arXiv subjects

Bertrand Iooss

Publications and source records attributed to Bertrand Iooss.

At least 37 records · Page 2Linked to original sources

arXiv2019

Shapley effects for sensitivity analysis with correlated inputs: comparisons with Sobol' indices, numerical estimation and applications

The global sensitivity analysis of a numerical model aims to quantify, by means of sensitivity indices estimate, the contributions of each uncertain input variable to the model output uncertainty. The so-called Sobol' indices, which are based on the functional variance analysis, present a difficult interpretation in the presence of statistical dependence between inputs. The Shapley effect was recently introduced to overcome this problem as they allocate the mutual contribution (due to correlation and interaction) of a group of inputs to each individual input within the group.In this paper, using several new analytical results, we study the effects of linear correlation between some Gaussian input variables on Shapley effects, and compare these effects to classical first-order and total Sobol' indices.This illustrates the interest, in terms of sensitivity analysis setting and interpretation, of the Shapley effects in the case of dependent inputs. For the practical issue of computationally demanding computer models, we show that the substitution of the original model by a metamodel (here, kriging) makes it possible to estimate these indices with precision at a reasonable computational cost.

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arXiv2019

Background Error Covariance Iterative Updating with Invariant Observation Measures for Data Assimilation

In order to leverage the information embedded in the background state and observations, covariance matrices modelling is a pivotal point in data assimilation algorithms. These matrices are often estimated from an ensemble of observations or forecast differences. Nevertheless, for many industrial applications the modelling still remains empirical based on some form of expertise and physical constraints enforcement in the absence of historical observations or predictions. We have developed two novel robust adaptive assimilation methods named CUTE (Covariance Updating iTerativE) and PUB (Partially Updating BLUE). These two non-parametric methods are based on different optimization objectives, both capable of sequentially adapting background error covariance matrices in order to improve assimilation results under the assumption of a good knowledge of the observation error covariances. We have compared these two methods with the standard approach using a misspecified background matrix in a shallow water twin experiments framework with a linear observation operator. Numerical experiments have shown that the proposed methods bear a real advantage both in terms of posterior error correlation identification and assimilation accuracy.

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arXiv2019

Optimal Uncertainty Quantification of a risk measurement from a thermal-hydraulic code using Canonical Moments

We study an industrial computer code related to nuclear safety. A major topic of interest is to assess the uncertainties tainting the results of a computer simulation. In this work we gain robustness on the quantification of a risk measurement by accounting for all sources of uncertainties tainting the inputs of a computer code. To that extent, we evaluate the maximum quantile over a class of distributions defined only by constraints on their moments. Two options are available when dealing with such complex optimization problems: one can either optimize under constraints; or preferably, one should reformulate the objective function. We identify a well suited parameterization to compute the optimal quantile based on the theory of canonical moments. It allows an effective, free of constraints, optimization.

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arXiv2019

Large scale in transit computation of quantiles for ensemble runs

The classical approach for quantiles computation requires availability of the full sample before ranking it. In uncertainty quantification of numerical simulation models, this approach is not suitable at exascale as large ensembles of simulation runs would need to gather a prohibitively large amount of data. This problem is solved thanks to an on-the-fly and iterative approach based on the Robbins-Monro algorithm. This approach relies on Melissa, a file avoiding, adaptive, fault-tolerant and elastic framework. On a validation case producing 11 TB of data, which consists in 3000 fluid dynamics parallel simulations on a 6M cell mesh, it allows on-line computation of spatio-temporal maps of percentiles.

