Convergence of a greedy algorithm for high-dimensional convex nonlinear problems

Mathematical Models and Methods in Applied Sciences

Volumn 21, Issue 12, 2011, Pages 2433-2467

  Cancès, Eric ^a   Ehrlacher, Virginie ^a   Lelièvre, Tony ^a  

^a UNIVERSITÉ PARIS EST   (France)

Author keywords

Greedy algorithm; high dimension; obstacle problem; uncertainty quantification

Indexed keywords

CONVEX PROBLEMS; ENERGY FUNCTIONALS; GLOBAL MINIMA; GREEDY ALGORITHMS; HIGH DIMENSIONS; HIGH-DIMENSIONAL; LIPSCHITZ; NONLINEAR PROBLEMS; OBSTACLE PROBLEMS; TENSOR PRODUCTS; UNCERTAINTY PROPAGATION; UNCERTAINTY QUANTIFICATIONS;

ALGORITHMS; TENSORS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 80255141549     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202511005799     Document Type: Article

References (16)

1. A. Ammar, B. Mokdad, F. Chinesta and 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-Newtonian Fluid Mech. 139 (2002) 153-176. (Pubitemid 44793705)
2. H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numer. 13 (2004) 147-269. (Pubitemid 38773594)
3. W. E. W. Ren and E. Van den-Eijnden, A string method for the study of rare events, Phys. Rev. B 66 (2002) 052301.
4. M. Fortin and R. Glowinski, Méthodes de Lagrangien Augmenté - Application à la Résolution Numérique de Problèmes aux Limites (Dunod, 1982).
5. R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements - A Spectral Approach (Springer, 1991).
6. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (Dunod, 1976).
7. R. Glowinski, J. L. Lions and R. Trémolières, Analyse Numérique des Inéquations Variationnelles - Théorie Générale et Premières Applications (Society for Industrial and Applied Mathematics, 1989).
8. C. Grossmann, H.-G. Roos andM. Stynes, Numerical Treatment of Partial Differential Equations (Springer, 2007).
9. C. Le Bris, T. Lelièvre and Y. Maday, Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations, Constructive Approximation 30 (2009) 621-651.
10. A. Nouy, Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Arch. Comput. Methods Engnrg. 16 (2009) 251-285.
11. A. Nouy and O. Le Maitre, Generalized spectral decomposition for stochastic nonlinear problems, J. Comput. Phys. 228 (2009) 202-235.
12. L. Schwartz, Théorie des Distributions (Hermann, 1966).
13. Scilab, http://www.scilab.org.
14. V. N. Temlyakov, Greedy approximation, Acta Numer. 17 (2008) 235-409.
15. S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov. Math. Dokl. 3 (1963) 240-243.
16. T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high dimensions, M2AN Math. Model. Numer. Anal. 38 (2004) 93-127. (Pubitemid 38254563)