arXiv subjects

Emmanuel Vazquez

Publications and source records attributed to Emmanuel Vazquez.

31 records · Page 2Linked to original sources

arXiv2016

Design of a commercial aircraft environment control system using Bayesian optimization techniques

In this paper, we present the application of a recently developed algorithm for Bayesian multi-objective optimization to the design of a commercial aircraft environment control system (ECS). In our model, the ECS is composed of two cross-flow heat exchangers, a centrifugal compressor and a radial turbine, the geometries of which are simultaneously optimized to achieve minimal weight and entropy generation of the system. While both objectives impact the overall performance of the aircraft, they are shown to be antagonistic and a set of trade-off design solutions is identified. The algorithm used for optimizing the system implements a Bayesian approach to the multi-objective optimization problem in the presence of non-linear constraints and the emphasis is on conducting the optimization using a limited number of system simulations. Noteworthy features of this particular application include a non-hypercubic design domain and the presence of hidden constraints due to simulation failures.

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arXiv2016

Gaussian process modeling for stochastic multi-fidelity simulators, with application to fire safety

To assess the possibility of evacuating a building in case of a fire, a standard method consists in simulating the propagation of fire, using finite difference methods and takes into account the random behavior of the fire, so that the result of a simulation is non-deterministic. The mesh fineness tunes the quality of the numerical model, and its computational cost. Depending on the mesh fineness, one simulation can last anywhere from a few minutes to several weeks. In this article, we focus on predicting the behavior of the fire simulator at fine meshes, using cheaper results, at coarser meshes. In the literature of the design and analysis of computer experiments, such a problem is referred to as multi-fidelity prediction. Our contribution is to extend to the case of stochastic simulators the Bayesian multi-fidelity model proposed by Picheny and Ginsbourger (2013) and Tuo et al. (2014).

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arXiv2016

A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria---the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions---as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization.

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arXiv2015

The Informational Approach to Global Optimization in presence of very noisy evaluation results. Application to the optimization of renewable energy integration strategies

We consider the problem of global optimization of a function f from very noisy evaluations. We adopt a Bayesian sequential approach: evaluation points are chosen so as to reduce the uncertainty about the position of the global optimum of f, as measured by the entropy of the corresponding random variable (Informational Approach to Global Optimization, Villemonteix et al., 2009). When evaluations are very noisy, the error coming from the estimation of the entropy using conditional simulations becomes non negligible compared to its variations on the input domain. We propose a solution to this problem by choosing evaluation points as if several evaluations were going to be made at these points. The method is applied to the optimization of a strategy for the integration of renewable energies into an electrical distribution network.

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arXiv2014

A new integral loss function for Bayesian optimization

We consider the problem of maximizing a real-valued continuous function $f$ using a Bayesian approach. Since the early work of Jonas Mockus and Antanas Žilinskas in the 70's, the problem of optimization is usually formulated by considering the loss function $\max f - M_n$ (where $M_n$ denotes the best function value observed after $n$ evaluations of $f$). This loss function puts emphasis on the value of the maximum, at the expense of the location of the maximizer. In the special case of a one-step Bayes-optimal strategy, it leads to the classical Expected Improvement (EI) sampling criterion. This is a special case of a Stepwise Uncertainty Reduction (SUR) strategy, where the risk associated to a certain uncertainty measure (here, the expected loss) on the quantity of interest is minimized at each step of the algorithm. In this article, assuming that $f$ is defined over a measure space $(\mathbb{X}, λ)$, we propose to consider instead the integral loss function $\int_{\mathbb{X}} (f - M_n)_{+}\, dλ$, and we show that this leads, in the case of a Gaussian process prior, to a new numerically tractable sampling criterion that we call $\rm EI^2$ (for Expected Integrated Expected Improvement). A numerical experiment illustrates that a SUR strategy based on this new sampling criterion reduces the error on both the value and the location of the maximizer faster than the EI-based strategy.

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arXiv2012

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.

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arXiv2012

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.

