Greedy Approximation of High-Dimensional OrnsteinUhlenbeck Operators
We investigate the convergence of a nonlinear approximation method introduced by Ammar etal. (J. Non-Newtonian Fluid Mech. 139:153176, 2006) for the numerical solution of high-dimensional FokkerPlanck equations featuring in NavierStokesFokkerPlanck systems that arise in kinetic models of dilute polymers. In the case of Poissons equation on a rectangular domain in ?2, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelivre and Maday (Const. Approx. 30:621651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173187, 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 OrnsteinUhlenbeck operator of the kind that appears in FokkerPlanck equations that arise in beadspring 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 ? Nd , where each set D i , i=1,,N, is a bounded open ball in ? d , d=2,3.
nonlinear approximation fokkerplanck equation greedy algorithm 65n15 65d15 41a63 41a25