Mireille Bossy - Academia.edu (original) (raw)

Papers by Mireille Bossy

In a collaboration with the french national agency for development of ecol- ogy and energy contro... more In a collaboration with the french national agency for development of ecol- ogy and energy control (ADEME), we intend to build a new numerical method to com- pute small scale phenomena in atmospheric models, getting rid of any mesh refinement. In an existing mesh, we virtually drop some particles that are moved thanks to a sys- tem of Stochastic Dierential Equations adapted from S.B. Pope. We then estimate local values of the required fields, thanks to the computation of a mean value over an ensemble of particles. We are thus using Monte-Carlo methods, and aim to study their convergence rates, and finally compare them to classical refinement methods. One (con- tastable) constraint of Pope's model is the uniform value of the density ( = cst). That is, the particles have to be uniformly distributed at every time step. This particular problem is the framework of our CEMRACS project. We aim to use D.P. Bertsekas Auction Algorithm in order to move a given cloud of particles to a new ...

Physical Review Fluids

Non-spherical particles transported by an anisotropic turbulent flow preferentially align with th... more Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for small, inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear, and can thereby be integrated analytically. Reproducing the single-time orientation statistics obtained numerically requires to take account of the mean shear, of anisotropic velocity gradient fluctuations, and of the presence of persistent rotating structures that combine together to bias cumulative Lagrangian statistics. The model is then used to address two-time statistics. The notion of tumbling rate is extended to diffusive dynamics by introducing the stationary probability flux of the rods unfolded angle, which provides information on the overall, cumulated rotation of the particle. The model is found to reproduce the long-term effects of an average shear on the mean and the variance of the fibres angular increment. Still, for intermediate times, the model fails catching violent fluctuations of the rods rotation that are due to trapping events in coherent, long-living eddies.

Suspension of anisotropic particles can be found in various applications, e.g. industrial manufac... more Suspension of anisotropic particles can be found in various applications, e.g. industrial manufacturing processes or natural phenomena (micro-organism locomotion, ice crystal formation in clouds). Microscopic ellipsoidal bodies suspended in a turbulent fluid flow rotate in response to the velocity gradient of the flow. Understanding their orientation is important since it can affect the optical or rheological properties of the suspension (e.g. polymeric fluids). In this work, the orientation dynamics of rod-like tracer particles, i.e. long ellipsoidal particles (in the limit to infinity of the aspect-ratio) is studied. The size of the rod is assumed smaller than the Kolmogorov length scale but sufficiently large that its Brownian motion need not be considered. As a result, the local flow around a particle can be considered as inertia-free and Stokes flow solutions can be used to relate particle rotational dynamics to the local velocity gradient tensor A ij = ∂u i ∂x j. The orientati...

We consider a model of an electricity market in which S suppliers offer electricity: each supplie... more We consider a model of an electricity market in which S suppliers offer electricity: each supplier Si offers a maximum quantity qi at a fixed price pi. The response of the market to these offers is the quantities bought from the suppliers. The objective of the market is to satisfy its demand at minimal price. We investigate two cases. In the first case, each of the suppliers strives to maximize its market share on the market; in the second case each supplier strives to maximize its profit. We show that in both cases some Nash equilibrium exists. Nevertheless a close analysis of the equilibrium for profit maximization shows that it is not realistic. This raises the difficulty to predict the behavior of a market where the suppliers are known to be mainly interested by profit maximization.

Springer Proceedings in Mathematics & Statistics, 2018

We present a stochastic Lagrangian approach for atmospheric boundary layer simulation. Based on a... more We present a stochastic Lagrangian approach for atmospheric boundary layer simulation. Based on a turbulent-fluid-particle model, a stochastic Lagrangian particle approach could be an advantageous alternative for some applications, in particular in the context of down-scaling simulation and wind farm simulation. This paper presents two recent advances in this direction, first the analysis of an optimal rate of convergence result for the particle approximation method that grounds the space discretisation of the Lagrangian model, and second a preliminary illustration of our methodology based on the simulation of a Zephyr ENR wind farm of six turbines.