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arXiv2011

Bayesian optimization using sequential Monte Carlo

We consider the problem of optimizing a real-valued continuous function $f$ using a Bayesian approach, where the evaluations of $f$ are chosen sequentially by combining prior information about $f$, which is described by a random process model, and past evaluation results. The main difficulty with this approach is to be able to compute the posterior distributions of quantities of interest which are used to choose evaluation points. In this article, we decide to use a Sequential Monte Carlo (SMC) approach.

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arXiv2011

Sequential search based on kriging: convergence analysis of some algorithms

Let $\FF$ be a set of real-valued functions on a set $\XX$ and let $S:\FF \to \GG$ be an arbitrary mapping. We consider the problem of making inference about $S(f)$, with $f\in\FF$ unknown, from a finite set of pointwise evaluations of $f$. We are mainly interested in the problems of approximation and optimization. In this article, we make a brief review of results concerning average error bounds of Bayesian search methods that use a random process prior about $f$.

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arXiv2010

Convergence properties of the expected improvement algorithm

This paper has been withdrawn from the arXiv. It is now published by Elsevier in the Journal of Statistical Planning and Inference, under the modified title "Convergence properties of the expected improvement algorithm with fixed mean and covariance functions". See http://dx.doi.org/10.1016/j.jspi.2010.04.018 An author-generated post-print version is available from the HAL repository of SUPELEC at http://hal-supelec.archives-ouvertes.fr/hal-00217562 Abstract : "This paper deals with the convergence of the expected improvement algorithm, a popular global optimization algorithm based on a Gaussian process model of the function to be optimized. The first result is that under some mild hypotheses on the covariance function k of the Gaussian process, the expected improvement algorithm produces a dense sequence of evaluation points in the search domain, when the function to be optimized is in the reproducing kernel Hilbert space generated by k. The second result states that the density property also holds for P-almost all continuous functions, where P is the (prior) probability distribution induced by the Gaussian process."

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arXiv2009

Pointwise consistency of the kriging predictor with known mean and covariance functions

This paper deals with several issues related to the pointwise consistency of the kriging predictor when the mean and the covariance functions are known. These questions are of general importance in the context of computer experiments. The analysis is based on the properties of approximations in reproducing kernel Hilbert spaces. We fix an erroneous claim of Yakowitz and Szidarovszky (J. Multivariate Analysis, 1985) that the kriging predictor is pointwise consistent for all continuous sample paths under some assumptions.

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arXiv2007

An informational approach to the global optimization of expensive-to-evaluate functions

In many global optimization problems motivated by engineering applications, the number of function evaluations is severely limited by time or cost. To ensure that each evaluation contributes to the localization of good candidates for the role of global minimizer, a sequential choice of evaluation points is usually carried out. In particular, when Kriging is used to interpolate past evaluations, the uncertainty associated with the lack of information on the function can be expressed and used to compute a number of criteria accounting for the interest of an additional evaluation at any given point. This paper introduces minimizer entropy as a new Kriging-based criterion for the sequential choice of points at which the function should be evaluated. Based on \emph{stepwise uncertainty reduction}, it accounts for the informational gain on the minimizer expected from a new evaluation. The criterion is approximated using conditional simulations of the Gaussian process model behind Kriging, and then inserted into an algorithm similar in spirit to the \emph{Efficient Global Optimization} (EGO) algorithm. An empirical comparison is carried out between our criterion and \emph{expected improvement}, one of the reference criteria in the literature. Experimental results indicate major evaluation savings over EGO. Finally, the method, which we call IAGO (for Informational Approach to Global Optimization) is extended to robust optimization problems, where both the factors to be tuned and the function evaluations are corrupted by noise.

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arXiv2006

Estimation of the volume of an excursion set of a Gaussian process using intrinsic Kriging

Assume that a Gaussian process $ξ$ is predicted from $n$ pointwise observations by intrinsic Kriging and that the volume of the excursion set of $ξ$ above a given threshold $u$ is approximated by the volume of the predictor. The first part of this paper gives a bound on the convergence rate of the approximated volume. The second part describes an algorithm that constructs a sequence of points to yield a fast convergence of the approximation. The estimation of the volume of an excursion set is a highly relevant problem for the industrial world since it corresponds to the estimation of the failure probability of a system that is known only through sampled observations.

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