International audienceThis 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 $\RR^d$ 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

Responses of many real-world problems can only be evaluated perturbed by noise. In order to make an efficient optimization of these problems possible, intelligent optimization strategies successfully coping with noisy evaluations are required. In this article, a comprehensive review of existing kriging-based methods for the optimization of noisy functions is provided. In summary, ten methods for choosing the sequential samples are described using a unified formalism. They are compared on analytical benchmark problems, whereby the usual assumption of homoscedastic Gaussian noise made in the underlying models is meet. Different problem configurations (noise level, maximum number of observations, initial number of observations) and setups (covariance functions, budget, initial sample size) are considered. It is found that the choices of the initial sample size and the covariance function are not critical. The choice of the method, however, can result in significant differences in the performance. In particular, the three most intuitive criteria are found as poor alternatives. Although no criterion is found consistently more efficient than the others, two specialized methods appear more robust on average.

This paper addresses the issue of designing experiments for a metamodel that needs to

This article addresses the issue of kriging-based optimization of stochastic simulators. Many of these simulators depend on factors that tune the level of precision of the response, the gain in accuracy being at a price of computational time. The contribution of this work is two-fold: firstly, we propose a quantile-based criterion for the sequential design of experiments, in the fashion of the classical Expected Improvement criterion, which allows an elegant treatment of heterogeneous response precisions. Secondly, we present a procedure for the allocation of the computational time given to each measurement, allowing a better distribution of the computational effort and increased efficiency. Finally, the optimization method is applied to an original application in nuclear criticality safety. This article has supplementary material online.

Stepwise Uncertainty Reduction (SUR) strategies aim at constructing a sequence of points for evaluating a function f in such a way that the residual uncertainty about a quantity of interest progressively decreases to zero. Using such strategies in the framework of Gaussian process modeling has been shown to be efficient for estimating the volume of excursion of f above a fixed threshold. However, SUR strategies remain cumbersome to use in practice because of their high computational complexity, and the fact that they deliver a single point at each iteration. In this paper we introduce several multi-point sampling criteria, allowing the selection of batches of points at which f can be evaluated in parallel. Such criteria are of particular interest when f is costly to evaluate and several CPUs are simultaneously available. We also manage to drastically reduce the computational cost of these strategies through the use of closed form formulas. We illustrate their performances in various numerical experiments, including a nuclear safety test case. Basic notions about kriging, auxiliary problems, complexity calculations, R code and datas are available online as supplementary materials.