arXiv subjects

Mathieu Le Provost

Publications and source records attributed to Mathieu Le Provost.

7 recordsLinked to original sources

arXiv2025

A friendly introduction to triangular transport

Decision making under uncertainty is a cross-cutting challenge in science and engineering. Most approaches to this challenge employ probabilistic representations of uncertainty. In complicated systems accessible only via data or black-box models, however, these representations are rarely known. We discuss how to characterize and manipulate such representations using triangular transport maps, which approximate any complex probability distribution as a transformation of a simple, well-understood distribution. The particular structure of triangular transport guarantees many desirable mathematical and computational properties that translate well into solving practical problems. Triangular maps are actively used for density estimation, (conditional) generative modelling, Bayesian inference, data assimilation, optimal experimental design, and related tasks. While there is ample literature on the development and theory of triangular transport methods, this manuscript provides a detailed introduction for scientists interested in employing measure transport without assuming a formal mathematical background. We build intuition for the key foundations of triangular transport, discuss many aspects of its practical implementation, and outline the frontiers of this field.

stat.CO↗

arXiv2024

Preserving linear invariants in ensemble filtering methods

Data assimilation combines dynamical models with observations to improve state estimates. Ensemble filters sequentially assimilate observations by updating a set of samples over time, alternating between a forecast and an analysis step. Accurate and robust predictions often require preserving critical invariants such as mass, stoichiometric balance of chemical species, and electrical charge. While modern numerical solvers maintain these invariants, existing invariant-preserving analysis steps are limited to Gaussian settings. Furthermore, they can be incompatible with regularization techniques such as inflation and covariance tapering. In this work, we focus on preserving linear invariants in non-Gaussian filtering problems. Leveraging tools from measure transport theory, we introduce a novel class of nonlinear ensemble filters that preserve any desired linear invariants. Notably, we recover a constrained formulation of the Kalman filter for the special case of the Gaussian setting. We also demonstrate how to combine preserving invariants with regularization techniques in the ensemble Kalman filter. Numerical experiments illustrate the benefits of preserving linear invariants in both ensemble Kalman filters and transport-based nonlinear ensemble filters.

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arXiv2023

An adaptive ensemble filter for heavy-tailed distributions: tuning-free inflation and localization

Heavy tails is a common feature of filtering distributions that results from the nonlinear dynamical and observation processes as well as the uncertainty from physical sensors. In these settings, the Kalman filter and its ensemble version - the ensemble Kalman filter (EnKF) - that have been designed under Gaussian assumptions result in degraded performance. t-distributions are a parametric family of distributions whose tail-heaviness is modulated by a degree of freedom $\nu$. Interestingly, Cauchy and Gaussian distributions correspond to the extreme cases of a t-distribution for $\nu = 1$ and $\nu = \infty$, respectively. Leveraging tools from measure transport (Spantini et al., SIAM Review, 2022), we present a generalization of the EnKF whose prior-to-posterior update leads to exact inference for t-distributions. We demonstrate that this filter is less sensitive to outlying synthetic observations generated by the observation model for small $\nu$. Moreover, it recovers the Kalman filter for $\nu = \infty$. For nonlinear state-space models with heavy-tailed noise, we propose an algorithm to estimate the prior-to-posterior update from samples of joint forecast distribution of the states and observations. We rely on a regularized expectation-maximization (EM) algorithm to estimate the mean, scale matrix, and degree of freedom of heavy-tailed \textit{t}-distributions from limited samples (Finegold and Drton, arXiv preprint, 2014). Leveraging the conditional independence of the joint forecast distribution, we regularize the scale matrix with an $l1$ sparsity-promoting penalization of the log-likelihood at each iteration of the EM algorithm. By sequentially estimating the degree of freedom at each analysis step, our filter can adapt its prior-to-posterior update to the tail-heaviness of the data. We demonstrate the benefits of this new ensemble filter on challenging filtering problems.

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arXiv2023

Bayesian inference of vorticity in unbounded flow from limited pressure measurements

We study the instantaneous inference of an unbounded planar flow from sparse noisy pressure measurements. The true flow field comprises one or more regularized point vortices of various strength and size. We interpret the true flow's measurements with a vortex estimator, also consisting of regularized vortices, and attempt to infer the positions and strengths of this estimator assuming little prior knowledge. The problem often has several possible solutions, many due to a variety of symmetries. To deal with this ill-posedness and to quantify the uncertainty, we develop the vortex estimator in a Bayesian setting. We use Markov-chain Monte Carlo and a Gaussian mixture model to sample and categorize the probable vortex states in the posterior distribution, tailoring the prior to avoid spurious solutions. Through experiments with one or more true vortices, we reveal many aspects of the vortex inference problem. With fewer sensors than states, the estimator infers a manifold of equally-possible states. Using one more sensor than states ensures that no cases of rank deficiency arise. Uncertainty grows rapidly with distance when a vortex lies outside of the vicinity of the sensors. Vortex size cannot be reliably inferred, but the position and strength of a larger vortex can be estimated with a much smaller one. In estimates of multiple vortices their individual signs are discernible because of the non-linear coupling in the pressure. When the true vortex state is inferred from an estimator of fewer vortices, the estimate approximately aggregates the true vortices where possible.

