arXiv subjects

Olivier Le Maître

Publications and source records attributed to Olivier Le Maître.

6 recordsLinked to original sources

arXiv2026

Sequential sparse Gaussian process quantile regression

Quantile regression aims to estimate the conditional quantiles of a response variable from observed data. In a Bayesian setting, Gaussian process quantile regression provides uncertainty quantification but faces significant computational challenges due to the nonconjugacy of the asymmetric Laplace likelihood and the cost of posterior inference. We develop a sparse Gaussian process framework in which the quantile function is represented through a reduced set of inducing variables and posterior inference is performed using a Laplace approximation. A decomposition of the predictive uncertainty into conditional-prior and posterior-induced variance components is then exploited to drive two complementary adaptive mechanisms: inducing-input infilling and data acquisition. These mechanisms are combined within a sequential algorithm that allocates computational effort toward the dominant source of predictive uncertainty and adaptively controls model complexity. Numerical experiments on benchmark problems demonstrate the accuracy of the Laplace approximation, the benefits of variance-based inducing-input placement, and the effectiveness of the proposed sequential enrichment strategy compared with predefined data-acquisition strategies.

cs.LG↗

arXiv2026

Covariance-Informed Subspace: an Adaptive Gradient-Free Input Dimension Reduction Method for Bayesian Inference

This paper addresses the challenge of dimension reduction (DR) in Bayesian inference of high-resolution two-or three-dimensional fields, where a priori parametrizations require a large number of terms. The underlying idea is common to state-of-the-art methods in which the parameter space is decomposed into two subspaces, one informed by the likelihood and one constrained by the prior. DR techniques generally use gradient information from the log-likelihood to derive the corresponding subspaces. However, the gradient may be unavailable or expensive to compute accurately, for instance in the case of simulation-based inference. Inspired by approaches based on likelihood-informed subspaces, we develop a new DR method tailored for settings where gradient computation is not feasible. More specifically, we propose a gradient-free indicator for determining whether a direction is informed by the data. This indicator is derived from the posterior-to-prior covariance ratio introduced in Spantini et al. (2015). We show that, in the linear Gaussian case, this indicator combined with an approximate likelihood leads to a better posterior approximation. The method is then extended to nonlinear cases, and strategies to approximate the posterior covariance are detailed. We demonstrate the effectiveness of this DR through two high-dimensional inference problems arising from groundwater and atmospheric applications.

math.NA↗

arXiv2025

Preconditioners based on Voronoi quantizers of random variable coefficients for stochastic elliptic partial differential equations

A preconditioning strategy is proposed for the iterative solve of large numbers of linear systems with parameter-dependent matrix and right-hand side which arise during the computation of solution statistics of stochastic elliptic partial differential equations with random and spatially variable coefficients sampled by Monte Carlo. Building on the assumption that a truncated Karhunen-Loève expansion of a known transform of the random coefficient is available, we introduce a compact approximation of the random coefficient in the form of a Voronoi quantizer. The number of Voronoi cells, each of which is represented by a centroidal coefficient, is set to the prescribed number of preconditioners. Upon sampling the random coefficient, the linear system assembled with a given realization of the coefficient is solved using a Krylov subspace iterative solver with the preconditioner whose centroidal coefficient is the closest to the realization. We consider different ways to define and obtain the centroidal coefficients, and we investigate the properties of the induced preconditioning strategies in terms of average number of solver iterations for sequential simulations, and of load balancing for parallel simulations. Another approach, which is based on deterministic grids on the system of stochastic coordinates of the truncated representation of the random coefficient, is proposed with a stochastic dimension that increases with the number of preconditioners. This approach allows to bypass the need for preliminary computations in order to determine the optimal stochastic dimension of the truncated approximation of the random coefficient for a given number of preconditioners.

math.NA↗

arXiv2025

Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling

This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Lo{è}ve(KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only onthe field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusionproblem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters. A third application to a 2D anisotropic groundwater flow problem illustrates the method on a more complex geometry.

math.NA↗

arXiv2024

Ensemble Data Assimilation for Particle-based Methods

This study presents a novel approach to applying data assimilation techniques for particle-based simulations using the Ensemble Kalman Filter. While data assimilation methods have been effectively applied to Eulerian simulations, their application in Lagrangian solution discretizations has not been properly explored. We introduce two specific methodologies to address this gap. The first methodology employs an intermediary Eulerian transformation that combines a projection with a remeshing process. The second is a purely Lagrangian scheme designed for situations where remeshing is not appropriate. The second is a purely Lagrangian scheme that is applicable when remeshing is not adapted. These methods are evaluated using a one-dimensional advection-diffusion model with periodic boundaries. Performance benchmarks for the one-dimensional scenario are conducted against a grid-based assimilation filter Subsequently, assimilation schemes are applied to a non-linear two-dimensional incompressible flow problem, solved via the Vortex-In-Cell method. The results demonstrate the feasibility of applying these methods in more complex scenarios, highlighting their effectiveness in both the one-dimensional and two-dimensional contexts.

math.NA↗

arXiv2019

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.

math.NA↗