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1. Global optimization using Sobol indices

Author

Janon, Alexandre

Subjects
Mathematics - Optimization and Control, Statistics - Applications, Statistics - Computation
Abstract

We propose and assess a new global (derivative-free) optimization algorithm, inspired by the LIPO algorithm, which uses variance-based sensitivity analysis (Sobol indices) to reduce the number of calls to the objective function. This method should be efficient to optimize costly functions satisfying the sparsity-of-effects principle.

Published
2019

2. Global sensitivity analysis for the boundary control of an open channel

Author

Janon, Alexandre, Nodet, Maëlle, Prieur, Christophe, and Prieur, Clémentine

Subjects
Mathematics - Analysis of PDEs, Mathematics - Optimization and Control, Mathematics - Statistics Theory, Statistics - Applications
Abstract

The goal of this paper is to solve the global sensitivity analysis for a particular control problem. More precisely, the boundary control problem of an open-water channel is considered, where the boundary conditions are defined by the position of a down stream overflow gate and an upper stream underflow gate. The dynamics of the water depth and of the water velocity are described by the Shallow Water equations, taking into account the bottom and friction slopes. Since some physical parameters are unknown, a stabilizing boundary control is first computed for their nominal values, and then a sensitivity anal-ysis is performed to measure the impact of the uncertainty in the parameters on a given to-be-controlled output. The unknown physical parameters are de-scribed by some probability distribution functions. Numerical simulations are performed to measure the first-order and total sensitivity indices.

Published
2014

3. On the consistency of Sobol indices with respect to stochastic ordering of model parameters

Author

Cousin, Areski, Janon, Alexandre, Maume-Deschamps, Véronique, and Niang, Ibrahima

Subjects
Mathematics - Statistics Theory, Mathematics - Probability
Abstract

In the past decade, Sobol's variance decomposition have been used as a tool - among others - in risk management. We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol's indices in uncertainty quantification, as one indicator among others.

Published
2014

4. Randomized pick-freeze for sparse Sobol indices estimation in high dimension

Author

De Castro, Yohann and Janon, Alexandre

Subjects
Statistics - Computation, Mathematics - Statistics Theory, Statistics - Applications, Statistics - Methodology
Abstract

This article investigates a new procedure to estimate the influence of each variable of a given function defined on a high-dimensional space. More precisely, we are concerned with describing a function of a large number $p$ of parameters that depends only on a small number $s$ of them. Our proposed method is an unconstrained $\ell_{1}$-minimization based on the Sobol's method. We prove that, with only $\mathcal O(s\log p)$ evaluations of $f$, one can find which are the relevant parameters.

Published
2014

5. Pick and freeze estimation of sensitivity indices for models with dependent and dynamic input processes

Author

Grandjacques, Mathilde, Janon, Alexandre, Delinchant, Benoit, and Adrot, Olivier

Subjects
Statistics - Applications
Abstract

This paper addresses sensitivity analysis for dynamic models, linking dependent inputs to observed outputs. The usual method to estimate Sobol indices are based on the independence of input variables. We present a method to overpass this constraint when inputs are Gaussian processes of high dimension in a time related framework. Our proposition leads to a generalization of Sobol indices when inputs are both dependant and dynamic. The method of estimation is a modification of the Pick and Freeze simulation scheme. First we study the general Gaussian cases and secondly we detail the case of stationary models. We then apply the results to an example of heat exchanges inside a building.

Published
2014

6. Sensitivity analysis for multidimensional and functional outputs

Author

Gamboa, Fabrice, Janon, Alexandre, Klein, Thierry, and Lagnoux, Agnès

Subjects
Statistics - Applications, Mathematics - Statistics Theory, Statistics - Methodology
Abstract

Let $X:=(X_1, \ldots, X_p)$ be random objects (the inputs), defined on some probability space $(\Omega,{\mathcal{F}}, \mathbb P)$ and valued in some measurable space $E=E_1\times\ldots \times E_p$. Further, let $Y:=Y = f(X_1, \ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\mathbb R$ ), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme., Comment: Fixed missing references

Published
2013

7. Goal-oriented error estimation for the reduced basis method, with application to sensitivity analysis

Author

Janon, Alexandre, Nodet, Maëlle, and Prieur, Clémentine

Subjects
Mathematics - Analysis of PDEs, Mathematics - Numerical Analysis, Statistics - Computation
Abstract

The reduced basis method is a powerful model reduction technique designed to speed up the computation of multiple numerical solutions of parametrized partial differential equations. We consider a quantity of interest, which is a linear functional of the PDE solution. A new probabilistic error bound for the reduced model is proposed. It is efficiently and explicitly computable, and we show on different examples that this error bound is sharper than existing ones. We include application of our work to sensitivity analysis studies.

