The Mathematical Institute, University of Oxford, Eprints Archive

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

Figueroa, Leonardo and Suli, Endre (2011) Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift. Technical Report. Foundations of Computational Mathematics. (Submitted)

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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 $\mathbb{R}^2$, 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.

Item Type:Technical Report (Technical Report)
Subjects:A - C > Approximations and expansions
H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1058
Deposited By:Lotti Ekert
Deposited On:17 Apr 2011 08:12
Last Modified:17 Apr 2011 08:12

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