arXiv subjects

Patrick Héas

Publications and source records attributed to Patrick Héas.

12 recordsLinked to original sources

arXiv2025

Non-linear reduced modeling of dynamical systems using kernel methods and low-rank approximation

Reduced modeling of a computationally demanding dynamical system aims at approximating its trajectories, while optimizing the trade-off between accuracy and computational complexity. In this work, we propose to achieve such an approximation by first embedding the trajectories in a reproducing kernel Hilbert space (RKHS), which exhibits appealing approximation and computational capabilities, and then solving the associated reduced model problem. More specifically, we propose a new efficient algorithm for data-driven reduced modeling of non-linear dynamics based on linear approximations in a RKHS. This algorithm takes advantage of the closed-form solution of a low-rank constraint optimization problem while exploiting advantageously kernel-based computations. Reduced modeling with this algorithm reveals a gain in approximation accuracy, as shown by numerical simulations, and in complexity with respect to existing approaches.

stat.ML↗

arXiv2024

Adaptive Reduced Multilevel Splitting

This paper considers the classical problem of sampling with Monte Carlo methods a target rare event distribution defined by a score function that is very expensive to compute. We assume we can build using evaluations of the true score, an approximate surrogate score certified with error bounds. This work proposes a fully adaptive algorithm to sequentially sample surrogate rare event distributions with increasing target levels. An essential contribution consists in sampling at each iteration the surrogate rare event at a critical level corresponding to a specific cost. This cost is related to importance sampling for a target for a given budget. The critical level is calculated solely from the reduced score and its error bound From a practical point of view, sampling the proposal sequence is performed by extending the framework of the popular adaptive multilevel splitting algorithm to the use of score approximations. Numerical experiments evaluate the proposed importance sampling algorithm in terms of computational complexity versus squared error. In particular, we investigate the performance of the algorithm when simulating rare events related to the solution of a parametric PDE, which is approximated by a reduced basis.

stat.CO↗

arXiv2023

Chilled Sampling for Uncertainty Quantification: A Motivation From A Meteorological Inverse Problem

Atmospheric motion vectors (AMVs) extracted from satellite imagery are the only wind observations with good global coverage. They are important features for feeding numerical weather prediction (NWP) models. Several Bayesian models have been proposed to estimate AMVs. Although critical for correct assimilation into NWP models, very few methods provide a thorough characterization of the estimation errors. The difficulty of estimating errors stems from the specificity of the posterior distribution, which is both very high dimensional, and highly ill-conditioned due to a singular likelihood. Motivated by this difficult inverse problem, this work studies the evaluation of the (expected) estimation errors using gradient-based Markov Chain Monte Carlo (MCMC) algorithms. The main contribution is to propose a general strategy, called here chilling, which amounts to sampling a local approximation of the posterior distribution in the neighborhood of a point estimate. From a theoretical point of view, we show that under regularity assumptions, the family of chilled posterior distributions converges in distribution as temperature decreases to an optimal Gaussian approximation at a point estimate given by the Maximum A Posteriori, also known as the Laplace approximation. Chilled sampling therefore provides access to this approximation generally out of reach in such high-dimensional nonlinear contexts. From an empirical perspective, we evaluate the proposed approach based on some quantitative Bayesian criteria. Our numerical simulations are performed on synthetic and real meteorological data. They reveal that not only the proposed chilling exhibits a significant gain in terms of accuracy of the point estimates and of their associated expected errors, but also a substantial acceleration in the convergence speed of the MCMC algorithms.

stat.ME↗

arXiv2022

Entropy minimizing distributions are worst-case optimal importance proposals

Importance sampling of target probability distributions belonging to a given convex class is considered. Motivated by previous results, the cost of importance sampling is quantified using the relative entropy of the target with respect to proposal distributions. Using a reference measure as a reference for cost, we prove under some general conditions that the worst-case optimal proposal is precisely given by the distribution minimizing entropy with respect to the reference within the considered convex class of distributions. The latter conditions are in particular satisfied when the convex class is defined using a push-forward map defining atomless conditional measures. Applications in which the optimal proposal is Gibbsian and can be practically sampled using Monte Carlo methods are discussed.

math.NA↗

arXiv2021

Low-Rank Dynamic Mode Decomposition: An Exact and Tractable Solution

This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algorithms are all sub-optimal. This paper shows that there exists a closed-form solution, which is computed in polynomial time, and characterises the l2-norm of the optimal approximation error. The paper also proposes low-complexity algorithms building reduced models from this optimal solution, based on singular value decomposition or eigen value decomposition. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.

