arXiv subjects

Didier Rullière

Publications and source records attributed to Didier Rullière.

15 recordsLinked to original sources

arXiv2026

Optimal Linear Interpolation under Differential Information: application to the prediction of perfect flows

Approximation of functions satisfying partial differential equations (PDEs) is paramount for simulation of physical fluid flows and other problems in physics. Recently, physics-informed machine learning approaches have proven useful as a data-driven complement to numerical models for partial differential equations, bringing faster responses and allowing us to capitalize on past observations. However, their efficiency and convergence depend on the availability of vast training datasets. For sparse observations, Gaussian process regression or Kriging has emerged as a powerful interpolation model, offering principled estimates and uncertainty quantification. Several attempts have been made to condition Gaussian processes on linear PDEs via artificial or collocation observations and kernel design.These methods suffer from scalability issues in higher dimensions and limited generalizability. The aim of this study is to explore the extension of the Kriging predictor in the presence of linear PDE information at a finite number of collocation points. Two approaches are proposed: 1) A collocated co-Kriging with primary observations of the physical field and auxiliary differential observations; 2) A constrained Kriging optimization problem strongly satisfying linear PDE constraints at the points of prediction through a Lagrangian formulation. Numerical experiments are given for ordinary differential equations, 2D harmonic PDEs and an application to perfect flows around a cylinder. This work highlights a trade-off between the computational efficiency of the Lagrange multipliers approach and the strict interpolation of observations.

math.NA↗

arXiv2025

Augmented Quantization: Mixture Models for Risk-Oriented Sensitivity Analysis

A central question in risk analysis is to identify the factors that drive the system toward a specific hazardous outcome, such as the exceedance of a given threshold. When relying on numerical simulators, we propose to study the distribution of the inputs, transformed into uniform variables via their cumulative distributions, conditionally on the occurrence of the hazardous event. To represent this multivariate conditional distribution for sensitivity analysis, we introduce an original quantization approach based on estimating a mixture of Dirac and local uniform distributions. For each marginal of this mixture, a Dirac component indicates a strong influence of the corresponding variable, whereas a uniform component with wide support reflects weak influence. A notable advantage of this method is its ability to identify the regions of the input space that most strongly influence the occurrence of the risk event, while also capturing the joint effects of multiple variables. However, learning mixture models typically relies on likelihood-based methods, which are not well suited to mixtures involving singular or Dirac components. To address this, we propose an \emph{Augmented Quantization} method, a reformulation of the classical quantization problem based on the p-Wasserstein distance, which can be computed in very general distribution spaces. The performance of Augmented Quantization in estimating such mixture models is first demonstrated on analytical toy problems, and then applied to sensitivity analysis of both an analytical function and a practical flooding case study on a section of the Loire River.

stat.AP↗

arXiv2024

FunQuant: A R package to perform quantization in the context of rare events and time-consuming simulations

Quantization summarizes continuous distributions by calculating a discrete approximation. Among the widely adopted methods for data quantization is Lloyd's algorithm, which partitions the space into Voronoï cells, that can be seen as clusters, and constructs a discrete distribution based on their centroids and probabilistic masses. Lloyd's algorithm estimates the optimal centroids in a minimal expected distance sense, but this approach poses significant challenges in scenarios where data evaluation is costly, and relates to rare events. Then, the single cluster associated to no event takes the majority of the probability mass. In this context, a metamodel is required and adapted sampling methods are necessary to increase the precision of the computations on the rare clusters.

stat.CO↗

arXiv2023

Quantizing rare random maps: application to flooding visualization

Visualization is an essential operation when assessing the risk of rare events such as coastal or river floodings. The goal is to display a few prototype events that best represent the probability law of the observed phenomenon, a task known as quantization. It becomes a challenge when data is expensive to generate and critical events are scarce, like extreme natural hazard. In the case of floodings, each event relies on an expensive-to-evaluate hydraulic simulator which takes as inputs offshore meteo-oceanic conditions and dyke breach parameters to compute the water level map. In this article, Lloyd's algorithm, which classically serves to quantize data, is adapted to the context of rare and costly-to-observe events. Low probability is treated through importance sampling, while Functional Principal Component Analysis combined with a Gaussian process deal with the costly hydraulic simulations. The calculated prototype maps represent the probability distribution of the flooding events in a minimal expected distance sense, and each is associated to a probability mass. The method is first validated using a 2D analytical model and then applied to a real coastal flooding scenario. The two sources of error, the metamodel and the importance sampling, are evaluated to quantify the precision of the method.

stat.AP↗

arXiv2021

Estimation of multivariate generalized gamma convolutions through Laguerre expansions

The generalized gamma convolutions class of distributions appeared in Thorin's work while looking for the infinite divisibility of the log-Normal and Pareto distributions. Although these distributions have been extensively studied in the univariate case, the multivariate case and the dependence structures that can arise from it have received little interest in the literature. Furthermore, only one projection procedure for the univariate case was recently constructed, and no estimation procedures are available. By expanding the densities of multivariate generalized gamma convolutions into a tensorized Laguerre basis, we bridge the gap and provide performant estimation procedures for both the univariate and multivariate cases. We provide some insights about performance of these procedures, and a convergent series for the density of multivariate gamma convolutions, which is shown to be more stable than Moschopoulos's and Mathai's univariate series. We furthermore discuss some examples.

