Multilevel Monte Carlo covariance estimation for the computation of Sobol' indices

International audienceCrude and quasi Monte Carlo (MC) sampling techniques are common tools dedicated to estimating statistics (expectation, variance, covariance) of a random quantity of interest.We focus here on the uncertainty quantification framework where the quantity of interest is the output of a numerical simulator fed with uncertain input parameters.Then, sampling the output involves running the simulator for different samples of the inputs, which may be computationally time-consuming.To reduce the cost of sampling, a first approach consists in replacing the numerical simulator by a surrogate model that is cheaper to evaluate, thus making it possible to generate more samples of the output and therefore leading to a lower sampling error.However, this approach adds to the sampling error an unavoidable model error.Another approach, which does not introduce any model error, is the so-called multilevel MC (MLMC) method.Given a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost, MLMC combines samples obtained at different levels to construct an estimator at a reduced cost compared to standard MC sampling.In this paper, we derive and analyze multilevel covariance estimators and adapt the MLMC convergence theorem in terms of the corresponding covariances and fourth order moments. We propose a multilevel algorithm driven by a target cost as an alternative to typical algorithms driven by a target accuracy.These results are used in a sensitivity analysis context in order to derive a multilevel estimation of Sobol' indices, whose building blocks can be written as covariance terms in a pick-and-freeze formulation.These contributions are successfully tested on an initial value problem with random parameters

Sensitivity analysis with dependence and variance-based measures for spatio-temporal numerical simulators

International audienceIn a case of radioactive release in the environment, modeling the radionuclide atmospheric dispersion is particularly useful for emergency response procedures and risk assessment. For this, the CEA has developed a numerical simulator, called Ceres-Mithra, to predict spatial maps of radionuclide concentrations at different instants. This computer code depends on many uncertain scalar and temporal parameters, describing the radionuclide, release or weather characteristics. The purpose is to detect the input parameters the uncertainties of which highly affect the predicted concentrations and to quantify their influences. To this end, we present various measures for the sensitivity analysis of a spatial model. Some of them lead to as many analyses as spatial locations (site sensitivity indices) while others consider a single one, with respect to the whole spatial domain (block sensitivity indices). For both categories, variance-based and dependence measures are considered, based on recent literature. All of these sensitivity measures are applied to the CM computer code and compared to each other, showing the complementarity of block and site sensitivity analyses. Finally, a sensitivity analysis summarizing the input uncertainty contribution over the entirety of the spatio-temporal domain is proposed

New improvements in the use of dependence measures for sensitivity analysis and screening

International audiencePhysical phenomena are commonly modeled by numerical simulators. Such codes can take as input a high number of uncertain parameters and it is important to identify their influences via a global sensitivity analysis (GSA). However, these codes can be time consuming which prevents a GSA based on the classical Sobol' indices, requiring too many simulations. This is especially true as the number of inputs is important. To address this limitation, we consider recent advances in dependence measures, focusing on the distance correlation and the Hilbert-Schmidt independence criterion (HSIC). Our objective is to study these indices and use them for a screening purpose. Numerical tests reveal some differences between dependence measures and classical Sobol' indices, and preliminary answers to "What sensitivity indices to what situation?" are derived. Then, two approaches are proposed to use the dependence measures for a screening purpose. The first one directly uses these indices with independence tests; asymptotic tests and their spectral extensions exist and are detailed. For a higher accuracy in presence of small samples, we propose a non-asymptotic version based on bootstrap sampling. The second approach is based on a linear model associating two simulations, which explains their output difference as a weighed sum of their input differences. From this, a bootstrap method is proposed for the selection of the influential inputs. We also propose a heuristic approach for the calibration of the HSIC Lasso method. Numerical experiments are performed and show the potential of these approaches for screening when many inputs are not influential

Mixture of polynomial chaos expansions for uncertainty propagation

HydrodynamicsAbstrac

Spatio-temporal and multifidelity surrogate models : Application in thermal engineering