Published
2013

8. Asymptotic normality and efficiency of two Sobol index estimators

Author

Janon, Alexandre, Klein, Thierry, Lagnoux-Renaudie, Agnes, Nodet, Maëlle, and Prieur, Clémentine

Subjects
Mathematics - Statistics Theory, Statistics - Applications, Statistics - Computation
Abstract

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs: we present two estimators and state a central limit theorem for each. We show that one of these estimators has an optimal asymptotic variance. We also generalize our results to the case where the true output is not observable, and is replaced by a noisy version.

Published
2013

9. Statistical inference for Sobol pick freeze Monte Carlo method

Author

Gamboa, Fabrice, Janon, Alexandre, Klein, Thierry, Lagnoux-Renaudie, Agnes, and Prieur, Clémentine

Subjects
Statistics - Applications, Mathematics - Statistics Theory, Statistics - Methodology
Abstract

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs. We study asymptotic and non-asymptotic properties of two estimators of Sobol indices. These properties are applied to significance tests and estimation by confidence intervals.

Published
2013

10. Multifidelity variance reduction for pick-freeze Sobol index estimation

Author

Janon, Alexandre

Subjects
Mathematics - Statistics Theory, Statistics - Applications, Statistics - Computation
Abstract

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index, which can be estimated using a large sample of evaluations of the output. We propose a variance reduction technique, based on the availability of a fast approximation of the output, which can enable significant computational savings when the output is costly to evaluate.

Published
2013

11. Sensitivity indices for multivariate outputs

Author

Gamboa, Fabrice, Janon, Alexandre, Klein, Thierry, and Lagnoux, Agnès

Subjects
Statistics - Applications, Statistics - Methodology
Abstract

We define and study a generalization of Sobol sensitivity indices for the case of a vector output.

Published
2013

13. Certified metamodels for sensitivity indices estimation

Author

Janon, Alexandre, Nodet, Maëlle, and Prieur, Clémentine

Subjects
Mathematics - Analysis of PDEs, Mathematics - Statistics Theory
Abstract

Global sensitivity analysis of a numerical code, more specifically estimation of Sobol indices associated with input variables, generally requires a large number of model runs. When those demand too much computation time, it is necessary to use a reduced model (metamodel) to perform sensitivity analysis, whose outputs are numerically close to the ones of the original model, while being much faster to run. In this case, estimated indices are subject to two kinds of errors: sampling error, caused by the computation of the integrals appearing in the definition of the Sobol indices by a Monte-Carlo method, and metamodel error, caused by the replacement of the original model by the metamodel. In cases where we have certified bounds for the metamodel error, we propose a method to quantify both types of error, and we compute confidence intervals for first-order Sobol indices., Comment: 5e Biennale Fran\c{c}aise des Math\'ematiques Appliqu\'ees (2011)

Published
2011

14. Confidence intervals for sensitivity indices using reduced-basis metamodels

Author

Janon, Alexandre, Nodet, Maëlle, and Prieur, Clémentine

Subjects
Statistics - Computation, Mathematics - Statistics Theory
Abstract

Global sensitivity analysis is often impracticable for complex and time demanding numerical models, as it requires a large number of runs. The reduced-basis approach provides a way to replace the original model by a much faster to run code. In this paper, we are interested in the information loss induced by the approximation on the estimation of sensitivity indices. We present a method to provide a robust error assessment, hence enabling significant time savings without sacrifice on precision and rigourousness. We illustrate our method with an experiment where computation time is divided by a factor of nearly 6. We also give directions on tuning some of the parameters used in our estimation algorithms.

Published
2011

15. Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values

Author

Janon, Alexandre, Nodet, Maëlle, and Prieur, Clémentine

Subjects
Mathematics - Analysis of PDEs
Abstract

We present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. Our numerical experiments show significant computational savings, as well as efficiency of the error bound.

