arXiv subjects

Jérôme Morio

Publications and source records attributed to Jérôme Morio.

13 recordsLinked to original sources

arXiv2026

Stochastic Modeling of Composite Interfaces: Sensitivity to Spatial Correlation and Bayesian Identification from Standard Fracture Tests

To enable a numerical handling of uncertainties in composite structures, this work presents a stochastic finite-element framework aimed at improving the reliability assessment of aerospace composites, with particular attention to stiffener debonding. By representing interface variability between laminate parts with spatially correlated random fields, the method aims at considering scattering effect at a higher scale of simulation and testing. A parametric study carried out on standardized Mode I and Mode II fracture tests reveals that the correlation length is the primary driver of observed variability, while the regularity of the covariance kernel has only a marginal impact. To guarantee industrial relevance, we demonstrate that this key parameter can be extracted from experimental fracture data using an Approximate Bayesian Computation approach. The proposed methodology therefore offers a robust route to high-fidelity virtual testing and to the predictive management of uncertainties in the design of damage-tolerant composite airframes.

stat.AP↗

arXiv2026

Control variates for variance-reduced ratio of means estimators

The control variates method is a classical variance reduction technique for Monte Carlo estimators that exploits correlated auxiliary variables without introducing bias. In many applications, the quantity of interest can be expressed as a ratio of expectations. We propose a variance-reduced estimator for such ratios, which applies control variates to both the numerator and the denominator. The control variates coefficients are optimized jointly to minimize the approximated variance of the resulting estimator. This approach guarantees variance reduction and naturally extends to approximate control variates. Simulation studies show significant variance reduction, particularly when correlations between variables and control variates are strong. The practical value of the method is illustrated on multi-fidelity applications: estimating a proportion in an aircraft design use case and a conditional value-at-risk in an electromagnetic dataset.

math.ST↗

arXiv2026

High-dimensional reliability-oriented Shapley effect estimation with Normalizing Flows

This article presents a new estimation scheme for the reliability-oriented Shapley effects when there is a large number of correlated input variables in the model, using a unique sample of failure points. To do so, we first propose a new writing of the reliability-oriented closed Sobol indices involving the marginal densities conditionally to the failure, which may be high-dimensional. Then, we propose to estimate these densities with the available failing samples using Normalizing Flows, powerful tools from generative modeling that enable the estimation of complex high-dimensional densities. In addition, we provide an error estimation procedure relying on the same sample of failing points, which constitutes a new contribution for the estimation of target Shapley effects. Finally, we illustrate our methodology on numerical use-cases, discuss insightful features of our approach and provide prospects for the future.

stat.ME↗

arXiv2026

Efficient Estimation of A-basis and B-Basis Value under Epistemic Uncertainty using Importance Sampling and Control Variates

In aerospace certification and other safety-critical domains, conservative quantile estimation such as A- and B-basis values is essential to guarantee reliability. While these metrics are traditionally derived from experimental campaigns, this work focuses on their estimation using a validated deterministic numerical model. The problem is formulated under mixed aleatory-epistemic uncertainty, accounting for limited material data, finite sampling effects, and surrogate modeling errors. We propose a methodology for estimating conservative design quantiles with statistical guarantees under mixed uncertainties. The proposed method leverages importance sampling and control variates to achieve accurate and efficient estimates within a fixed computational budget. One key point is the surrogate model's role solely as a variance reduction device, which guarantees unbiased and consistent quantile estimation. By explicitly integrating all sources of uncertainty, the proposed framework provides a numerical alternative to estimate A-basis and B-Basis. Furthermore, Sobol-based sensitivity indices are obtained at no additional cost, offering insight into the dominant epistemic sources. Numerical experiments on structural models demonstrate the method's reliability and computational efficiency. In particular, the application to large-scale industrial simulations confirms its suitability for aerospace certification workflows and highlights its relevance for real world engineering environments.

stat.AP↗

arXiv2026

Importance sampling for Sobol' indices estimation

We propose a new importance sampling framework for the estimation and analysis of Sobol' indices. We focus on the estimation of the conditional second-moment quantity underlying these indices, which is the most challenging term to estimate. We show that this quantity, originally defined under a reference input distribution, can be estimated from samples drawn under auxiliary distributions by reweighting the model outputs. We derive the optimal sampling distribution that minimises the asymptotic variance of efficient estimators and demonstrate its impact on estimation. Beyond variance reduction, the framework also supports distributional sensitivity analysis through reverse importance sampling.

