arXiv subjects

Charlie Sire

Publications and source records attributed to Charlie Sire.

6 recordsLinked to original sources

arXiv2026

Uncertainty Quantification of Spline Predictors on Compact Riemannian Manifolds

To predict smooth physical phenomena from observations, spline interpolation provides an interpretable framework by minimizing an energy functional associated with the Laplacian operator. This work proposes a methodology to construct a spline predictor on a compact Riemannian manifold, while quantifying the uncertainty inherent in the classical deterministic solution. Our approach leverages the equivalence between spline interpolation and universal kriging with a specific covariance kernel. By adopting a Gaussian random field framework, we generate stochastic simulations that reflect prediction uncertainty. However, on compact manifolds, the covariance kernel depends on the generally unknown spectrum of the Laplace-Beltrami operator. To address this, we introduce a finite element approximation based on a triangulation of the manifold. This leads to the use of intrinsic Gaussian Markov Random Fields (GMRF) and allows for the incorporation of anisotropies through local modifications of the Riemannian metric. The method is validated using a temperature study on a sphere, where the operator's spectrum is known, and is further extended to a test case on a cylindrical surface.

stat.CO↗

arXiv2025

Spline Interpolation on Compact Riemannian Manifolds

Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with smooth variations and local anisotropies via domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach.

stat.CO↗

arXiv2024

Bayesian Calibration for Prediction in a Multi-Output Transposition Context

Numerical simulations are widely used to predict the behavior of physical systems, with Bayesian approaches being particularly well suited for this purpose. However, experimental observations are necessary to calibrate certain simulator parameters for the prediction. In this work, we use a multi-output simulator to predict all its outputs, including those that have never been experimentally observed. This situation is referred to as the transposition context. To accurately quantify the discrepancy between model outputs and real data in this context, conventional methods cannot be applied, and the Bayesian calibration must be augmented by incorporating a joint model error across all outputs. To achieve this, the proposed method is to consider additional numerical input parameters within a hierarchical Bayesian model, which includes hyperparameters for the prior distribution of the calibration variables. This approach is applied on a computer code with three outputs that models the Taylor cylinder impact test with a small number of observations. The outputs are considered as the observed variables one at a time, to work with three different transposition situations. The proposed method is compared with other approaches that embed model errors to demonstrate the significance of the hierarchical formulation.

stat.ME↗

arXiv2023

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↗

arXiv2023

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 Vorono\"i 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↗

arXiv2022

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.

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