Published
2010

18. Statistical inference for Sobol pick-freeze Monte Carlo method

Author

Gamboa, Fabrice, Janon, Alexandre, Klein, Thierry, Lagnoux-Renaudie, Agnes, Prieur, Clémentine, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), Centre National de la Recherche Scientifique (CNRS), Laboratoire de Sciences Actuarielle et Financière (SAF), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, Mathematics and computing applied to oceanic and atmospheric flows (AIRSEA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
FOS: Computer and information sciences, Statistics and Probability, Monte Carlo method, 0211 other engineering and technologies, Mathematics - Statistics Theory, Statistics Theory (math.ST), 02 engineering and technology, Statistics - Applications, 01 natural sciences, Methodology (stat.ME), 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Statistics, FOS: Mathematics, Statistical inference, Applications (stat.AP), Sensitivity (control systems), 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Statistics - Methodology, Mathematics, Variable (mathematics), [STAT.AP]Statistics [stat]/Applications [stat.AP], 021110 strategic, defence & security studies, Mathematical model, Estimator, Sobol sequence, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Confidence interval, Statistics, Probability and Uncertainty, [STAT.ME]Statistics [stat]/Methodology [stat.ME]
Abstract

International audience; Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs. We study asymptotic and non-asymptotic properties of two estimators of Sobol indices. These properties are applied to significance tests and estimation by confidence intervals.

Published
2015

21. Normalité asymptotique et efficacité dans l'estimation des indices de Sobol

Author

Janon, Alexandre, Klein, Thierry, Lagnoux, Agnès, Nodet, Maëlle, Prieur, Clémentine, Modelling, Observations, Identification for Environmental Sciences (MOISE), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), Centre National de la Recherche Scientifique (CNRS), Laboratoire de Statistique et Probabilités (LSP), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), Société Française de Statistique (SFDS), ANR-09-COSI-0015,COSTA-BRAVA,Analyse de dynamiques spatio-temporelles complexes par réduction de modèle et analyses de sensibilité(2009), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; De nombreux modèles mathématiques font intervenir plusieurs paramètres qui ne sont pas tous connus précisément. L'analyse de sensibilité globale se propose de sélectionner les paramètres d'entrée dont l'incertitude a le plus d'impact sur la variabilité d'une quantité d'intérêt, sortie du modèle. Un des outils statistiques pour quantifier l'influence de chacune des entrées sur la sortie est l'indice de sensibilité de Sobol. Nous considérons l'estimation statistique de cet indice à l'aide d'un nombre fini d'échantillons de sorties du modèle: nous présentons deux estimateurs de cet indice et énonçons un théorème central limite pour chacun d'eux. Nous démontrons que l'un de ces deux estimateurs est optimal en terme de variance asymptotique. Nous généralisons également nos résultats au cas où la vraie sortie du modèle n'est pas observée, mais où seule une version dégradée (bruitée) de la sortie est disponible.

Published
2012

22. Sensitivity analysis and model reduction : application to oceanography

Author

Janon, Alexandre, STAR, ABES, Modelling, Observations, Identification for Environmental Sciences (MOISE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Université de Grenoble, Clémentine Prieur, and Maëlle Nodet

Subjects
[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Model reduction, Calcul scientifique, Statistics, Analyse de sensibilité, Statistiques, Réduction de dimension, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], Sensitivity analysis, Scientific computing
Abstract

