Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain $\Omega \subset R^d$, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data: a square-integrable and divergence-free initial velocity datum $u_0$ for the Navier-Stokes equation and a nonnegative initial probability density function $\psi_0$ for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian M

Numerical approximation of corotational dumbbell models for dilute polymers

We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω in R d, d=2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. In the case of a corotational drag term we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system

Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (cf. J. Non-Newtonian Fluid Mech. 139:153--176, 2006) for the numerical solution of high-dimensional Fokker--Planck equations featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. In this paper, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris, Leli\`evre and Maday to the technically more complicated case where the Laplace operator is replaced by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D = D_1 x ... x D_N contained in $\mathbb{R}^{N d}$, where each set D_i, i=1,...,N, is a bounded open ball in $\mathbb{R}^d$, d = 2, 3

Deep Residual Learning via Large Sample Mean-Field Stochastic Optimization

We study a class of stochastic optimization problems of the mean-field type arising in the optimal training of a deep residual neural network. We consider the sampling problem arising from a continuous layer idealization, and establish the existence of optimal relaxed controls when the training set has finite size. The core of our paper is to prove the Gamma-convergence of the sequence of sampled objective functionals, i.e., show that as the size of the training set grows large, the minimizer of the sampled relaxed problem converges to that of the limiting optimization problem. We connect the limit of the large sampled objective functional to the unique solution, in the trajectory sense, of a nonlinear Fokker-Planck-Kolmogorov (FPK) equation in a random environment. We construct an example to show that, under mild assumptions, the optimal network weights can be numerically computed by solving a second-order differential equation with Neumann boundary conditions in the sense of distributions

Wealth distribution in presence of debts. A Fokker--Planck description

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker--Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached

Kinetic equations with Maxwell boundary conditions

We prove global stability results of {\sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz\`es-Guiraud information in a crucial way