Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators
By Leonardo E. Figueroa, Endre Süli. 2011
48 citations recorded by the source corpus.
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=D1×⋯×DN contained in ℝNd, where each set Di, i=1,…,N, is a bounded open ball in ℝd, d=2,3.
Subfields: Applied Mathematics, Fluid Flow and Transfer Processes, Statistical and Nonlinear Physics
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Citations in the corpus
- A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Convergence results in this direction can be found in [3,39,40,84–86, A number of improvements to the simple scheme outlined above have been proposed to increase its convergence speed, see, e.g., [4, 95, 200, 202].”
- Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “As such reduced grid-based methods, we would like to mention the sparse grids technique [13], as well as the initial tensor-structured approaches: the greedy algorithm in the canonical tensor format [14, 15, 16, 17, 18], and the solver in the Tucker format, based on the manifold dynamics via the pro”
- Greedy algorithms for high-dimensional non-symmetric linear problems (2012, background): “Although the proof is given in [10], we recall it here for the sake of self-containedness.”
- Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “A particular model, also referred to as the Hookean spring model [11], is defined by”