Separated representations at the heart of Proper Generalized Decomposition are constructed incrementally by minimizing the problem residual. However, the modes involved in the resulting decomposition do not exhibit a clear multi-scale character. In order to recover a multi-scale description of the solution within a separated representation framework, we study the use of wavelets for approximating the functions involved in the separated representation of the solution. We will prove that such an approach allows separating the different scales as well as taking profit from its multi-resolution behavior for defining adaptive strategies.
Accepté le :
Publié le :
@article{CRMECA_2018__346_7_485_0,
author = {Angel Leon and Anais Barasinski and Emmanuelle Abisset-Chavanne and Elias Cueto and Francisco Chinesta},
title = {Wavelet-based multiscale proper generalized decomposition},
journal = {Comptes Rendus. M\'ecanique},
pages = {485--500},
publisher = {Elsevier},
volume = {346},
number = {7},
year = {2018},
doi = {10.1016/j.crme.2018.04.013},
language = {en},
}
TY - JOUR AU - Angel Leon AU - Anais Barasinski AU - Emmanuelle Abisset-Chavanne AU - Elias Cueto AU - Francisco Chinesta TI - Wavelet-based multiscale proper generalized decomposition JO - Comptes Rendus. Mécanique PY - 2018 SP - 485 EP - 500 VL - 346 IS - 7 PB - Elsevier DO - 10.1016/j.crme.2018.04.013 LA - en ID - CRMECA_2018__346_7_485_0 ER -
%0 Journal Article %A Angel Leon %A Anais Barasinski %A Emmanuelle Abisset-Chavanne %A Elias Cueto %A Francisco Chinesta %T Wavelet-based multiscale proper generalized decomposition %J Comptes Rendus. Mécanique %D 2018 %P 485-500 %V 346 %N 7 %I Elsevier %R 10.1016/j.crme.2018.04.013 %G en %F CRMECA_2018__346_7_485_0
Angel Leon; Anais Barasinski; Emmanuelle Abisset-Chavanne; Elias Cueto; Francisco Chinesta. Wavelet-based multiscale proper generalized decomposition. Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 485-500. doi : 10.1016/j.crme.2018.04.013. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.04.013/
[1] A short review in model order reduction based on Proper Generalized Decomposition, Arch. Comput. Methods Eng., Volume 18 (2011), pp. 395-404
[2] Parametric PGD based computational vademecum for efficient design, optimization and control, Arch. Comput. Methods Eng., Volume 20 (2013) no. 1, pp. 31-59
[3] Separated Representations and PGD Based Model Reduction: Fundamentals and Applications (F. Chinesta; P. Ladevèze, eds.), CISM–Springer, 2014
[4] Model order reduction, Encyclopedia of Computational Mechanics, Wiley, 2016
[5] A priori hyperreduction method: an adaptive approach, J. Comput. Phys., Volume 202 (2005), pp. 346-366
[6] Model Reduction Using Proper Orthogonal Decomposition, Lecture Notes, Institute of Mathematics and Scientific Computing, University of Graz, Austria, 2011
[7] A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., Volume 57 (2015) no. 4, pp. 483-531
[8] Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Pappalardo Monographs in Mechanical Engineering, 2007 http://augustine.mit.edu (online at)
[9] Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics, Arch. Comput. Methods Eng., Volume 15 (2008) no. 3, pp. 229-275
[10] The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables, C. R. Acad. Sci. Paris, Ser. IIb, Volume 309 (1989), pp. 1095-1099
[11] Nonlinear Computational Structural Mechanics. New Approaches and Non-incremental Methods of Calculation, Springer-Verlag, 1999
[12] A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newton. Fluid Mech., Volume 139 (2006), pp. 153-176
[13] A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space–time separated representation, J. Non-Newton. Fluid Mech., Volume 144 (2007), pp. 98-121
[14] Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms, Comput. Methods Appl. Mech. Eng., Volume 197 (2008), pp. 4718-4736
[15] Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity, Comput. Methods Appl. Mech. Eng., Volume 201 (2012), pp. 1-12
[16] Recent advances and new challenges in the use of the Proper Generalized Decomposition for solving multidimensional models, Arch. Comput. Methods Eng., Volume 17 (2010) no. 4, pp. 327-350
[17] The Proper Generalized Decomposition for Advanced Numerical Simulations. A Primer, Springerbriefs, Springer, 2014
[18] An error estimator for separated representations of highly multidimensional models, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1872-1880
[19] On the verification of model reduction methods based on the proper generalized decomposition, Comput. Methods Appl. Mech. Eng., Volume 200 (2011), pp. 2032-2047
[20] A separated representation of an error indicator for the mesh refinement process under the Proper Generalized Decomposition framework, Comput. Mech., Volume 55 (2015) no. 2, pp. 251-266
[21] Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numer. Math., Volume 121 (2012) no. 3, pp. 503-530
[22] Wavelet Methods for Dynamical Problems with Application to Metallic, Composite, and Nano-Composite Structures, CRC Press, Taylor & Francis, 2010
[23] Orthonormal basis of compactly supported wavelets, Commun. Pure Appl. Math., Volume 41 (1988) no. 7, pp. 909-996
[24] Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math., Volume 32 (2000) no. 3, pp. 247-254
[25] The wavelet-Galerkin method for solving PDE's with spatially dependent variables, Numer. Methods Acoust. Vib., Volume 326 (2012), p. R33
[26] A three-step wavelet Galerkin method for parabolic and hyperbolic partial differential equations, Int. J. Comput. Math., Volume 83 (2006) no. 1, pp. 143-157
[27] Wavelet Galerkin method for numerical solution of nonlinear integral equations, Appl. Math. Comput., Volume 167 (2005) no. 2, pp. 1119-1129
[28] The evaluation of connection coefficients of compactly supported wavelets, Proceedings of the French–USA Workshop on Wavelets and Turbulence, Springer, 1991
[29] Non-intrusive sparse subspace learning for parametrized problems, Arch. Comput. Methods Eng. (2017) | DOI
Cité par Sources :
Commentaires - Politique
Incremental dynamic mode decomposition: A reduced-model learner operating at the low-data limit
Agathe Reille; Nicolas Hascoet; Chady Ghnatios; ...
C. R. Méca (2019)
Algebraic PGD for tensor separation and compression: An algorithmic approach
Pedro Díez; Sergio Zlotnik; Alberto García-González; ...
C. R. Méca (2018)
Numerical approaches on driven elastic interfaces in random media
Ezequiel E. Ferrero; Sebastian Bustingorry; Alejandro B. Kolton; ...
C. R. Phys (2013)