math.ST↗

arXiv2025

Variance-reduced extreme value index estimators using control variates in a semi-supervised setting

The estimation of the Extreme Value Index (EVI) is fundamental in extreme value analysis but suffers from high variance due to reliance on only a few extreme observations. We propose a control variates based transfer learning approach in a semi-supervised framework, where a small set of coupled target and source observations is combined with abundant unpaired source data. By expressing the Hill estimator of the target EVI as a ratio of means, we apply approximate control variates to both numerator and denominator, with jointly optimized coefficients that guarantee variance reduction without introducing bias. We show theoretically and through simulations that the asymptotic relative variance reduction of the transferred Hill estimator is proportional to the tail dependence between the target and source variables and independent of their EVI values. Thus, substantial variance reduction can be achieved even without similarity in tail heaviness of the target and source distributions. The proposed approach can be extended to other EVI estimators expressed with ratio of means, as demonstrated on the moment estimator. The practical value of the proposed method is illustrated on multi-fidelity water surge and ice accretion datasets.

stat.ME↗

arXiv2025

Affine invariant interacting Langevin dynamics in Markov chain importance sampling for rare event estimation

This work considers the framework of Markov chain importance sampling~(MCIS), in which one employs a Markov chain Monte Carlo~(MCMC) scheme to sample particles approaching the optimal distribution for importance sampling, prior to estimating the quantity of interest through importance sampling. In rare event estimation, the optimal distribution admits a non-differentiable log-density, thus gradient-based MCMC can only target a smooth approximation of the optimal density. We propose a new gradient-based MCIS scheme for rare event estimation, called affine invariant interacting Langevin dynamics for importance sampling~(ALDI-IS), in which the affine invariant interacting Langevin dynamics~(ALDI) is used to sample particles according to the smoothed zero-variance density. We establish a non-asymptotic error bound when importance sampling is used in conjunction with samples independently and identically distributed according to the smoothed optiaml density to estimate a rare event probability, and an error bound on the sampling bias when a simplified version of ALDI, the unadjusted Langevin algorithm, is used to sample from the smoothed optimal density. We show that the smoothing parameter of the optimal density has a strong influence and exhibits a trade-off between a low importance sampling error and the ease of sampling using ALDI. We perform a numerical study of ALDI-IS and illustrate this trade-off phenomenon on standard rare event estimation test cases.

math.ST↗

arXiv2023

Variational autoencoder with weighted samples for high-dimensional non-parametric adaptive importance sampling

Probability density function estimation with weighted samples is the main foundation of all adaptive importance sampling algorithms. Classically, a target distribution is approximated either by a non-parametric model or within a parametric family. However, these models suffer from the curse of dimensionality or from their lack of flexibility. In this contribution, we suggest to use as the approximating model a distribution parameterised by a variational autoencoder. We extend the existing framework to the case of weighted samples by introducing a new objective function. The flexibility of the obtained family of distributions makes it as expressive as a non-parametric model, and despite the very high number of parameters to estimate, this family is much more efficient in high dimension than the classical Gaussian or Gaussian mixture families. Moreover, in order to add flexibility to the model and to be able to learn multimodal distributions, we consider a learnable prior distribution for the variational autoencoder latent variables. We also introduce a new pre-training procedure for the variational autoencoder to find good starting weights of the neural networks to prevent as much as possible the posterior collapse phenomenon to happen. At last, we explicit how the resulting distribution can be combined with importance sampling, and we exploit the proposed procedure in existing adaptive importance sampling algorithms to draw points from a target distribution and to estimate a rare event probability in high dimension on two multimodal problems.

cs.LG↗

arXiv2022

Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variates

Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they require the knowledge of the values of the expectations to estimate. Motivated by the form of these optimal estimators and some interesting properties, we therefore propose an adaptive algorithm. The general idea is to adaptively update the parameters of the estimators for approaching the optimal ones. We suggest then a quantitative stopping criterion that exploits the trade-off between approaching these optimal parameters and having a sufficient budget left. This left budget is then used to draw a new independent sample from the final sampling distribution, allowing to get unbiased estimators of the expectations. We show how to apply our procedure to sensitivity analysis, by estimating Sobol' indices and quantifying the impact of the input distributions. Finally, realistic test cases show the practical interest of the proposed algorithm, and its significant improvement over estimating the expectations separately.