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arXiv2018

Advanced methodology for uncertainty propagation in computer experiments with large number of inputs

In the framework of the estimation of safety margins in nuclear accident analysis, a quantitative assessment of the uncertainties tainting the results of computer simulations is essential. Accurate uncertainty propagation (estimation of high probabilities or quantiles) and quantitative sensitivity analysis may call for several thousand of code simulations. Complex computer codes, as the ones used in thermal-hydraulic accident scenario simulations, are often too cpu-time expensive to be directly used to perform these studies. A solution consists in replacing the computer model by a cpu inexpensive mathematical function, called a metamodel, built from a reduced number of code simulations. However, in case of high dimensional experiments (with typically several tens of inputs), the metamodel building process remains difficult. To face this limitation, we propose a methodology which combines several advanced statistical tools: initial space-filling design, screening to identify the non-influential inputs, Gaussian process (Gp) metamodel building with the group of influential inputs as explanatory variables. The residual effect of the group of non-influential inputs is captured by another Gp metamodel. Then, the resulting joint Gp metamodel is used to accurately estimate Sobol' sensitivity indices and high quantiles (here $95\%$-quantile).The efficiency of the methodology to deal with a large number of inputs and reduce the calculation budget is illustrated on a thermal-hydraulic calculation case simulating with the CATHARE2 code a Loss Of Coolant Accident scenario in a Pressurized Water Reactor. A predictive Gp metamodel is built with only a few hundred of code simulations and allows the calculation of the Sobol' sensitivity indices. This Gp also provides a more accurate estimation of the 95%-quantile and associated confidence interval than the empirical approach, at equal calculation budget. Moreover, on this test case, the joint Gp approach outperforms the simple Gp.

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arXiv2018

Optimal Uncertainty Quantification on moment class using canonical moments

We gain robustness on the quantification of a risk measurement by accounting for all sources of uncertainties tainting the inputs of a computer code. We evaluate the maximum quantile over a class of distributions defined only by constraints on their moments. The methodology is based on the theory of canonical moments that appears to be a well-suited framework for practical optimization.

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arXiv2017

Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes

A functional risk curve gives the probability of an undesirable event as a function of the value of a critical parameter of a considered physical system. In several applicative situations, this curve is built using phenomenological numerical models which simulate complex physical phenomena. To avoid cpu-time expensive numerical models, we propose to use Gaussian process regression to build functional risk curves. An algorithm is given to provide confidence bounds due to this approximation. Two methods of global sensitivity analysis of the models' random input parameters on the functional risk curve are also studied. In particular, the PLI sensitivity indices allow to understand the effect of misjudgment on the input parameters' probability density functions.

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arXiv2017

Sensitivity analysis using perturbed-law based indices for quantiles and application to an industrial case

In this paper, we present perturbed law-based sensitivity indices and how to adapt them for quantile-oriented sensitivity analysis. We exhibit a simple way to compute these indices in practice using an importance sampling estimator for quantiles. Some useful asymptotic results about this estimator are also provided. Finally, we apply this method to the study of a numerical model which simulates the behaviour of a component in a hydraulic system in case of severe transient solicitations. The sensitivity analysis is used to assess the impact of epistemic uncertainties about some physical parameters on the output of the model.

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arXiv2017

Probabilistic risk bounds for the characterization of radiological contamination

The radiological characterization of contaminated elements (walls, grounds, objects) from nuclear facilities often suffers from a too small number of measurements. In order to determine risk prediction bounds on the level of contamination, some classic statistical methods may then reveal unsuited as they rely upon strong assumptions (e.g. that the underlying distribution is Gaussian) which cannot be checked. Considering that a set of measurements or their average value arise from a Gaussian distribution can sometimes lead to erroneous conclusion, possibly underconservative. This paper presents several alternative statistical approaches which are based on much weaker hypotheses than Gaussianity. They result from general probabilistic inequalities and order-statistics based formula. Given a data sample, these inequalities make it possible to derive prediction intervals for a random variable, which can be directly interpreted as probabilistic risk bounds. For the sake of validation, they are first applied to synthetic data samples generated from several known theoretical distributions. In a second time, the proposed methods are applied to two data sets obtained from real radiological contamination measurements.