[ Research paper thumbnail of Weak convergence analysis of the symmetrized Euler scheme for one dimensional SDEs with diffusion coefficient |x|^a, a in [1/2,1)](https://attachments.academia-assets.com/97480576/thumbnails/1.jpg)

arXiv: Probability, 2015

In this paper, we are interested in the time discrete approximation of Ef(X(T)) when X is the sol... more In this paper, we are interested in the time discrete approximation of Ef(X(T)) when X is the solution of a stochastic differential equation with a diffusion coefficient function of the form |x|^a. We propose a symmetrized version of the Euler scheme, applied to X. The symmetrized version is very easy to simulate on a computer. For smooth functions f, we prove the Feynman Kac representation u(t,x) = E_{t,x} f(X(T)), for u(t,x) solving the associated Kolmogorov PDE and we obtain the upper-bounds on the spatial derivatives of u up to the order four. Then we show that the weak error of our symmetrized scheme is of order one, as for the classical Euler scheme.

Applied Mathematical Modelling, 2021

This article presents a new data-driven spatial decomposition algorithm that allows the splitting... more This article presents a new data-driven spatial decomposition algorithm that allows the splitting of a domain containing point particles into elementary cells, each cell containing a spatially-uniform distribution of particles. For that purpose, the algorithm relies on the use of statistical information for the spatial distribution of particles and then extracts an optimal spatial decomposition. After evaluating the convergence and accuracy of the algorithm on homogeneous and inhomogeneous cases, this optimal spatial decomposition is applied to study the case of particle agglomeration. Indeed, in CFD context, recent developments on numerical simulations of particle agglomeration in complex and turbulent flows increasingly resort to Euler-Lagrange approaches. These methods are coupled with population balance equation (PBE)-like algorithms to compute agglomeration inside each cell of the Eulerian mesh. One of the key issues with such approaches is related to the spatially-uniform condition, i.e. agglomeration should be computed on a set of particles that are uniformly distributed locally in each cell. Yet, CFD simulations in realistic industrial/environmental cases often involve non-homogeneous concentrations of particles (due to local injection or accumulation in specific regions). We show that more accurate and mesh-independent predictions of particle agglomeration are made possible by the application of this new data-driven spatial decomposition algorithm.

SSRN Electronic Journal, 2017

We analyze optimal investment strategies under the drawdown constraint that the wealth process ne... more We analyze optimal investment strategies under the drawdown constraint that the wealth process never falls below a fixed fraction of its running maximum. We derive optimal allocation programs by solving numerically the Hamilton-Jacobi-Bellman equation that characterizes the finite horizon expected utility maximization problem, for investors with power utility as well as S-shaped utility. Using stochastic simulations, we find that, according to utility maximization, implementing the drawdown constraint can be gainful in optimal portfolios for the power utility, for some market configurations and investment horizons. However, our study reveals that the optimal strategy with drawdown constraint is not the preferred investment for the S-shaped utility investor, who rather prefers the equivalent optimal strategy without constraint. Indeed, the latter investment being similar to a partial portfolio insurance, the additional drawdown constraint does not appear valuable for this investor in optimal portfolios.

ESAIM: Proceedings and Surveys, 2015

The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the no... more The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g. [7], [27]). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized equation on real size biomolecules, that also require efficient computational geometry algorithms. We compare our results with the deterministic solver APBS. In the second part, we prove a new probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte Carlo algorithm is also derived and tested on a simple test case. Résumé. Le potentielélectrostatique autour d'une biomolécule peutêtre calculé grâceà l'équation de Poisson-Boltzmann, une EDP elliptique non-linéaire sous forme divergence. Des méthodes de Monte-Carlo, dédiéesà sa version linéarisée, combinent la marche sur les sphères avec différents schémas de replacementà la frontière de la molécule (voir [7], [27]). La première partie de cet article est consacréeà l'étude et la comparaison de différentes méthodes de replacement pour l'EDP linéarisée, dans le cas de géométries réalistes de biomolécules. Dans la seconde partie, nous donnons une nouvelle interprétation probabiliste de l'EDP de Poisson-Boltzmann non linéaire. Nous en déduisons une méthode de Monte-Carlo originale qui est testée sur un cas test simple.

