Partial Differential Equations/Numerical Analysis
A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices
Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 593-598.

We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB–ANOVA–RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB–ANOVA–RB procedure computationally more feasible.

Nous considérons une méthode permettant dʼévaluer en temps réel, de manière efficace, une fonctionnelle basée sur la solution numérique dʼune équation aux dérivées partielles dépendant dʼun grand nombre de paramètres. Nous présentons un résultat qui permet dʼaméliorer la méthode en trois étapes, RB–ANOVA–RB. Cette dernière est une combinaison de la méthode des bases réduites (RB) et de la méthode dʼexpansion dʼanalyse des variations (ANOVA). Le but est de pouvoir compresser lʼespace des paramètres sans affecter la précision de notre fonctionnelle. Dans un premier temps, nous calculons une première approximation (grossière) RB de notre problème, en considérant toutes les composantes des paramètres. Ensuite, nous utilisons lʼapproximation obtenue pour calculer lʼexpansion ANOVA de la fonctionnelle afin de déterminer lʼinfluence de chacune des composantes de nos paramètres sur ce dernier et fixer les moins influentes. Finalement, une deuxième approximation (fine) RB est faite sur le modèle ne contenant que les composantes les plus importantes. Le résultat que nous présentons ici donne un critère pour éviter le calcul de termes, basé sur lʼinteraction des composantes des paramètres dans la forme bilinéaire, permettant ainsi de diminuer drastiquement les coûts computationnels liés à lʼexpansion ANOVA.

Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.023

Denis Devaud 1; Andrea Manzoni 2, 3; Gianluigi Rozza 2, 3

1 EPFL SMA, CH-1015, Lausanne, Switzerland
2 EPFL MATHICSE-CMCS, Station 8-MA, CH-1015, Lausanne, Switzerland
3 SISSA, International School for Advanced Studies, Mathlab, Via Bonomea 265, 34136 Trieste, Italy
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Denis Devaud; Andrea Manzoni; Gianluigi Rozza. A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 593-598. doi : 10.1016/j.crma.2013.07.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.07.023/

[1] Y. Cao; Z. Chen; M. Gunzburger ANOVA expansions and efficient sampling methods for parameter dependent nonlinear PDEs, Int. J. Numer. Anal. Model., Volume 6 (2009), pp. 256-273

[2] J.L. Eftang; B. Stamm Parameter multi-domain hp empirical interpolation, Int. J. Numer. Methods Eng., Volume 90 (2012) no. 4, pp. 412-428

[3] J.L. Eftang; A.T. Patera; E.M. Rønquist An “hp” certified reduced basis method for parametrized elliptic partial differential equations, SIAM J. Sci. Comput., Volume 32 (2010) no. 6, pp. 3170-3200

[4] Z. Gao; J.S. Hesthaven On ANOVA expansions and strategies for choosing the anchor point, Appl. Math. Comput., Volume 217 (2010) no. 7, pp. 3274-3285

[5] M. Gunzburger; A. Labovsky An efficient and accurate numerical method for high-dimensional stochastic PDEs, SIAM J. Sci. Comput. (2013) (submitted for publication)

[6] J.S. Hesthaven; S. Zhang On the use of ANOVA expansions in reduced basis methods for high-dimensional parametric partial differential equations, Scientific Computing Group, Brown University, Providence, RI, USA, 2011 (Technical Report 2011-31 submitted for publication)

[7] A. Quarteroni; G. Rozza; A. Manzoni Certified reduced basis approximation for parametrized partial differential equations in industrial applications, J. Math. Ind., Volume 1 (2011) no. 3

[8] G. Rozza; D.B.P. Huynh; A.T. Patera Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., Volume 15 (2008), pp. 229-275

[9] I.M. Sobol Theorems and examples on high dimensional model representation, Reliab. Eng. Syst. Saf., Volume 79 (2003), pp. 187-193

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