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arXiv2017

An efficient methodology for the analysis and modeling of computer experiments with large number of inputs

Complex computer codes are often too time expensive to be directly used to perform uncertainty, sensitivity, optimization and robustness analyses. A widely accepted method to circumvent this problem consists in replacing cpu-time expensive computer models by cpu inexpensive mathematical functions, called metamodels. For example, the Gaussian process (Gp) model has shown strong capabilities to solve practical problems , often involving several interlinked issues. However, in case of high dimensional experiments (with typically several tens of inputs), the Gp metamodel building process remains difficult, even unfeasible, and application of variable selection techniques cannot be avoided. In this paper, we present a general methodology allowing to build a Gp metamodel with large number of inputs in a very efficient manner. While our work focused on the Gp metamodel, its principles are fully generic and can be applied to any types of metamodel. The objective is twofold: estimating from a minimal number of computer experiments a highly predictive metamodel. This methodology is successfully applied on an industrial computer code.

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arXiv2016

Poincaré inequalities on intervals -- application to sensitivity analysis

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kind of sensitivity indices are linked by a Poincaré type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper to compute. This allows performing a low-cost screening of unessential variables. The efficiency of this screening then highly depends on the accuracy of the upper bounds in Poincaré inequalities. The novelty in the questions concern the wide range of probability distributions involved, which are often truncated on intervals. After providing an overview of the existing knowledge and techniques, we add some theory about Poincaré constants on intervals, with improvements for symmetric intervals. Then we exploit the spectral interpretation for computing exact value of Poincaré constants of any admissible distribution on a given interval. We give semi-analytical results for some frequent distributions (truncated exponential, triangular, truncated normal), and present a numerical method in the general case. Finally, an application is made to a hydrological problem, showing the benefits of the new results in Poincaré inequalities to sensitivity analysis.

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arXiv2016

Stochastic simulators based optimization by Gaussian process metamodels -- Application to maintenance investments planning issues

This paper deals with the optimization of industrial asset management strategies, whose profitability is characterized by the Net Present Value (NPV) indicator which is assessed by a Monte Carlo simulator. The developed method consists in building a metamodel of this stochastic simulator, allowing to get, for a given model input, the NPV probability distribution without running the simulator. The present work is concentrated on the emulation of the quantile function of the stochastic simulator by interpolating well chosen basis functions and metamodeling their coefficients (using the Gaussian process metamodel). This quantile function metamodel is then used to treat a problem of strategy maintenance optimization (four systems installed on different plants), in order to optimize an NPV quantile. Using the Gaussian process framework, an adaptive design method (called QFEI) is defined by extending in our case the well known EGO algorithm. This allows to obtain an "optimal" solution using a small number of simulator runs.

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arXiv2016

Model Assisted Probability of Detection curves: New statistical tools and progressive methodology

The Probability Of Detection (POD) curve is a standard tool in several industries to evaluate the performance of Non Destructive Testing (NDT) procedures for the detection of harmful defects for the inspected structure. Due to new capabilities of NDT process numerical simulation , Model Assisted Probability of Detection (MAPOD) approaches have also been recently developed. In this paper, a generic and progressive MAPOD methodology is proposed. Limits and assumptions of the classical methods are enlightened, while new metamodel-based methods are proposed. They allow to access to relevant information based on sensitivity analysis of MAPOD inputs. Applications are performed on Eddy Current Non Destructive Examination numerical data.

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arXiv2016

Bregman superquantiles. Estimation methods and applications

In this work, we extend some quantities introduced in "Optimization of conditional value-at-risk" of R.T Rockafellar and S. Uryasev to the case where the proximity between real numbers is measured by using a Bregman divergence. This leads to the definition of the Bregman superquantile. Axioms of a coherent measure of risk discussed in "Coherent approches to risk in optimization under uncertainty" of R.T Rockafellar are studied in the case of Bregman superquantile. Furthermore, we deal with asymptotic properties of a Monte Carlo estimator of the Bregman superquantile.

