Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators

Journal article

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 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. 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 x ... x D_N contained in R^(N d), where each set D_i, i = 1, ..., N, is a bounded open ball in R^d, d = 2, 3.

Authors: Figueroa, L; Süli, E

• Publication status: Published
• Journal: J. Foundations of Computational Mathematics, 2012
• Volume: 12
• Issue: 5
• Pages: 573-623
• Publication date: 2011-03-03
• DOI: 10.1007/s10208-012-9122-z
• EISSN: 1615-3383
• ISSN: 1615-3375
• Language: English
• Keywords: math.AP; 65N15, 65D15, 41A63, 41A25; math.NA