math.ST↗

arXiv2021

Properties and comparison of some Kriging sub-model aggregation methods

Kriging is a widely employed technique, in particular for computer experiments, in machine learning or in geostatistics. An important challenge for Kriging is the computational burden when the data set is large. This article focuses on a class of methods aiming at decreasing this computational cost, consisting in aggregating Kriging predictors based on smaller data subsets. It proves that aggregation methods that ignore the covariancebetween sub-models can yield an inconsistent final Kriging prediction. In contrast, a theoretical study of the nested Kriging method shows additional attractive properties for it: First, this predictor is consistent, second it can be interpreted as an exact conditional distribution for a modified process and third, the conditional covariances given the observations can be computed efficiently. This article also includes a theoretical and numerical analysis of how the assignment of the observation points to the sub-models can affect the prediction ability of the aggregated model. Finally, the nested Kriging method is extended to measurement errors and to universal Kriging.

math.ST↗

arXiv2021

Dependence structure estimation using Copula Recursive Trees

We construct the COpula Recursive Tree (CORT) estimator: a flexible, consistent, piecewise linear estimator of a copula, leveraging the patchwork copula formalization and various piecewise constant density estimators. While the patchwork structure imposes a grid, the CORT estimator is data-driven and constructs the (possibly irregular) grid recursively from the data, minimizing a chosen distance on the copula space. The addition of the copula constraints makes usual density estimators unusable, whereas the CORT estimator is only concerned with dependence and guarantees the uniformity of margins. Refinements such as localized dimension reduction and bagging are developed, analyzed, and tested through simulated data.

math.ST↗

arXiv2018

Asymptotic multivariate expectiles

In [16], a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of these multivariate asymptotic expectiles, in the Fr{é}chet attraction domain case, with asymptotic independence, or in the comonotonic case.

q-fin.RM↗

arXiv2017

Nested Kriging predictions for datasets with large number of observations

This work falls within the context of predicting the value of a real function at some input locations given a limited number of observations of this function. The Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such a problem but the method suffers from its computational burden when the number of observation points is large. We introduce in this article nested Kriging predictors which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. Contrarily to some other methods it can be shown that the proposed aggregation method is consistent. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with $10^4$ observations in a 6-dimensional space.

stat.ML↗

arXiv2016

Multivariate extensions of expectiles risk measures

This paper is devoted to the introduction and study of a new family of multivariate elicitable risk measures. We call the obtained vector-valued measures multivariate expectiles. We present the different approaches used to construct our measures. We discuss the coherence properties of these multivariate expectiles. Furthermore, we propose a stochastic approximation tool of these risk measures.

stat.ME↗

arXiv2016

Kriging of financial term-structures

Due to the lack of reliable market information, building financial term-structures may be associated with a significant degree of uncertainty. In this paper, we propose a new term-structure interpolation method that extends classical spline techniques by additionally allowing for quantification of uncertainty. The proposed method is based on a generalization of kriging models with linear equality constraints (market-fit conditions) and shape-preserving conditions such as monotonicity or positivity (no-arbitrage conditions). We define the most likely curve and show how to build confidence bands. The Gaussian process covariance hyper-parameters under the construction constraints are estimated using cross-validation techniques. Based on observed market quotes at different dates, we demonstrate the efficiency of the method by building curves together with confidence intervals for term-structures of OIS discount rates, of zero-coupon swaps rates and of CDS implied default probabilities. We also show how to construct interest-rate surfaces or default probability surfaces by considering time (quotation dates) as an additional dimension.

q-fin.CP↗

arXiv2015

Impact of dependence on some multivariate risk indicators

The minimization of some multivariate risk indicators may be used as an allocation method, as proposed in Cénac et al. [6]. The aim of capital allocation is to choose a point in a simplex, according to a given criterion. In a previous paper [17] we proved that the proposed allocation technique satisfies a set of coherence axioms. In the present one, we study the properties and asymptotic behavior of the allocation for some distribution models. We analyze also the impact of the dependence structure on the allocation using some copulas.

math.PR↗

arXiv2015

A risk management approach to capital allocation

The European insurance sector will soon be faced with the application of Solvency 2 regulation norms. It will create a real change in risk management practices. The ORSA approach of the second pillar makes the capital allocation an important exercise for all insurers and specially for groups. Considering multi-branches firms, capital allocation has to be based on a multivariate risk modeling. Several allocation methods are present in the literature and insurers practices. In this paper, we present a new risk allocation method, we study its coherence using an axiomatic approach, and we try to define what the best allocation choice for an insurance group is.

q-fin.RM↗

arXiv2010

An extension of Davis and Lo's contagion model

The present paper provides a multi-period contagion model in the credit risk field. Our model is an extension of Davis and Lo's infectious default model. We consider an economy of n firms which may default directly or may be infected by other defaulting firms (a domino effect being also possible). The spontaneous default without external influence and the infections are described by not necessarily independent Bernoulli-type random variables. Moreover, several contaminations could be required to infect another firm. In this paper we compute the probability distribution function of the total number of defaults in a dependency context. We also give a simple recursive algorithm to compute this distribution in an exchangeability context. Numerical applications illustrate the impact of exchangeability among direct defaults and among contaminations, on different indicators calculated from the law of the total number of defaults. We then examine the calibration of the model on iTraxx data before and during the crisis. The dynamic feature together with the contagion effect seem to have a significant impact on the model performance, especially during the recent distressed period.

q-fin.RM↗

arXiv2009

Les Générateurs de Scénarios Économiques : quelle utilisation en assurance?

In this paper, we present the principal components of an economic scenario generator (ESG), both for the theoretical design and for practical implementation. The choice of these components should be linked to the ultimate vocation of the economic scenario generator, which can be either a tool for pricing financial products or a tool for projection and risk management. We then develop a study on some performance measure indicators of the ESG as an input for the decision-making process, namely the indicators of stability and bias absence. Finally, a numerical application illustrates the main ideas of the paper.

q-fin.RM↗