Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators
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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.
KeywordsNonlinear approximation Greedy algorithm Fokker–Planck equation
Mathematics Subject Classification65N15 65D15 41A63 41A25
The first author acknowledges a doctoral scholarship from the Chilean government’s Comisión Nacional de Investigación Científica y Tecnológica. The second author was supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).
We are grateful to Professor Marco Marletta (Cardiff University) for helpful hints regarding the Liouville transformation. We also wish to express our gratitude to Professors Claude Le Bris and Tony Lelièvre (CERMICS, École des Ponts ParisTech) and the anonymous referee for helpful and constructive suggestions.
- 2.A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newton. Fluid Mech. 139(3), 153–176 (2006). doi: 10.1016/j.jnnfm.2006.07.007. http://www.sciencedirect.com/science/article/pii/S0377025706001662. zbMATHCrossRefGoogle Scholar
- 3.A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part II: Transient simulation using space-time separated representations, J. Non-Newton. Fluid Mech. 144(2–3), 98–121 (2007). doi: 10.1016/j.jnnfm.2007.03.009. http://www.sciencedirect.com/science/article/pii/S0377025707000821. zbMATHCrossRefGoogle Scholar
- 4.A. Ammar, M. Normandin, F. Daim, D. González, E. Cueto, F. Chinesta, Non incremental strategies based on separated representations: applications in computational rheology, Commun. Math. Sci. 8(3), 671–695 (2010). http://projecteuclid.org/getRecord?id=euclid.cms/1282747135. MathSciNetzbMATHGoogle Scholar
- 10.R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, 2nd edn. Kinetic Theory, vol. 2 (Wiley, New York, 1987). Google Scholar
- 16.F. Chinesta, A. Ammar, A. Leygue, R. Keunings, An overview of the proper generalized decomposition with applications in computational rheology, J. Non-Newton. Fluid Mech. 166(11), 578–592 (2011). doi: 10.1016/j.jnnfm.2010.12.012. http://www.sciencedirect.com/science/article/pii/S0377025711000061. zbMATHCrossRefGoogle Scholar
- 18.A. Cohen, A Padé approximant to the inverse Langevin function, Rheol. Acta 30(3), 270–273 (1991). doi: 10.1007/BF00366640. http://www.springerlink.com/content/t240j86m14247501. CrossRefGoogle Scholar
- 19.R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. I (Interscience Publishers, Inc., New York, 1953). Google Scholar
- 22.L.E. Figueroa, E. Süli, Greedy approximation of high-dimensional Ornstein–Uhlenbeck operators. Tech. rep., 2012. arXiv:1103.0726v2 [math.NA].
- 40.G.M. Taščijan, The spectral asymptotic behavior of elliptic boundary value problems with weak degeneracy, in Proceedings of the Sixth Winter School on Mathematical Programming and Related Questions (Drogobych 1973), Functional Analysis and Its Applications, pp. 277–293 Akad. Nauk SSSR Central. Èkonom.-Mat. Inst., Moscow, 1975) (Russian). Google Scholar