Reduced basis method for the rapid and reliable solution of partial differential equations

International Congress of Mathematicians, ICM 2006

Volumn 3, Issue , 2006, Pages 1255-1270

  Maday, Yvon ^a,b  

^a SORBONNE UNIVERSITÉ   (France)

^b BROWN UNIVERSITY   (United States)

Author keywords

A posteriori error estimation; Galerkin approximation; Offline online procedures; Output bounds; Parametrized partial differential equations; Reduced basis

Indexed keywords

A-POSTERIORI ERROR ESTIMATIONS; GALERKIN APPROXIMATIONS; OFFLINE-ONLINE PROCEDURES; OUTPUT BOUNDS; PARAMETRIZED PARTIAL DIFFERENTIAL EQUATIONS; REDUCED-BASIS;

ALGORITHMS; CHEMICAL ANALYSIS; COMPUTER AIDED ANALYSIS; COMPUTER AIDED DESIGN; DISCRETE EVENT SIMULATION;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 84875826962     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (20)

1. Buffa, A., Maday, Y., Patera, A. T., Prud'homme, C., Turinici, G. In progress, 2006.
2. Cancès, E., Le Bris, C., Maday, Y., Nguyen, N. C., Patera, A. T., Pau, G., Turinici, G. In progress, 2006.
3. Cancès, E., Le Bris, C., Maday, Y., Turinici, G., Towards reduced basis approaches in ab initio electronic structure computations. J. Sci. Comput. 17 (1-4) (2002), 461-469.
4. Charpentier, I., Maday, Y., Patera, A. T., Bounds evaluation for ouputs of eigenvalue problems approximated by the overlapping modal synthesis method. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 909-914.
5. Charpentier, I., de Vuyst, F., Maday, Y., The overlapping component mode synthesis: The shifted eigenmodes strategy and the case of self-adjoint operators with discontinuous coefficients. In Proceedings of the Ninth International Conference on Domain Decomposition Methods (ed. by P. E. Bjørstad, S. Magne, S. Espedal and D. E. Keyes), 1998, 583-596.
6. Grepl, M. A., Maday, Y., Nguyen, N. C., Patera, A. T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Math. Model. Numer. Anal., submitted.
7. Grepl, M. A., Patera, A. T., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Math. Model. Numer. Anal. 39 (1) (2005), 157-181.
8. Kolmogoroff, A., Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. of Math. 37 (1963), 107-110.
9. Løvgren, A. E., Maday, Y., Rønquist, E. M., A reduced basis element method for the steady Stokes problem. Math. Model. Numer. Anal., to appear.
10. Løvgren, A. E., Maday, Y., Rønquist, E. M., The reduced basis element method for fluid flows. Adv. Math. Fluid Mech., 2006, to appear.
11. Machiels, L., Maday, M., Oliveira, I. B., Patera, A. T., Rovas, D. V., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 153-158.
12. Maday, Y., Patera, A. T., Turinici, G., A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations. J. Sci. Comput. 17 (1-4) (2002), 437-446.
13. Maday, Y., and Rønquist, E. M., The reduced-basis element method: Application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004), 240-258.
14. Pinkus, A., n-Widths in Approximation Theory. Ergeb. Math. Grenzgeb. (3) 7, Springer-Verlag, Berlin 1985.
15. Prud'homme, C., Contributions aux simulations temps-réel fiables et certains aspects du calcul scientifique. Mémoire d'Habilitation à Diriger les Recherches, Université Pierre et Marie Curie, Paris, 2005.
16. Prud'homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engrg. 124 (2002), 70-80.
17. Rovas, D. V., Machiels, L., Maday, Y., Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal., Advance Access published on March 6, (2006).
18. Sen, S., Veroy, K., Huynh, D. B. P., Deparis, S., Nguyen, N. C., Patera, A. T., Natural norm, a posteriori error estimators for reduced basis approximations. J. Comp. Phys., 2006, to appear.
19. Veroy, K., Prud'homme, C., Patera, A. T., Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris Ser. I Math. 337 (2003), 619-624.
20. Veroy, K., Patera, A. T., Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Meth. Fluids 47 (2005), 773-788.