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

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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 , subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelievre 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, Lelievre 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 , where each set D_i, i=1,...,N, is a bounded open ball in , d = 2, 3.

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• Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift. (deposited 05 Mar 2011 13:41)
  • Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift. (deposited 17 Apr 2011 08:12) [Currently Displayed]