Institute of Computational and Data Science (ICDS), Penn State（航空航天工程系与计算数据科学研究所，宾夕法尼亚州立大学）；Department of Statistics, Virginia Tech（统计系，弗吉尼亚理工大学）
Directional subset simulation method for reliability analysis
Georg Nemetschek Institute, Technical University of Munich（慕尼黑工业大学）；Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes（巴黎萨克雷大学）
In this paper, we propose a sequential directional importance sampling (SDIS) method for rare event estimation. SDIS expresses a small failure probability in terms of a sequence of auxiliary failure probabilities, defined by magnifying the input variability. The first probability in the sequence is estimated with Monte Carlo simulation in Cartesian coordinates, and all the subsequent ones are computed with directional importance sampling in polar coordinates. Samples from the directional importance sampling densities used to estimate the intermediate probabilities are drawn in a sequential man…
In the last fifteen the subset sampling method has often been used in reliability problems as a tool for calculating small probabilities. This method is extrapolating from an initial Monte Carlo estimate for the probability content of a failure domain found by a suitable higher level of the original limit state function. Then iteratively conditional probabilities are estimated for failures domains decreasing to the original failure domain. But there are assumptions not immediately obvious about the structure of the failure domains which must be fulfilled that the method works properly. Here ex…
The Tristructural isotropic (TRISO)-coated particle fuel is a robust nuclear fuel proposed to be used for multiple modern nuclear technologies. Therefore, characterizing its safety is vital for the reliable operation of nuclear technologies. However, the TRISO fuel failure probabilities are small and the computational model is time consuming to evaluate them using traditional Monte Carlo-type approaches. In the paper, we present a multifidelity active learning approach to efficiently estimate small failure probabilities given an expensive computational model. Active learning suggests the next …
The problem of estimating the probability p=P(g(X<0) is considered when X represents a multivariate stochastic input of a monotone function g. First, a heuristic method to bound p is formally described, involving a specialized design of numerical experiments. Then a statistical estimation of p is considered based on a sequential stochastic exploration of the input space. A maximum likelihood estimator of p based on successive dependent Bernoulli data is defined and its theoretical convergence properties are studied. Under intuitive or mild conditions, the estimation is faster and more robust t…
The objective of this work is to quantify the uncertainty in probability of failure estimates resulting from incomplete knowledge of the probability distributions for the input random variables. We propose a framework that couples the widely used Subset simulation (SuS) with Bayesian/information theoretic multi-model inference. The process starts with data used to infer probability distributions for the model inputs. Often such data sets are small. Multi-model inference is used to assess uncertainty associated with the model-form and parameters of these random variables in the form of model pr…
We describe an adaptive Markov chain Monte Carlo method suitable for the estimation of rare failure probabilities in complex probabilistic models. This method, the Accelerated Weight Histogram (AWH) method, has its origin in statistical physics (Lidmar, 2012) and has successfully been applied to molecular dynamics simulations in biophysics. Here we introduce it in the context of structural reliability and demonstrate its usefulness for calculation of failure probabilities in some selected problems of varying degrees of complexity and compare with other established techniques, e.g., subset simu…
In applications of imprecise probability, analysts must compute lower (or upper) expectations, defined as the infimum of an expectation over a set of parameter values. Monte Carlo methods consistently approximate expectations at fixed parameter values, but can be costly to implement in grid search to locate minima over large subsets of the parameter space. We investigate the use of stochastic iterative root-finding methods for efficiently computing lower expectations. In two examples we illustrate the use of various stochastic approximation methods, and demonstrate their superior performance i…
Rare event probability estimation is an important topic in reliability analysis. Stochastic methods, such as importance sampling, have been developed to estimate such probabilities but they often fail in high dimension. In this paper, we propose a new cross-entropy-based importance sampling algorithm to improve rare event probability estimation in high dimension. We focus on the cross-entropy method with Gaussian auxiliary distributions and we suggest to update the Gaussian covariance matrix only in a one-dimensional subspace. For that purpose, the main idea is to consider the projection in th…
