GREEDY APPROXIMATION OF HIGH-DIMENSIONAL ORNSTEIN-UHLENBECK OPERATORS WITH UNBOUNDED DRIFT
Leonardo E. Figueroa, ENDRE E. SÜLI · Oxford University Research Archive (ORA) (University of Oxford) · 2011
Abstract. 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 R 2, subject to a homogeneous Dirichlet boundary condition, the mathematicalanalysis ofthe algorithmwascarriedoutrecentlyby Le Bris, Lelièvre andMaday (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, Lelièvre 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 = D1 ×···×DN contained in R Nd, where each set Di, i = 1,...,N, is a bounded open ball in R d, d = 2,3. 1.