Comptes Rendus
no. 7
Model reduction, data-based and advanced discretization in computational mechanics
A door to model reduction in high-dimensional parameter space
[Vers des modèles réduits dans les espaces de très grande dimension]
Charles Paillet1  ; David Néron1 ; Pierre Ladevèze1
1 LMT (ENS Paris-Saclay, CNRS, Université Paris-Saclay), 61, avenue du Président-Wilson, 94235 Cachan cedex, France
Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 524-531.

Les modèles réduits, en particulier ceux basés sur la Proper Generalized Decomposition (PGD) sont des outils de conception qui s'apprêtent à révolutionner la simulation numérique. Malheureusement, pour les problèmes à grand nombre de paramètres, la « malédiction de la dimensionnalité » semble être une limitation majeure. Nous proposons, avec la parameter-multiscale PGD, une solution à ce problème basée sur le principe de Saint-Venant. Un cas test comprenant jusqu'à mille paramètres est présenté dans cet article et prouve que la méthode permet bien de s'affranchir de la « malédiction de la dimensionnalité ».

Model reduction techniques such as Proper Generalized Decomposition (PGD) are decision-making tools that are about to revolutionize many domains. Unfortunately, their computation is still problematic for problems involving many parameters, for which one has to face the “curse of dimensionality”. An answer to this challenge is given in solid mechanics by the so-called “parameter-multiscale PGD”, which is based on Saint-Venant's principle. In this article, a model problem composed of up to a thousand parameters is presented, showing that the method is able to overcome the “curse of dimensionality”.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2018.04.009
Keywords: Reduced Order Model (ROM), PGD, Multiparametric
Mot clés : Réduction de modèles, PGD, Multiparamétrique
Charles Paillet 1 ; David Néron 1 ; Pierre Ladevèze 1

1 LMT (ENS Paris-Saclay, CNRS, Université Paris-Saclay), 61, avenue du Président-Wilson, 94235 Cachan cedex, France 
@article{CRMECA_2018__346_7_524_0,
     author = {Charles Paillet and David N\'eron and Pierre Ladev\`eze},
     title = {A door to model reduction in high-dimensional parameter space},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {524--531},
     publisher = {Elsevier},
     volume = {346},
     number = {7},
     year = {2018},
     doi = {10.1016/j.crme.2018.04.009},
     language = {en},
}
TY  - JOUR
AU  - Charles Paillet
AU  - David Néron
AU  - Pierre Ladevèze
TI  - A door to model reduction in high-dimensional parameter space
JO  - Comptes Rendus. Mécanique
PY  - 2018
SP  - 524
EP  - 531
VL  - 346
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crme.2018.04.009
LA  - en
ID  - CRMECA_2018__346_7_524_0
ER  - 
%0 Journal Article
%A Charles Paillet
%A David Néron
%A Pierre Ladevèze
%T A door to model reduction in high-dimensional parameter space
%J Comptes Rendus. Mécanique
%D 2018
%P 524-531
%V 346
%N 7
%I Elsevier
%R 10.1016/j.crme.2018.04.009
%G en
%F CRMECA_2018__346_7_524_0
Charles Paillet; David Néron; Pierre Ladevèze. A door to model reduction in high-dimensional parameter space. Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 524-531. doi : 10.1016/j.crme.2018.04.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.04.009/

[1] Separated Representations and PGD-Based Model Reduction: Fundamentals and Applications, vol. CISM 554 (F. Chinesta; P. Ladevèze, eds.), Springer, 2014

[2] P. Ladevèze On algorithm family in structural mechanics, C. R. Acad. Sci. Paris, Ser. IIb, Volume 300 (1985) no. 2, pp. 41-44

[3] P. Ladevèze The large time increment method for the analyse of structures with nonlinear constitutive relation described by internal variables, C. R. Acad. Sci. Paris, Ser. IIb, Volume 309 (1989) no. 2, pp. 1095-1099 (in French)

[4] P. Ladevèze On reduced models in nonlinear solid mechanics, Eur. J. Mech. A, Solids, Volume 60 (2016), pp. 227-237

[5] P. Ladevèze Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation, Springer-Verlag, New York, 1999

[6] P. Ladevèze New Methods with Conditioner for PGD Reduced Order Models, LMT Cachan, France, 2014 Technical report (in French)

[7] A. Falco; W. Hackbusch; A. Nouy (2015), pp. 1-50 (work document)

[8] W. Hackbusch Tensor Spaces and Numerical Tensor Calculus, Springer, 2012 (series in ed)

[9] M. Billaud-Friess; A. Nouy; O. Zahm A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems, ESAIM: Math. Model. Numer. Anal., Volume 48 (2014) no. 6, pp. 1777-1806

[10] S. Holtz; T. Rohwedder; R. Schneider On manifolds of tensors of fixed TT-rank, Numer. Math., Volume 120 (2012) no. 4, pp. 701-731

[11] I.V. Oseledets Tensor-train decomposition, SIAM J. Sci. Comput., Volume 33 (2011) no. 5, pp. 2295-2317

[12] P. Ladevèze A New Method for the ROM Computation: The Parameter-Multiscale PGD, LMT Cachan, 2016 Technical report (in French)

[13] P. Ladevèze; C. Paillet; D. Néron Extended-PGD model reduction for nonlinear solid mechanics problems involving many parameters, Comput. Methods Appl. Sci., Volume 46 (2018), pp. 201-220

[14] P. Ladevèze; J.-C. Passieux; D. Néron The LATIN multiscale computational method and the proper generalized decomposition, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1287-1296

[15] P. Ladevèze New Variational Formulations for Discontinuous Approximations, LMT Cachan, 2011 Technical report (in French)

[16] P. Ladevèze; H. Riou On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics, Comput. Methods Appl. Mech. Eng., Volume 278 (2014), pp. 729-743

[17] A. Ammar; B. Mokdad; F. Chinesta; R. Keunings A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations, J. Non-Newton. Fluid Mech., Volume 144 (2007) no. 2–3, pp. 98-121

Cité par Sources :

Commentaires - Politique