stat.ML↗

arXiv2020

Selecting Reduced Models in the Cross-Entropy Method

This paper deals with the estimation of rare event probabilities using importance sampling (IS), where an optimal proposal distribution is computed with the cross-entropy (CE) method. Although, IS optimized with the CE method leads to an efficient reduction of the estimator variance, this approach remains unaffordable for problems where the repeated evaluation of the score function represents a too intensive computational effort. This is often the case for score functions related to the solution of a partial differential equation (PDE) with random inputs. This work proposes to alleviate computation by the parsimonious use of a hierarchy of score function approximations in the CE optimization process. The score function approximation is obtained by selecting the surrogate of lowest dimensionality, whose accuracy guarantees to pass the current CE optimization stage. The selection relies on certified upper bounds on the error norm. An asymptotic analysis provides some theoretical guarantees on the efficiency and convergence of the proposed algorithm. Numerical results demonstrate the gain brought by the method in the context of pollution alerts and a system modeled by a PDE.

stat.CO↗

arXiv2018

Reduced-Order Modeling Of Hidden Dynamics

The objective of this paper is to investigate how noisy and incomplete observations can be integrated in the process of building a reduced-order model. This problematic arises in many scientific domains where there exists a need for accurate low-order descriptions of highly-complex phenomena, which can not be directly and/or deterministically observed. Within this context, the paper proposes a probabilistic framework for the construction of "POD-Galerkin" reduced-order models. Assuming a hidden Markov chain, the inference integrates the uncertainty of the hidden states relying on their posterior distribution. Simulations show the benefits obtained by exploiting the proposed framework.

stat.ML↗

arXiv2018

Optimal Low-Rank Dynamic Mode Decomposition

Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative domains, e.g. for climate prediction, to study molecular dynamics or micro-electromechanical devices. This low-rank extension takes the form of a non-convex optimization problem. To the best of our knowledge, only sub-optimal algorithms have been proposed in the literature to compute the solution of this problem. In this paper, we prove that there exists a closed-form optimal solution to this problem and design an effective algorithm to compute it based on Singular Value Decomposition (SVD). A toy-example illustrates the gain in performance of the proposed algorithm compared to state-of-the-art techniques.

stat.ML↗

arXiv2017

Reduced Modeling of Unknown Trajectories

This paper deals with model order reduction of parametrical dynamical systems. We consider the specific setup where the distribution of the system's trajectories is unknown but the following two sources of information are available: \textit{(i)} some "rough" prior knowledge on the system's realisations; \textit{(ii)} a set of "incomplete" observations of the system's trajectories. We propose a Bayesian methodological framework to build reduced-order models (ROMs) by exploiting these two sources of information. We emphasise that complementing the prior knowledge with the collected data provably enhances the knowledge of the distribution of the system's trajectories. We then propose an implementation of the proposed methodology based on Monte-Carlo methods. In this context, we show that standard ROM learning techniques, such e.g. Proper Orthogonal Decomposition or Dynamic Mode Decomposition, can be revisited and recast within the probabilistic framework considered in this paper.~We illustrate the performance of the proposed approach by numerical results obtained for a standard geophysical model.

stat.ME↗

arXiv2016

An Efficient Algorithm for Video Super-Resolution Based On a Sequential Model

In this work, we propose a novel procedure for video super-resolution, that is the recovery of a sequence of high-resolution images from its low-resolution counterpart. Our approach is based on a "sequential" model (i.e., each high-resolution frame is supposed to be a displaced version of the preceding one) and considers the use of sparsity-enforcing priors. Both the recovery of the high-resolution images and the motion fields relating them is tackled. This leads to a large-dimensional, non-convex and non-smooth problem. We propose an algorithmic framework to address the latter. Our approach relies on fast gradient evaluation methods and modern optimization techniques for non-differentiable/non-convex problems. Unlike some other previous works, we show that there exists a provably-convergent method with a complexity linear in the problem dimensions. We assess the proposed optimization method on {several video benchmarks and emphasize its good performance with respect to the state of the art.}

cs.CV↗

arXiv2014

Self-similar prior and wavelet bases for hidden incompressible turbulent motion

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.

stat.AP↗

arXiv2007

Supervised learning on graphs of spatio-temporal similarity in satellite image sequences

High resolution satellite image sequences are multidimensional signals composed of spatio-temporal patterns associated to numerous and various phenomena. Bayesian methods have been previously proposed in (Heas and Datcu, 2005) to code the information contained in satellite image sequences in a graph representation using Bayesian methods. Based on such a representation, this paper further presents a supervised learning methodology of semantics associated to spatio-temporal patterns occurring in satellite image sequences. It enables the recognition and the probabilistic retrieval of similar events. Indeed, graphs are attached to statistical models for spatio-temporal processes, which at their turn describe physical changes in the observed scene. Therefore, we adjust a parametric model evaluating similarity types between graph patterns in order to represent user-specific semantics attached to spatio-temporal phenomena. The learning step is performed by the incremental definition of similarity types via user-provided spatio-temporal pattern examples attached to positive or/and negative semantics. From these examples, probabilities are inferred using a Bayesian network and a Dirichlet model. This enables to links user interest to a specific similarity model between graph patterns. According to the current state of learning, semantic posterior probabilities are updated for all possible graph patterns so that similar spatio-temporal phenomena can be recognized and retrieved from the image sequence. Few experiments performed on a multi-spectral SPOT image sequence illustrate the proposed spatio-temporal recognition method.

cs.CV↗