Benjamin Jourdain

Existence of solution for a micro–macro model of polymeric fluid: the FENE model

Abstract We analyse a non-linear micro–macro model of polymeric fluids in the case of a shear flow. More precisely, we consider the FENE dumbbell model, which models polymers by nonlinear springs, accounting for the finite extensibility of the polymer chain. We prove the existence of a unique solution to the stochastic differential equation which rules the evolution of a representative polymer in the flow and next deduce a local-in-time existence and uniqueness result on the system coupling the stochastic differential equation and the momentum equation on the fluid.

A stochastic approach for the numerical simulation of the general dynamics equation for aerosols

We present in this article a stochastic algorithm based mainly on [Monte Carlo Methods and Applications 5(1) (1999) 1; Stochastic particle approximations for Smoluchowskis coagulation equation. Technical Report, Weierstrass-Institut for Applied Analysis and Stochastics, 2000. Preprint No. 585] applied to the integration of the General Dynamics Equation (GDE) for aerosols. This algorithm is validated by comparison with analytical solutions of the coagulation-condensation model and may provide an accurate reference solution in cases for which no analytical solution is available.

NUMERICAL ANALYSIS OF MICRO–MACRO SIMULATIONS OF POLYMERIC FLUID FLOWS: A SIMPLE CASE

We present in this paper the numerical analysis of a simple micro–macro simulation of a polymeric fluid flow, namely the shear flow for the Hookean dumbbells model. Although restricted to this academic case (which is however used in practice as a test problem for new numerical strategies to be applied to more sophisticated cases), our study can be considered as a first step towards that of more complicated models. Our main result states the convergence of the fully discretized scheme (finite element in space, finite difference in time, plus Monte Carlo realizations) towards the coupled solution of a partial differential equation/stochastic differential equation system.

Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures

In this paper, we explain how to associate a nonlinear martingale problem with some nonlinear parabolic evolution equations starting at bounded signed measures. Our approach generalizes the classical link made when the initial condition is a probability measure. It consists in giving to each sample-path a signed weight which depends on the initial position. After dealing with the classical McKean-Vlasov equation as an introductory example, we are interested in a viscous scalar conservation law. We prove uniqueness for the corresponding nonlinear martingale problem and then obtain existence thanks to a propagation of chaos result for a system of weakly interacting diffusion processes. Last, we study the behavior of the associated fluctuations and present numerical results which confirm the theoretical rate of convergence.

Propagation of chaos and fluctuations for a moderate model with smooth initial data

Abstract In this paper, we are interested in a stochastic differential equation which is nonlinear in the following sense: both the diffusion and the drift coefficients depend locally on the density of the time marginal of the solution. When the law of the initial data has a smooth density with respect to Lebesgue measure, we prove existence and uniqueness for this equation. Under more restrictive assumptions on the density, we approximate the solution by a system of n moderately interacting diffusion processes and obtain a trajectorial propagation of chaos result. Finally, we study the fluctuations associated with the convergence of the empirical measure of the system to the law of the solution of the nonlinear equation. In this situation, the convergence rate is different from √ n .

Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf

In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincare inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.

Robust Adaptive Importance Sampling for Normal Random Vectors

Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly non independent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator : in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor going from 3 to 15.

Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with

Exact retrospective Monte Carlo computation of arithmetic average Asian options

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Stochastic flow approach to Dupire’s formula

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