Cette thèse porte sur la construction de modèles de substitution en régimes transitoire et permanent pour la simulation thermique, en présence de peu d'observations et de plusieurs sorties.Nous proposons dans un premier temps une construction robuste de perceptron multicouche bouclé afin d'approcher une dynamique spatio-temporelle. Ce modèle de substitution s'obtient par une moyennisation de réseaux de neurones issus d'une procédure de validation croisée, dont le partitionnement des observations associé permet d'ajuster les paramètres de chacun de ces modèles sur une base de test sans perte d'information. De plus, la construction d'un tel perceptron bouclé peut être distribuée selon ses sorties. Cette construction est appliquée à la modélisation de l'évolution temporelle de la température en différents points d'une armoire aéronautique.Nous proposons dans un deuxième temps une agrégation de modèles par processus gaussien dans un cadre multifidélité où nous disposons d'un modèle d'observation haute-fidélité complété par plusieurs modèles d'observation de fidélités moindres et non comparables. Une attention particulière est portée sur la spécification des tendances et coefficients d'ajustement présents dans ces modèles. Les différents krigeages et co-krigeages sont assemblés selon une partition ou un mélange pondéré en se basant sur une mesure de robustesse aux points du plan d'expériences les plus fiables. Cette approche est employée pour modéliser la température en différents points de l'armoire en régime permanent.Nous proposons dans un dernier temps un critère pénalisé pour le problème de la régression hétéroscédastique. Cet outil est développé dans le cadre des estimateurs par projection et appliqué au cas particulier des ondelettes de Haar. Nous accompagnons ces résultats théoriques de résultats numériques pour un problème tenant compte de différentes spécifications du bruit et de possibles dépendances dans les observations.This PhD thesis deals with the construction of surrogate models in transient and steady states in the context of thermal simulation, with a few observations and many outputs.First, we design a robust construction of recurrent multilayer perceptron so as to approach a spatio-temporal dynamic. We use an average of neural networks resulting from a cross-validation procedure, whose associated data splitting allows to adjust the parameters of these models thanks to a test set without any information loss. Moreover, the construction of this perceptron can be distributed according to its outputs. This construction is applied to the modelling of the temporal evolution of the temperature at different points of an aeronautical equipment.Then, we proposed a mixture of Gaussian process models in a multifidelity framework where we have a high-fidelity observation model completed by many observation models with lower and no comparable fidelities. A particular attention is paid to the specifications of trends and adjustement coefficients present in these models. Different kriging and co-krigings models are put together according to a partition or a weighted aggregation based on a robustness measure associated to the most reliable design points. This approach is used in order to model the temperature at different points of the equipment in steady state.Finally, we propose a penalized criterion for the problem of heteroscedastic regression. This tool is build in the case of projection estimators and applied with the Haar wavelet. We also give some numerical results for different noise specifications and possible dependencies in the observations

Estimation des mesures de sensibilité globale basées sur les dérivées à partir d'un métamodèle par processus gaussien

International audiencePhysical phenomena are often studied using numerical simulators. Such computer codes are function of uncertain input parameters and a global sensitivity analysis (GSA) can be performed to identify their impacts on the simulator outputs. Sobol' indices, based on output variance decomposition, are commonly used to perform quantitative GSA. For many years now, other tools have been studied, closer to physical practices such as the derivative-based global sensitivity measures (DGSM). However, numerical simulators rarely provide the output gradient and DGSM estimation is not directly possible. To address this limitation, we propose to estimate the DGSMs using a Gaussian process metamodel (GPM) which approximates the simulator. Based on this GPM, we propose two DGSM estimators: a plug-in one defined by the DGSM of the GPM predictor and another one defined by the expectation of the DGSM associated to the full-GPM. The latter is equal to the first one completed by a variance term and can be accompanied by a confidence interval. For Gaussian kernel and uniform input laws, analytical formula are given for both DGSM estimators. For all other situations, Monte-Carlo methods for the expectation approximations are proposed: a propagative version of the Gibbs sampler and a chi-square approximation. Moreover, a significance test for the full-GPM based estimator is proposed for screening. The convergence of the two GPM-based DGSM estimators and the Monte-Carlo approaches are compared on analytical tests. Finally, we apply our work to an environmental application.Les phénomènes physiques sont souvent étudiés au moyen de simulateurs numériques. Ces codes de calcul sont fonction de paramètres d'entrée incertains et une analyse de sensibilité globale (ASG) peut être réalisée afin d'identifier leurs impacts sur les sorties. Les indices de Sobol, basés sur une décomposition de la variance des sorties, sont communément utilisés en ASG quantitative. Depuis plusieurs années, d'autres outils sont étudiés, plus proches du sens physique, tels que les mesures de sensibilité globale basées sur les dérivées (DGSM). Cependant, les simulateurs numériques fournissent rarement le gradient de sortie et l'estimation des DGSMs n'est pas possible directement. Pour pallier cette limitation, nous proposons d'estimer les DGSMs à partir d'un métamodèle par processus gaussien (MPG) approximant le simulateur. Au moyen de ce MPG, nous proposons deux estimateurs : un de type plug-in défini par le DGSM du prédicteur MPG, et un autre défini par l'espérance du DGSM associé à l'ensemble du MPG. Ce dernier est égal au premier complété par un terme de variance et peut s'accompagner d'un intervalle de confiance. Pour des noyaux gaussiens et des lois uniformes sur les entrées, des formules analytiques sont données pour les deux estimateurs. Pour les autres cas, des méthodes de Monte-Carlo pour l'approximation des espérances sont proposées : une version propagative de l'échantillonneur de Gibbs et une approximation par chi-deux. De plus, un test de significativité pour le second estimateur est proposé en criblage. La convergence des deux estimateurs et les approches par Monte-Carlo sont comparées sur des cas analytiques. Pour finir, nous appliquons nos travaux à une application environnementale

A scalable problem to benchmark robust multidisciplinary design optimization techniques

A scalable problem to benchmark robust multidisciplinary design optimization algorithms (RMDO) is proposed. This allows the user to choose the number of disciplines, the dimensions of the coupling and design variables and the extent of the feasible domain. After a description of the mathematical background, a deterministic version of the scalable problem is defined and the conditions on the existence and uniqueness of the solution are given. Then, this deterministic scalable problem is made uncertain by adding random variables to the coupling equations. Under classical assumptions, the existence and uniqueness of the solution of this RMDO problem is guaranteed. This solution can be easily computed with a quadratic programming algorithm and serves as a reference to assess the performances of RMDO algorithms. This scalable problem has been implemented in the open source software GEMSEO and tested with two techniques of statistics estimation: Monte-Carlo sampling and Taylor polynomials