Encyclopedia of Quantitative Finance, 2010

Journal of Computational Physics, 2022

We present the construction of an original stochastic model for the instantaneous turbulent kinet... more 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. Motivated by the need for wind energy industry of acquiring relevant statistical information of air motion at a local place, we adopt the Lagrangian description of fluid flows to derive, from the 3D+time equations of the physics, a 0D+time-stochastic model for the time series of the instantaneous turbulent kinetic energy at a given position. Specifically, based on the Lagrangian stochastic description of a generic fluid-particles, we derive a family of mean-field dynamics featuring the square norm of the turbulent velocity. By approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called Cox-Ingersoll-Ross process, which was previously suggested in the literature for modelling wind speed. We then propose a calibration procedure for the parameters employing both direct methods and Bayesian inference. In particular, we show the consistency of the estimators and validate the model through the quantification of uncertainty, with respect to the range of values given in the literature for some physical constants of turbulence modelling.

Abstract. We consider one-dimensional stochastic differential equations in the particular case of... more Abstract. We consider one-dimensional stochastic differential equations in the particular case of dif-fusion coefficient functions of the form |x|α, α ∈ [1/2, 1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

Springer Proceedings in Mathematics & Statistics, 2019

We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Di... more We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Méléard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Hölder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamics and a nondegenerate diffusion term in the position dynamics.

In the first part of this article, we present the main tools and definitions of Markov processes'... more In the first part of this article, we present the main tools and definitions of Markov processes' theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov's backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward SDEs (BSDEs). The second part of this article is devoted to the links between Markov processes and parabolic partial differential equations (PDEs). In particular, we give Feynman-Kac formula for linear PDEs, we present Feynman-Kac formula for BSDEs, and we give some examples of the correspondence between stochastic control problems and Hamilton-Jacobi-Bellman (HJB) equations and between optimal stopping problems and variational inequalities. Several examples of finan-* TOSCA project-team

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

L'objectif de ce travail est de modeliser la turbulence en eaux profondes et les effets du pr... more L'objectif de ce travail est de modeliser la turbulence en eaux profondes et les effets du processus d'extraction d'energie sur la dynamique de ces ecoulements turbulents. Nous utilisons le code SDM [Stochastic Downscaling Model] qui calcule les champs instantanes lagrangiens des particules en resolvant les equations stochastiques similaires a celles que l'on trouve dans [1]. SDM fournit egalement les champ euleriens moyennes sur une grille fixe, ce qui donne une description particulierement complete de l'ecoulement. La version « air » du code SDM est deja disponible pour modeliser les ecoulements a proximite d'eoliennes [2]. Notre objectif est d'adapter cette version pour la rendre applicable aux ecoulements marins. Durant la presentation, nous presentons les premieres modifications du code ainsi que trois cas de validations, principalement axes sur : (i) la generation de couches limites, (ii) les effets de la bathymetrie locale sur l'ecoulement, et ...

Mathematical Modelling and Numerical Analysis, 2010

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamic... more Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended Feynman-Kac formula for the Poisson-Boltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models. Résumé. Motivés par le développement de méthodes de Monte-Carlo efficaces pour deséquations aux dérivées partielles en dynamique moléculaire, nousétablissons une nouvelle interprétation probabiliste d'une famille d'opérateurs sous forme divergence età coefficients discontinus le long de l'interface de deux ouverts de R d. Cette famille d'opérateurs inclut le cas de l'équation de Poisson-Boltzmann linearisée utilisée pour le calcul de l'énergie libreélectrostatique d'une molécule. Plus précisément, nous construisons explicitement un processus de Markov dont le générateur infinitésimal appartientà cette famille d'opérateurs, en résolvant uneéquation différentielle stochastique avec un terme de temps local non standard définià partir de l'interface de discontinuité. Nousétendons ensuite la formule de Feynman-Kacà l'équation de Poisson-Boltzmann. Cette formule nous permet de justifier plusieurs algorithmes probabilistes approchant l'énergie libre d'une molécule. Nous analysons la vitesse de convergence de ces procédures de simulation et nous les comparons numériquement sur des modèles de molécules idéalisées.

A Lagrangian stochastic model is introduced in order to describe the local behavior of the wind. ... more A Lagrangian stochastic model is introduced in order to describe the local behavior of the wind. Based on some data given by a large scales meteorological model (MM5), and thanks to particles driven by stochastic differential equations, we propose a numerical method allowing to improve the MM5 simulations at small scales, without requiring too much additional computational cost.