physics.flu-dyn↗

arXiv2022

A low-rank ensemble Kalman filter for elliptic observations

We propose a regularization method for ensemble Kalman filtering (EnKF) with elliptic observation operators. Commonly used EnKF regularization methods suppress state correlations at long distances. For observations described by elliptic partial differential equations, such as the pressure Poisson equation (PPE) in incompressible fluid flows, distance localization cannot be applied, as we cannot disentangle slowly decaying physical interactions from spurious long-range correlations. This is particularly true for the PPE, in which distant vortex elements couple nonlinearly to induce pressure. Instead, these inverse problems have a low effective dimension: low-dimensional projections of the observations strongly inform a low-dimensional subspace of the state space. We derive a low-rank factorization of the Kalman gain based on the spectrum of the Jacobian of the observation operator. The identified eigenvectors generalize the source and target modes of the multipole expansion, independently of the underlying spatial distribution of the problem. Given rapid spectral decay, inference can be performed in the low-dimensional subspace spanned by the dominant eigenvectors. This low-rank EnKF is assessed on dynamical systems with Poisson observation operators, where we seek to estimate the positions and strengths of point singularities over time from potential or pressure observations. We also comment on the broader applicability of this approach to elliptic inverse problems outside the context of filtering.

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arXiv2020

The mean transport of inertial particles in viscous streaming flows

Viscous streaming has emerged as an effective method to transport, trap, and cluster inertial particles in a fluid. Previous work has shown that this transport is well described by the Maxey-Riley equation augmented with a term representing Saffman lift. However, in its straightforward application to viscous streaming flows, the equation suffers from severe numerical stiffness due to the wide disparity between the time scales of viscous response, oscillation period, and slow mean transport, posing a severe challenge for drawing physical insight on mean particle trajectories. In this work, we develop equations that directly govern the mean transport of particles in oscillatory viscous flows. The derivation of these equations relies on a combination of three key techniques. In the first, we develop an inertial particle velocity field via a small Stokes number expansion of the particle's deviation from that of the fluid. This expansion clearly reveals the primary importance of Faxén correction and Saffman lift in effecting the trapping of particles in streaming cells. Then, we apply Generalized Lagrangian Mean theory to unambiguously decompose the transport into fast and slow scales, and ultimately, develop the Lagrangian mean velocity field to govern mean transport. Finally, we carry out an expansion in small oscillation amplitude to simplify the governing equations and to clarify the hierarchy of first- and second-order influences, and particularly, the crucial role of Stokes drift in the mean transport. We demonstrate the final set of equations on the transport of both fluid and inertial particles in configurations involving one and two weakly oscillating cylinders. Notably, the new equations allow numerical time steps that are $O(10^3)$ larger than the existing approach with little sacrifice in accuracy, allowing more efficient predictions of transport.

physics.flu-dyn↗

arXiv2020

Ensemble-filtered vortex modeling of strongly disturbed aerodynamic flows

The task of dynamic flow estimation is to construct an approximation of an evolving flow---and particularly, its response to disturbances---using measurements from available sensors. Building from previous work by Darakananda et al.~(Phys Rev Fluids 2018), we further develop an ensemble Kalman filter (EnKF) framework for aerodynamic flows based on an ensemble of randomly-perturbed inviscid vortex models of flow about an infinitely-thin plate. In the forecast step, vortex elements in each ensemble member are advected by the flow and new elements are released from each edge of the plate; the elements are aggregated to maintain an efficient representation. The vortex elements and leading edge constraint are corrected in the analysis step by assimilating the surface pressure differences across the plate measured from the truth system. We show that the overall framework can be physically interpreted as a series of adjustments to the position and shape of an elliptical region of uncertainty associated with each vortex element. In this work, we compare the previously-used stochastic EnKF with the ensemble transform Kalman filter (ETKF), which uses a deterministic analysis step. We examine the response of the flat plate at $20^\circ$ in two perturbed flows, with truth data obtained from high-fidelity simulation at Reynolds number 500. In the first case, we apply a sequence of large-amplitude pulses near the leading edge of the plate to mimic flow actuation. In the second, we place the plate in a vortex street wake behind a cylinder. In both cases, we show that the vortex-based framework accurately estimates the pressure distribution and normal force, with no {\em a priori} knowledge of the perturbations. We show that the ETKF is consistently more robust than the stochastic EnKF. Finally, we examine the mapping from measurements to state update in the analysis step through SVD of the Kalman gain.

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