Several methods have been proposed in the literature to solve reliability-based optimization problems, where failure probabilities are design constraints. However, few methods address the problem of life-cycle cost or risk optimization, where failure probabilities are part of the objective function. Moreover, few papers in the literature address time-variant reliability problems in life-cycle cost or risk optimization formulations; in particular, because most often computationally expensive Monte Carlo simulation is required. This paper proposes a numerical framework for solving general risk o…
Rare events are events that are expected to occur infrequently, or more technically, those that have low probabilities (say, order of $10^{-3}$ or less) of occurring according to a probability model. In the context of uncertainty quantification, the rare events often correspond to failure of systems designed for high reliability, meaning that the system performance fails to meet some design or operation specifications. As reviewed in this section, computation of such rare-event probabilities is challenging. Analytical solutions are usually not available for non-trivial problems and standard Mo…
Let g : $Ω$ = [0, 1] d $\rightarrow$ R denote a Lipschitz function that can be evaluated at each point, but at the price of a heavy computational time. Let X stand for a random variable with values in $Ω$ such that one is able to simulate, at least approximately, according to the restriction of the law of X to any subset of $Ω$. For example, thanks to Markov chain Monte Carlo techniques, this is always possible when X admits a density that is known up to a normalizing constant. In this context, given a deterministic threshold T such that the failure probability p := P(g(X) > T) may be very low…
This paper addresses the problem of evaluating learning systems in safety critical domains such as autonomous driving, where failures can have catastrophic consequences. We focus on two problems: searching for scenarios when learned agents fail and assessing their probability of failure. The standard method for agent evaluation in reinforcement learning, Vanilla Monte Carlo, can miss failures entirely, leading to the deployment of unsafe agents. We demonstrate this is an issue for current agents, where even matching the compute used for training is sometimes insufficient for evaluation. To add…
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 a…
The estimation of small probabilities of failure from computer simulations is a classical problem in engineering, and the Subset Simulation algorithm proposed by Au & Beck (Prob. Eng. Mech., 2001) has become one of the most popular method to solve it. Subset simulation has been shown to provide significant savings in the number of simulations to achieve a given accuracy of estimation, with respect to many other Monte Carlo approaches. The number of simulations remains still quite high however, and this method can be impractical for applications where an expensive-to-evaluate computer model is …
The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit-state function, which depends on the solution of a partial differential equation (PDE). In many applications, the PDE cannot be solved analytically. We can only evaluate an approximation of the exact PDE solution. Therefore, the probability of rare events is estimated with respect to an approximation of the limit-state function. This leads to an approximation error in the estimate of the probability of rare events. Indeed, w…
An exponential-time exact algorithm is provided for the task of clustering n items of data into k clusters. Instead of seeking one partition, posterior probabilities are computed for summary statistics: the number of clusters, and pairwise co-occurrence. The method is based on subset convolution, and yields the posterior distribution for the number of clusters in O(n * 3^n) operations, or O(n^3 * 2^n) using fast subset convolution. Pairwise co-occurrence probabilities are then obtained in O(n^3 * 2^n) operations. This is considerably faster than exhaustive enumeration of all partitions.
We consider the problem of estimating a probability of failure $α$, defined as the volume of the excursion set of a function $f:\mathbb{X} \subseteq \mathbb{R}^{d} \to \mathbb{R}$ above a given threshold, under a given probability measure on $\mathbb{X}$. In this article, we combine the popular subset simulation algorithm (Au and Beck, Probab. Eng. Mech. 2001) and our sequential Bayesian approach for the estimation of a probability of failure (Bect, Ginsbourger, Li, Picheny and Vazquez, Stat. Comput. 2012). This makes it possible to estimate $α$ when the number of evaluations of $f$ is very li…
We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2β+1}) function evaluations suffice to achieve a given relative precision, where β is the number of bottleneck stations in the network. This …
This paper provides a detailed introductory description of Subset Simulation, an advanced stochastic simulation method for estimation of small probabilities of rare failure events. A simple and intuitive derivation of the method is given along with the discussion on its implementation. The method is illustrated with several easy-to-understand examples. For demonstration purposes, the MATLAB code for the considered examples is provided. The reader is assumed to be familiar only with elementary probability theory and statistics.