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arXiv2015

Stochastic simulators based optimization by Gaussian process metamodels - Application to maintenance investments planning issues

This paper deals with the construction of a metamodel (i.e. a simplified mathematical model) for a stochastic computer code (also called stochastic numerical model or stochastic simulator), where stochastic means that the code maps the realization of a random variable. The goal is to get, for a given model input, the main information about the output probability distribution by using this metamodel and without running the computer code. In practical applications, such a metamodel enables one to have estimations of every possible random variable properties, such as the expectation, the probability of exceeding a threshold or any quantile. The present work is concentrated on the emulation of the quantile function of the stochastic simulator by interpolating well chosen basis function and metamodeling their coefficients (using the Gaussian process metamodel). This quantile function metamodel is then used to treat a simple optimization strategy maintenance problem using a stochastic code, in order to optimize the quantile of an economic indicator. Using the Gaussian process framework, an adaptive design method (called QFEI) is defined by extending in our case the well known EGO algorithm. This allows to obtain an "optimal" solution using a small number of simulator runs.

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arXiv2015

Derivative based global sensitivity measures

The method of derivative based global sensitivity measures (DGSM) has recently become popular among practitioners. It has a strong link with the Morris screening method and Sobol' sensitivity indices and has several advantages over them. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is generally much lower than that for estimation of Sobol' sensitivity indices. This paper presents a survey of recent advances in DGSM concerning lower and upper bounds on the values of Sobol' total sensitivity indices $S\_{i}^{tot}$. Using these bounds it is possible in most cases to get a good practical estimation of the values of $S\_{i}^{tot} $. Several examples are used to illustrate an application of DGSM.

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arXiv2015

Open TURNS: An industrial software for uncertainty quantification in simulation

The needs to assess robust performances for complex systems and to answer tighter regulatory processes (security, safety, environmental control, and health impacts, etc.) have led to the emergence of a new industrial simulation challenge: to take uncertainties into account when dealing with complex numerical simulation frameworks. Therefore, a generic methodology has emerged from the joint effort of several industrial companies and academic institutions. EDF R&D, Airbus Group and Phimeca Engineering started a collaboration at the beginning of 2005, joined by IMACS in 2014, for the development of an Open Source software platform dedicated to uncertainty propagation by probabilistic methods, named OpenTURNS for Open source Treatment of Uncertainty, Risk 'N Statistics. OpenTURNS addresses the specific industrial challenges attached to uncertainties, which are transparency, genericity, modularity and multi-accessibility. This paper focuses on OpenTURNS and presents its main features: openTURNS is an open source software under the LGPL license, that presents itself as a C++ library and a Python TUI, and which works under Linux and Windows environment. All the methodological tools are described in the different sections of this paper: uncertainty quantification, uncertainty propagation, sensitivity analysis and metamodeling. A section also explains the generic wrappers way to link openTURNS to any external code. The paper illustrates as much as possible the methodological tools on an educational example that simulates the height of a river and compares it to the height of a dyke that protects industrial facilities. At last, it gives an overview of the main developments planned for the next few years.

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arXiv2014

A review on global sensitivity analysis methods

This chapter makes a review, in a complete methodological framework, of various global sensitivity analysis methods of model output. Numerous statistical and probabilistic tools (regression, smoothing, tests, statistical learning, Monte Carlo, \ldots) aim at determining the model input variables which mostly contribute to an interest quantity depending on model output. This quantity can be for instance the variance of an output variable. Three kinds of methods are distinguished: the screening (coarse sorting of the most influential inputs among a large number), the measures of importance (quantitative sensitivity indices) and the deep exploration of the model behaviour (measuring the effects of inputs on their all variation range). A progressive application methodology is illustrated on a scholar application. A synthesis is given to place every method according to several axes, mainly the cost in number of model evaluations, the model complexity and the nature of brought information.

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