stat.ME↗

arXiv2022

Shapley effect estimation in reliability-oriented sensitivity analysis with correlated inputs by importance sampling

Reliability-oriented sensitivity analysis aims at combining both reliability and sensitivity analyses by quantifying the influence of each input variable of a numerical model on a quantity of interest related to its failure. In particular, target sensitivity analysis focuses on the occurrence of the failure, and more precisely aims to determine which inputs are more likely to lead to the failure of the system. The Shapley effects are quantitative global sensitivity indices which are able to deal with correlated input variables. They have been recently adapted to the target sensitivity analysis framework. In this article, we investigate two importance-sampling-based estimation schemes of these indices which are more efficient than the existing ones when the failure probability is small. Moreover, an extension to the case where only an i.i.d. input/output N-sample distributed according to the importance sampling auxiliary distribution is proposed. This extension allows to estimate the Shapley effects only with a data set distributed according to the importance sampling auxiliary distribution stemming from a reliability analysis without additional calls to the numerical model. In addition, we study theoretically the absence of bias of some estimators as well as the benefit of importance sampling. We also provide numerical guidelines and finally, realistic test cases show the practical interest of the proposed methods.

math.ST↗

arXiv2022

Optimal projection to improve parametric importance sampling in high dimension

In this paper we propose a dimension-reduction strategy in order to improve the performance of importance sampling in high dimension. The idea is to estimate variance terms in a small number of suitably chosen directions. We first prove that the optimal directions, i.e., the ones that minimize the Kullback--Leibler divergence with the optimal auxiliary density, are the eigenvectors associated to extreme (small or large) eigenvalues of the optimal covariance matrix. We then perform extensive numerical experiments that show that as dimension increases, these directions give estimations which are very close to optimal. Moreover, we show that the estimation remains accurate even when a simple empirical estimator of the covariance matrix is used to estimate these directions. These theoretical and numerical results open the way for different generalizations, in particular the incorporation of such ideas in adaptive importance sampling schemes.

stat.CO↗

arXiv2021

Improvement of the cross-entropy method in high dimension for failure probability estimation through a one-dimensional projection without gradient estimation

Rare event probability estimation is an important topic in reliability analysis. Stochastic methods, such as importance sampling, have been developed to estimate such probabilities but they often fail in high dimension. In this paper, we propose a new cross-entropy-based importance sampling algorithm to improve rare event probability estimation in high dimension. We focus on the cross-entropy method with Gaussian auxiliary distributions and we suggest to update the Gaussian covariance matrix only in a one-dimensional subspace. For that purpose, the main idea is to consider the projection in the one-dimensional subspace spanned by the sample mean vector, which gives an influential direction for the variance estimation. This approach does not require any additional simulation budget compared to the basic cross-entropy algorithm and we show on different numerical test cases that it greatly improves its performance in high dimension.

stat.CO↗

arXiv2020

Variance based sensitivity analysis for Monte Carlo and importance sampling reliability assessment with Gaussian processes

Running a reliability analysis on engineering problems involving complex numerical models can be computationally very expensive, requiring advanced simulation methods to reduce the overall numerical cost. Gaussian process based active learning methods for reliability analysis have emerged as a promising way for reducing this computational cost. The learning phase of these methods consists in building a Gaussian process surrogate model of the performance function and using the uncertainty structure of the Gaussian process to enrich iteratively this surrogate model. For that purpose a learning criterion has to be defined. Then, the estimation of the probability of failure is typically obtained by a classification of a population evaluated on the final surrogate model. Hence, the estimator of the probability of failure holds two different uncertainty sources related to the surrogate model approximation and to the sampling based integration technique. In this paper, we propose a methodology to quantify the sensitivity of the probability of failure estimator to both uncertainty sources. This analysis also enables to control the whole error associated to the failure probability estimate and thus provides an accuracy criterion on the estimation. Thus, an active learning approach integrating this analysis to reduce the main source of error and stopping when the global variability is sufficiently low is introduced. The approach is proposed for both a Monte Carlo based method as well as an importance sampling based method, seeking to improve the estimation of rare event probabilities. Performance of the proposed strategy is then assessed on several examples.

stat.ML↗