Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (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 ℝ², subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre 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. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein–Uhlenbeck operator 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×⋯×D _N contained in ℝ ^Nd , where each set D _i , i=1,…,N, is a bounded open ball in ℝ ^d , d=2,3.