Sequential design of computer experiments for the estimation of a probability of failure

This paper deals with the problem of estimating the volume of the excursion set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian-theoretic formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of $f$ and aim at performing evaluations of $f$ as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure.Comment: This is an author-generated postprint version. The published version is available at http://www.springerlink.co

Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure

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 involved. We propose a new algorithm, called Bayesian Subset Simulation, that takes the best from the Subset Simulation algorithm and from sequential Bayesian methods based on kriging (also known as Gaussian process modeling). The performance of this new algorithm is illustrated using a test case from the literature. We are able to report promising results. In addition, we provide a numerical study of the statistical properties of the estimator.Comment: 11th International Probabilistic Assessment and Management Conference (PSAM11) and The Annual European Safety and Reliability Conference (ESREL 2012), Helsinki : Finland (2012

Differentiating the multipoint Expected Improvement for optimal batch design

This work deals with parallel optimization of expensive objective functions which are modeled as sample realizations of Gaussian processes. The study is formalized as a Bayesian optimization problem, or continuous multi-armed bandit problem, where a batch of q > 0 arms is pulled in parallel at each iteration. Several algorithms have been developed for choosing batches by trading off exploitation and exploration. As of today, the maximum Expected Improvement (EI) and Upper Confidence Bound (UCB) selection rules appear as the most prominent approaches for batch selection. Here, we build upon recent work on the multipoint Expected Improvement criterion, for which an analytic expansion relying on Tallis' formula was recently established. The computational burden of this selection rule being still an issue in application, we derive a closed-form expression for the gradient of the multipoint Expected Improvement, which aims at facilitating its maximization using gradient-based ascent algorithms. Substantial computational savings are shown in application. In addition, our algorithms are tested numerically and compared to state-of-the-art UCB-based batch-sequential algorithms. Combining starting designs relying on UCB with gradient-based EI local optimization finally appears as a sound option for batch design in distributed Gaussian Process optimization

Bounding rare event probabilities in computer experiments

We are interested in bounding probabilities of rare events in the context of computer experiments. These rare events depend on the output of a physical model with random input variables. Since the model is only known through an expensive black box function, standard efficient Monte Carlo methods designed for rare events cannot be used. We then propose a strategy to deal with this difficulty based on importance sampling methods. This proposal relies on Kriging metamodeling and is able to achieve sharp upper confidence bounds on the rare event probabilities. The variability due to the Kriging metamodeling step is properly taken into account. The proposed methodology is applied to a toy example and compared to more standard Bayesian bounds. Finally, a challenging real case study is analyzed. It consists of finding an upper bound of the probability that the trajectory of an airborne load will collide with the aircraft that has released it.Comment: 21 pages, 6 figure

Gaussian process surrogates for failure detection: a Bayesian experimental design approach

An important task of uncertainty quantification is to identify {the probability of} undesired events, in particular, system failures, caused by various sources of uncertainties. In this work we consider the construction of Gaussian {process} surrogates for failure detection and failure probability estimation. In particular, we consider the situation that the underlying computer models are extremely expensive, and in this setting, determining the sampling points in the state space is of essential importance. We formulate the problem as an optimal experimental design for Bayesian inferences of the limit state (i.e., the failure boundary) and propose an efficient numerical scheme to solve the resulting optimization problem. In particular, the proposed limit-state inference method is capable of determining multiple sampling points at a time, and thus it is well suited for problems where multiple computer simulations can be performed in parallel. The accuracy and performance of the proposed method is demonstrated by both academic and practical examples

Screening and metamodeling of computer experiments with functional outputs. Application to thermal-hydraulic computations

To perform uncertainty, sensitivity or optimization analysis on scalar variables calculated by a cpu time expensive computer code, a widely accepted methodology consists in first identifying the most influential uncertain inputs (by screening techniques), and then in replacing the cpu time expensive model by a cpu inexpensive mathematical function, called a metamodel. This paper extends this methodology to the functional output case, for instance when the model output variables are curves. The screening approach is based on the analysis of variance and principal component analysis of output curves. The functional metamodeling consists in a curve classification step, a dimension reduction step, then a classical metamodeling step. An industrial nuclear reactor application (dealing with uncertainties in the pressurized thermal shock analysis) illustrates all these steps