no. 12
Numerical analysis
Bounding stability constants for affinely parameter-dependent operators
[Des bornes inférieures pour les constantes de stabilité associées à des operateurs avec une dépendance affine des paramètres]

Présenté par : Olivier Pironneau

O'Connor, Robert1
1 RWTH Aachen University, Aachen, Germany
Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1236-1240.

Nous présentons de nouvelles méthodes pour borner les constantes de stabilité qui jouent un rôle essentiel dans les approximations par bases réduites. Notre méthode nous permet de borner les constantes dans tout un voisinage et non seulement en un nombre fini de points. Nous montrons aussi qu'on peut démontrer la stabilité de Liapounov dans le même cadre.

In this article we introduce new possibilities of bounding the stability constants that play a vital role in the reduced basis method. By bounding stability constants over a neighborhood we make it possible to guarantee stability at more than a finite number of points and to do that in the offline stage. We additionally show that Lyapunov stability of dynamical systems can be handled in the same framework.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.10.003
O'Connor, Robert 1

1 RWTH Aachen University, Aachen, Germany 
@article{CRMATH_2016__354_12_1236_0,
     author = {O'Connor, Robert},
     title = {Bounding stability constants for affinely parameter-dependent operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1236--1240},
     publisher = {Elsevier},
     volume = {354},
     number = {12},
     year = {2016},
     doi = {10.1016/j.crma.2016.10.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.10.003/}
}
TY  - JOUR
AU  - O'Connor, Robert
TI  - Bounding stability constants for affinely parameter-dependent operators
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1236
EP  - 1240
VL  - 354
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.10.003/
DO  - 10.1016/j.crma.2016.10.003
LA  - en
ID  - CRMATH_2016__354_12_1236_0
ER  - 
%0 Journal Article
%A O'Connor, Robert
%T Bounding stability constants for affinely parameter-dependent operators
%J Comptes Rendus. Mathématique
%D 2016
%P 1236-1240
%V 354
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.10.003/
%R 10.1016/j.crma.2016.10.003
%G en
%F CRMATH_2016__354_12_1236_0
O'Connor, Robert. Bounding stability constants for affinely parameter-dependent operators. Comptes Rendus. Mathématique, Tome 354 (2016) no. 12, pp. 1236-1240. doi : 10.1016/j.crma.2016.10.003. http://www.numdam.org/articles/10.1016/j.crma.2016.10.003/

[1] Chen, Y.; Hesthaven, J.S.; Maday, Y.; Rodríguez, J. Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem, ESAIM Math. Model. Num., Volume 43 (2009) no. 6, pp. 1099-1116 (10)

[2] Huynh, D.B.P.; Rozza, G.; Sen, S.; Patera, A.T. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 8, pp. 473-478

[3] Nguyen, N.C. Reduced-Basis Approximations and A Posteriori Error Bounds for Nonaffine and Nonlinear Partial Differential Equations: Application to Inverse Analysis, Singapore–MIT Alliance, June 2005 (PhD thesis)

[4] O'Connor, R. Lyapunov-based error bounds for the reduced-basis method, IFAC–PapersOnLine, Volume 49 (2016) no. 8, pp. 1-6

[5] R. O'Connor, M. Grepl, Offline error bounds for the reduced basis method, IGPM Preprint 452, June 2016.

[6] Rozza, G.; Huynh, D.B.P.; Patera, A.T. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., Volume 15 (2008) no. 3, pp. 229-275

[7] Veroy, K. Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications, Massachusetts Institute of Technology, Cambridge, MA, USA, 2003 (PhD thesis)

[8] Veroy, K.; Prud'homme, C.; Rovas, D.V.; Patera, A.T. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, Orlando, FL, USA, 23–26 June (2003) (AIAA Paper 2003-3847)

Cité par Sources :