Comptes Rendus
Model reduction, data-based and advanced discretization in computational mechanics
Wavelet-based multiscale proper generalized decomposition
Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 485-500.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2018.04.013
Mots clés : Wavelets, Proper Generalized Decomposition, Multi-resolution, Multi-scale PGD

Angel Leon 1 ; Anais Barasinski 1 ; Emmanuelle Abisset-Chavanne 2 ; Elias Cueto 3 ; Francisco Chinesta 4

1 GeM Institute, École centrale de Nantes, 1, rue de la Noë, BP 92101, 44321 Nantes cedex 3, France
2 High Performance Computing Institute & ESI GROUP Chair, École centrale de Nantes, 1, rue de la Noë, BP 92101, 44321 Nantes cedex 3, France
3 I3A, University of Zaragoza, Maria de Luna s/n, 50018 Zaragoza, Spain
4 PIMM Laboratory & ESI GROUP Chair, ENSAM ParisTech, 151, boulevard de l'Hôpital, 75013 Paris, France
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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] F. Chinesta; P. Ladevèze; E. Cueto A short review in model order reduction based on Proper Generalized Decomposition, Arch. Comput. Methods Eng., Volume 18 (2011), pp. 395-404

[2] F. Chinesta; A. Leygue; F. Bordeu; J.V. Aguado; E. Cueto; D. Gonzalez; I. Alfaro; A. Ammar; A. Huerta 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] F. Chinesta; A. Huerta; G. Rozza; K. Willcox Model order reduction, Encyclopedia of Computational Mechanics, Wiley, 2016

[5] D. Ryckelynck A priori hyperreduction method: an adaptive approach, J. Comput. Phys., Volume 202 (2005), pp. 346-366

[6] S. Volkwein Model Reduction Using Proper Orthogonal Decomposition, Lecture Notes, Institute of Mathematics and Scientific Computing, University of Graz, Austria, 2011

[7] P. Benner; S. Gugercin; K. Willcox A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., Volume 57 (2015) no. 4, pp. 483-531

[8] A.T. Patera; G. Rozza 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] 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: application to transport and continuum mechanics, Arch. Comput. Methods Eng., Volume 15 (2008) no. 3, pp. 229-275

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

[12] 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 modeling of complex fluids, J. Non-Newton. Fluid Mech., Volume 139 (2006), pp. 153-176

[13] 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 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] A. Nouy 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] B. Bognet; A. Leygue; F. Chinesta; A. Poitou; F. Bordeu 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] F. Chinesta; A. Ammar; E. Cueto 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] F. Chinesta; R. Keunings; A. Leygue The Proper Generalized Decomposition for Advanced Numerical Simulations. A Primer, Springerbriefs, Springer, 2014

[18] A. Ammar; F. Chinesta; P. Diez; A. Huerta An error estimator for separated representations of highly multidimensional models, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1872-1880

[19] P. Ladevèze; L. Chamoin 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] E. Nadal; A. Leygue; F. Chinesta; M. Beringhier; J.J. Rodenas; F.J. Fuenmayor 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] A. Falcó; A. Nouy Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces, Numer. Math., Volume 121 (2012) no. 3, pp. 503-530

[22] S. Gopalakrishnan; M. Mitra Wavelet Methods for Dynamical Problems with Application to Metallic, Composite, and Nano-Composite Structures, CRC Press, Taylor & Francis, 2010

[23] I. Daubechies Orthonormal basis of compactly supported wavelets, Commun. Pure Appl. Math., Volume 41 (1988) no. 7, pp. 909-996

[24] A. Avudainayagam; C. Vani Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math., Volume 32 (2000) no. 3, pp. 247-254

[25] S. Jones; M. Legrand The wavelet-Galerkin method for solving PDE's with spatially dependent variables, Numer. Methods Acoust. Vib., Volume 326 (2012), p. R33

[26] B.V.R. Kumar; M. Mehra 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] Y. Mahmoudi Wavelet Galerkin method for numerical solution of nonlinear integral equations, Appl. Math. Comput., Volume 167 (2005) no. 2, pp. 1119-1129

[28] A. Latto; H.L. Resnikoff; E. Tannenbaum The evaluation of connection coefficients of compactly supported wavelets, Proceedings of the French–USA Workshop on Wavelets and Turbulence, Springer, 1991

[29] D. Borzacchiello; J.V. Aguado; F. Chinesta Non-intrusive sparse subspace learning for parametrized problems, Arch. Comput. Methods Eng. (2017) | DOI

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