Dynamics of a planar Coulomb gas

We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincar{\'e} inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll -- Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.Comment: Minor revision for Annals of Applied Probabilit

The limiting move-to-front search-cost in law of large numbers asymptotic regimes

We explicitly compute the limiting transient distribution of the search-cost in the move-to-front Markov chain when the number of objects tends to infinity, for general families of deterministic or random request rates. Our techniques are based on a "law of large numbers for random partitions," a scaling limit that allows us to exactly compute limiting expectation of empirical functionals of the request probabilities of objects. In particular, we show that the limiting search-cost can be split at an explicit deterministic threshold into one random variable in equilibrium, and a second one related to the initial ordering of the list. Our results ensure the stability of the limiting search-cost under general perturbations of the request probabilities. We provide the description of the limiting transient behavior in several examples where only the stationary regime is known, and discuss the range of validity of our scaling limit.Comment: Published in at http://dx.doi.org/10.1214/09-AAP635 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Approches probabilistes d'un modèle d'interaction singulière et de l'équation de navier-stokes en dimension trois

PARIS-BIUSJ-Thèses (751052125) / SudocPARIS-BIUSJ-Physique recherche (751052113) / SudocSudocFranceF

A random space-time birth particle method for 2d vortex equations with external field

We consider incompressible 2d Navier-Stokes equations in the whole plane with external nonconservative forces fields. The initial data and external field are functions assumed to satisfy only slight integrability properties. We develop a probabilistic interpretation of these equations based on the associated vortex equation, in order to construct a numerical particle method to approximate the solutions. More precisely, we relate the vortex equation with additional term to a nonlinear process with random space-time birth, which provides a probabilistic description of the creation of vorticity. We then introduce interacting particle systems defined for a regularized interaction kernel, whose births are chosen randomly in time and space. By a coupling method, we show that these systems are approximations of the nonlinear process and obtain precise convergence estimates. From this result, we deduce a stochastic numerical particle method to obtain the vorticity and also to recover the velocity field. The results are either pathwise or of weak convergence, depending on the integrability of the data. We illustrate our results with simulations

A scalable online algorithm for passive seismic tomography in underground mines

Understanding and monitoring the seismic responses of rock masses to massive mining are crucial for safety and economic viability of ever larger and deeper underground operations. Seismic monitoring can be used to detect stress variations and hazardous instabilities, but its effectiveness requires accurate estimations of the nonhomogeneous propagation velocity of microseismic waves. While predetermined velocity models are not accurate enough and might bias hypocenter localization, using activesource seismic tomography methods to estimate the velocity field provides limited spatial coverage. Thus, passive seismic tomography using first-arrival traveltimes of mining-induced microseisms (of unknown hypocenters) constitutes a promising tool. However, available methods solving this high-dimensional statistical inverse problem do not scale well with the data set size and cannot easily refine or update estimations with new data. We have developed a novel passive seismic tomography method able to dynamically ylearn the nonhomogeneous velocity field from a streaming of noisy first-arrival times, online (in real time) or from catalogs. We have developed a new Bayesian approach that avoids linearizing the forward problem and allows for general 3D velocity models. This is combined with the use of the stochastic gradient descent (SGD) method, which underlies much of the recent progress in machine learning and provides increasing accuracy at a cost scaling linearly with the data set size. Moreover, we introduce an adaptive variant of SGD based on raypath density, which significantly improves the speed of the algorithm, and we implement a parallel version of our method enabling its systematic use in real applications. These include the design of optimal sensor locations, the dynamic update of velocity estimates in production conditions, and the real-time determination of hypocenters and their uncertainty. Our method's reach and effectiveness are illustrated with simulated seismic data on 3D checkerboards, using synthetic and real acquisition geometries, and on a dense 2D velocity grid.Basal-Conicyt PIA Fellowship AFB 170001 Center for Mathematical Modeling supercomputing infrastructure NLHPC (ECM-02), Conicy