Mathematical models seldom represent perfectly the reality of studied systems, due to, for instance, uncertainties on the parameters that define the system. In the context of geophysical fluids modelling, these parameters can be, e.g., the domain geometry, the initial state, the wind stress, the friction or viscosity coefficients. Sensitivity analysis aims at measuring the impact of each input parameter uncertainty on the model solution and, more specifically, to identify the ``sensitive'' parameters (or groups of parameters). Amongst the sensitivity analysis methods, we will focus on the Sobol indices method. The numerical computation of these indices require numerical solutions of the model for a large number of parameters' instances. However, many models (such as typical geophysical fluid models) require a large amount of computational time just to perform one run. In these cases, it is impossible (or at least not practical) to perform the number of runs required to estimate Sobol indices with the required precision. This leads to the replacement of the initial model by a emph{metamodel} (also called emph{response surface} or emph{surrogate model}), which is a model that approximates the original model, while having a significantly smaller time per run, compared to the original model. This thesis focuses on the use of metamodel to compute Sobol indices. More specifically, our main topic is the quantification of the metamodeling impact, in terms of Sobol indices estimation error. We also consider a method of metamodeling which leads to an efficient and rigorous metamodel, which can be used in the geophysical context., Les modèles mathématiques ont pour but de décrire le comportement d'un système. Bien souvent, cette description est imparfaite, notamment en raison des incertitudes sur les paramètres qui définissent le modèle. Dans le contexte de la modélisation des fluides géophysiques, ces paramètres peuvent être par exemple la géométrie du domaine, l'état initial, le forçage par le vent, ou les coefficients de frottement ou de viscosité. L'objet de l'analyse de sensibilité est de mesurer l'impact de l'incertitude attachée à chaque paramètre d'entrée sur la solution du modèle, et, plus particulièrement, identifier les paramètres (ou groupes de paramètres) og sensibles fg. Parmi les différentes méthodes d'analyse de sensibilité, nous privilégierons la méthode reposant sur le calcul des indices de sensibilité de Sobol. Le calcul numérique de ces indices de Sobol nécessite l'obtention des solutions numériques du modèle pour un grand nombre d'instances des paramètres d'entrée. Cependant, dans de nombreux contextes, dont celui des modèles géophysiques, chaque lancement du modèle peut nécessiter un temps de calcul important, ce qui rend inenvisageable, ou tout au moins peu pratique, d'effectuer le nombre de lancements suffisant pour estimer les indices de Sobol avec la précision désirée. Ceci amène à remplacer le modèle initial par un emph{métamodèle} (aussi appelé emph{surface de réponse} ou emph{modèle de substitution}). Il s'agit d'un modèle approchant le modèle numérique de départ, qui nécessite un temps de calcul par lancement nettement diminué par rapport au modèle original. Cette thèse se centre sur l'utilisation d'un métamodèle dans le cadre du calcul des indices de Sobol, plus particulièrement sur la quantification de l'impact du remplacement du modèle par un métamodèle en terme d'erreur d'estimation des indices de Sobol. Nous nous intéressons également à une méthode de construction d'un métamodèle efficace et rigoureux pouvant être utilisé dans le contexte géophysique.

Published
2012

23. Métamodèles base réduite pour l'analyse de sensibilité

Author

Janon, Alexandre, Nodet, Maëlle, Prieur, Clémentine, Modelling, Observations, Identification for Environmental Sciences (MOISE), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), Centre National de la Recherche Scientifique (CNRS), Atelier Modélisation du Cluster Environnement, Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

National audience; L'analyse de sensibilité globale requiert un grand nombre de lancements du code numérique implémentant le modèle considéré. Lorsque ce code est coûteux en temps de calcul, il est judicieux de remplacer le modèle d'origine par un métamodèle approchant le modèle de départ tout en étant plus rapide à simuler numériquement. Nous présenterons la méthode dite de la base réduite qui permet de construire des métamodèles efficaces, tout en disposant d'une majoration rigoureuse de l'erreur commise lors du remplacement du modèle par le métamodèle.

Published
2011

24. Métamodèles certifiés pour l'analyse de sensibilité

Author

Janon, Alexandre, Nodet, Maëlle, Prieur, Clémentine, Modelling, Observations, Identification for Environmental Sciences (MOISE), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), Centre National de la Recherche Scientifique (CNRS), Société Française de Statistique, Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], ComputingMilieux_MISCELLANEOUS
Abstract

National audience

Published
2011

25. Méthodes de quadrature multi-dimensionnelles et décomposition en polynômes de chaos Application à la quantification d'incertitude et aux méthodes stochastiques d'analyse de sensibilité

Author

Janon, Alexandre, Modelling, Observations, Identification for Environmental Sciences (MOISE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), and Centre National de la Recherche Scientifique (CNRS)

Subjects
[STAT.CO]Statistics [stat]/Computation [stat.CO], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Synthèse bibliographique.

Published
2010

28. Reduced-basis solutions of affinely-parametrized linear partial differential equations

Author

Janon, Alexandre, Modelling, Observations, Identification for Environmental Sciences (MOISE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Méthodes d'Analyse Stochastique des Codes et Traitements Numériques (GdR MASCOT-NUM), Centre National de la Recherche Scientifique (CNRS), and LJK

Subjects
reduced basis method, numerical solution, Computer Science::Information Retrieval, partial differential equation, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, MathematicsofComputing_NUMERICALANALYSIS, Mathematics::Metric Geometry, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], error analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This document presents the reduced basis method for the fast many-query computation of solutions of affinely-parametrized elliptic linear partial differential equations.

Published
2010

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