Convergence rate analysis of time discretization scheme for confined Lagrangian processes

In this paper, we propose and analyze the convergence of a time-discretization scheme for the motion of a particle when its instantaneous velocity is drifted by the known velocity of the carrying flow, and when the motion is taking into account the collision event with a boundary wall. We propose a symetrized version of the Euler scheme and prove a convergence of order one for the weak error. The regularity analysis of the associated Kolmogorov PDE is obtained by mixed variational and stochastic flow techniques for PDE problem with specular condition

Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain

We consider an incompressible kinetic Fokker Planck equation in the flat torus, which is a simplified version of the Lagrangian stochastic models for turbulent flows introduced by S.B. Pope in the context of computational fluid dynamics. The main difficulties in its treatment arise from a pressure type force that couples the Fokker Planck equation with a Poisson equation which strongly depends on the second order moments of the fluid velocity. In this paper we prove short time existence of analytic solutions in the one-dimensional case, for which we are able to use techniques and functional norms that have been recently introduced in the study of a related singular model.Comment: 32 page

On confined McKean Langevin processes satisfying the mean no-permeability boundary condition

We construct a confined Langevin type process aimed to satisfy a mean no-permeability condition at the boundary. This Langevin process lies in the class of conditional McKean Lagrangian stochastic models studied in [5]. The confined process considered here is a first construction of solutions to the class of Lagrangian stochastic equations with boundary condition issued by the so-called PDF methods for Computational Fluid Dynamics. We prove the well-posedness of the confined system when the state space of the Langevin process is a half-space

A Kramers' type law for the first collision-time of two self-stabilizing diffusions and of their particle approximations

The present work investigates the asymptotic behaviors, at the zero-noise limit, of the first collision-time and first collision-location related to a pair of self-stabilizing diffusions and of their related particle approximations. These asymptotic are considered in a peculiar framework where diffusions evolve in a double-wells landscape where collisions manifest due to the combined action of the Brownian motions driving each diffusion and the action of a selfstabilizing kernel. As the Brownian effects vanish, we show that first collision-times grow at an explicit exponential rate and that the related collision-locations persist at a special point in space. These results are mainly obtained by linking collision phenomena for diffusion processes with exit-time problems of random perturbed dynamical systems, and by exploiting Freidlin-Wentzell's LDP approach to solve these exit-time problems. Importantly, we consider two distinctive situations: the one-dimensional case (where true collisions can be directly studied) and the general multidimensional case (where collisions are required to be enlarged)

Modèles stochastiques lagrangiens de type McKean-Vlasov conditionnel et leur confinement

Dans cette thèse, nous nous intéressons aux aspects théoriques liés à une nouvelle classe d équations différentielles stochastiques appelées modèles stochastiques lagrangiens. Ces modèles ont notamment été introduits pour modéliser les propriétés de particules associées à des écoulements turbulents. Motivés par une application récente de ces modèles dans le cadre du développement de méthodes de raffinement d échelles pour la prévision météorologie, nous considérons également l introduction de conditions aux bords dans les dynamiques. Dans le cadre des équations non linéaires de type McKean, les modèles stochastiques lagrangiens désignent une classe particulière de dynamique non linéaire due à la présence dans les coefficients de distribution conditionnelle. Dans des cas simplifiés, nous établissons le caractère bien posé de ces dynamiques et leur approximation particulaire. Concernant l introduction de conditions aux bords, nous construisons un modèle stochastique confiné pour la condition prototype de non perméabilité en moyenne . Dans le cas où le domaine de confinement est l hyperplan, nous obtenons un résultat d existence et d unicité des dynamiques considérées, et montrons que la condition de bord est satisfaite. Pour des domaines généraux, nous étudions l équation de McKean-Vlasov-Fokker-Planck conditionnelle satisfaite par la loi des systèmes. Nous développons les notions de sur- et sous-solutions maxwelliennes, donnant l existence de bornes gaussiennes sur la solution de l équation.In this thesis, we are interested in theoretical aspects related to a new class of stochastic differential equations referred as Lagrangian stochastic models. These models have been introduced to model the properties of particles issued from turbulent flows. Motivated by a recent application of the Lagrangien models to the context of downscaling methods for weather forecasting, we also consider the introduction of boundary conditions in the dynamics. In the frame of nonlinear McKean equations, the Lagrangian stochastic models provide a particular case of non-linear dynamics due to the presence ion the coefficients of conditional distribution. For simplified cases, we establish a well-posedness result and particle approximations. In concern of boundary conditions, we construct a confined stochastic system within general domain for the prototypic mean no-permeability condition. In the case where the confinement domain is the hyper plane, we obtain existence and uniqueness results for the considered dynamics, and prove the accuracy of our model. For more general domains, we study the conditional McKean-Vlasov-Fokker-Planck equation satisfied by the law of the systems. We develop the notions of super- and sub-Maxwellians solutions, ensuring the existence of Gaussian bounds for the solution of the equation.NICE-BU Sciences (060882101) / SudocSudocFranceF

Local turbulent kinetic energy modelling based on Lagrangian stochastic approach in CFD and application to wind energy

International audienceIn order to better integrate the underlying meteorological processes with the developing technologies within wind energy industry, acquiring relevant statistical information of air motion at a local place, and quantifying the subsequent uncertainty of involved parameters in the models, are fundamental tasks. Special emphasis should be made on the growing interest in energy production forecasting and modelling for wind energy developments that rises the issue of accounting for the uncertain nature of the local forecast. Taking this into consideration, we present the construction of an original stochastic model for the instantaneous turbulent kinetic energy at a given point of a flow, and we validate estimator methods on this model with observational data examples from annual historic of a 10 Hz anemometer wind measurements. <br>More precisely, starting from the viewpoint of Lagrangian modelling of the wind in the boundary layer, we establish a mathematical link between 3D+time computational fluid dynamics (CDF) models for turbulent near-wall flows and stochastic time series models by deriving a family of mean-field dynamics featuring the square norm of the turbulent velocity. Then, by approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called Cox-Ingersoll-Ross stochastic model, which was previously suggested in the literature for modelling wind speed. Remarkably, our stochastic model for the instantaneous turbulent kinetic energy is parametrised by physical constants in CFD, which provides a more direct link between the stochastic nature of the underlying processes and the classical physics behind these phenomena. Nevertheless, these physical parameters may vary with the flow characteristics and situations, so we consider it relevant to adjust their values while constructing the forecasts. Such tuning of the physical parameters was previously proposed in the literature from a deterministic modelling context with RANS equations. We then propose a two-step procedure for the calibration of the parameters: a training stage where we construct a priori distribution for the parameter vector using direct methods and wind measurements, and a stage of refinement of the uncertainty distribution using Bayesian inference combined with Markov Chain Monte Carlo sample techniques. In particular, we show the accuracy of the calibration method and the performance of the calibrated model in predicting the wind distribution through the quantification of uncertainty

Stochastic Lagrangian Method for Downscaling Problems in Computational Fluid Dynamics

This work aims at introducing modelization, theoretical and numerical studies related to a new downscaling technique applied to computational fluid dynamics. Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor. The local model, compatible with the Navier-Stokes equations, is used for the small scale computation (downscaling) of the considered fluid. It is inspired by S.B. Pope’s works on turbulence, and consists in a so-called Langevin system of stochastic differential equations. We introduce this model and exhibit its links with classical RANS models. Well-posedness, as well as mean-field interacting particle approximations and boundary condition issues are addressed. We present the numerical discretization of the stochastic downscaling method and investigate the accuracy of the proposed algorithm on simplified situations