Partial differential equations/Numerical analysis
Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing
Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 901-906.

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

On présente une nouvelle méthode dʼapproximation non linéaire des variétés de solutions de problèmes dʼévolution non linéaires paramétrées, en particulier dans les régimes hyperboliques. Pour une action de groupe de Lie donnée sur lʼespace des solutions, le problème initial est reformulé comme une équation aux derivées partielles algébriques, en décomposant la solution en une partie sur le groupe et une partie sous forme spatiale. On impose ensuite des contraintes algébriques sur la décomposition. Dans la suite, on projette le système sur un espace de base réduite. On démontre que la méthode peut être évaluée « en ligne » de manière efficace, et on traite un exemple numérique montrant une perfomance améliorée si on la compare à la même méthode sans figeage.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.10.028
Ohlberger, Mario 1; Rave, Stephan 2

1 Institute for Computational and Applied Mathematics & Center for Nonlinear Science, University of Münster, Einsteinstr. 62, 48149 Münster, Germany
2 Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster, Germany
@article{CRMATH_2013__351_23-24_901_0,
     author = {Ohlberger, Mario and Rave, Stephan},
     title = {Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {901--906},
     publisher = {Elsevier},
     volume = {351},
     number = {23-24},
     year = {2013},
     doi = {10.1016/j.crma.2013.10.028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.10.028/}
}
TY  - JOUR
AU  - Ohlberger, Mario
AU  - Rave, Stephan
TI  - Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 901
EP  - 906
VL  - 351
IS  - 23-24
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.10.028/
DO  - 10.1016/j.crma.2013.10.028
LA  - en
ID  - CRMATH_2013__351_23-24_901_0
ER  - 
%0 Journal Article
%A Ohlberger, Mario
%A Rave, Stephan
%T Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing
%J Comptes Rendus. Mathématique
%D 2013
%P 901-906
%V 351
%N 23-24
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.10.028/
%R 10.1016/j.crma.2013.10.028
%G en
%F CRMATH_2013__351_23-24_901_0
Ohlberger, Mario; Rave, Stephan. Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 901-906. doi : 10.1016/j.crma.2013.10.028. http://www.numdam.org/articles/10.1016/j.crma.2013.10.028/

[1] Beyn, W.-J.; Thümmler, V. Freezing solutions of equivariant evolution equations, SIAM J. Appl. Dyn. Syst., Volume 3 (2004), pp. 85-116

[2] Drohmann, M.; Haasdonk, B.; Ohlberger, M. Reduced basis approximation for nonlinear parameterized evolution equations based on empirical operator interpolation, SIAM J. Sci. Comput., Volume 34 (2012), p. A937-A969

[3] Haasdonk, B. Convergence rates of the POD–Greedy method, M2AN Math. Model. Numer. Anal., Volume 47 (2013), pp. 859-873

[4] Haasdonk, B.; Ohlberger, M. Reduced basis method for finite volume approximations of parametrized linear evolution equations, M2AN Math. Model. Numer. Anal., Volume 42 (2008), pp. 277-302

[5] Haasdonk, B.; Ohlberger, M. Reduced basis method for explicit finite volume approximations of nonlinear conservation laws (Tadmor, E.; Liu, J.-G., eds.), Hyperbolic Problems: Theory, Numerics and Applications, Amer. Math. Soc., 2009, pp. 605-614

[6] Janon, A.; Nodet, M.; Prieur, C. Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values, ESAIM Math. Model. Numer. Anal., Volume 47 (2013), pp. 317-348

[7] Nguyen, N.-C.; Rozza, G.; Patera, A. Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgersʼ equation, Calcolo, Volume 46 (2009), pp. 157-185

[8] Rowley, C.W.; Kevrekidis, I.G.; Marsden, J.E.; Lust, K. Reduction and reconstruction for self-similar dynamical systems, Nonlinearity, Volume 16 (2003), pp. 1257-1275

Cited by Sources: