Nonlinear compressive reduced basis approximation for PDE’s

Albert Cohen; Charbel Farhat; Yvon Maday; Agustin Somacal

doi:10.5802/crmeca.191

Nonlinear compressive reduced basis approximation for PDE’s

Albert Cohen ¹; Charbel Farhat ^2,^3,⁴; Yvon Maday ¹ ; Agustin Somacal ¹

¹ Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France
² Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
³ Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA
⁴ Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USA

Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 357-374.

Abstract
Résumé

Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case.

Les techniques linéaires de réduction de modèles proposent, hors ligne, des sous-espaces de faible dimension adaptés à l’approximation des solutions d’une équation aux dérivées partielles paramétrée, dans le but d’effectuer des simulations numériques rapides en ligne. Ces méthodes, telles que la décomposition orthogonale appropriée (POD) ou les méthodes de base réduite (RB), sont très efficaces lorsque la famille de solutions a des valeurs propres de Karhunen–Loève ou des épaisseurs de Kolmogorov à décroissance rapide, reflétant la possibilité d’approximation par des espaces linéaires de dimension finie. D’autre part, elles deviennent inefficaces lorsque ces quantités ont une décroissance lente, en particulier pour les familles de solutions aux équations de transport hyperboliques avec des positions de choc dépendant des paramètres. L’objectif de ce travail est d’explorer la capacité de la réduction de modèle non linéaire à contourner cette situation particulière. À cette fin, nous décrivons d’abord des notions particulières d’épaisseurs non linéaires qui ont une décroissance substantiellement plus rapide pour les familles susmentionnées. Ensuite, nous discutons d’une approche systématique permettant d’obtenir de meilleures performances via une reconstruction non linéaire à partir des premières coordonnées d’une approximation de modèle réduit linéaire, ce qui nous permet de rester dans le même cadre “classique” de la réduction de modèle basée sur la projection. Nous analysons l’approche et rendons compte de ses performances pour un cas test univarié simple mais instructif.

Received: 2023-03-15
Revised: 2023-04-16
Accepted: 2023-04-20
Online First: 2023-09-28
Published online: 2024-04-26

DOI: 10.5802/crmeca.191

Keywords: non linear reduced basis, compressed sensing, solution manifold, machine learning, $m$-width
Mot clés : base réduite non linéaire, acquisition comprimée, variété des solutions, apprentissage automatique, $m$-épaisseur

Author's affiliations:

Albert Cohen ¹; Charbel Farhat ^{2, 3, 4}; Yvon Maday ¹; Agustin Somacal ¹

¹ Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France
² Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
³ Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA
⁴ Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Albert Cohen and Charbel Farhat and Yvon Maday and Agustin Somacal},
     title = {Nonlinear compressive reduced basis approximation for {PDE{\textquoteright}s}},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {357--374},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
     number = {S1},
     year = {2023},
     doi = {10.5802/crmeca.191},
     language = {en},
}

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Albert Cohen; Charbel Farhat; Yvon Maday; Agustin Somacal. Nonlinear compressive reduced basis approximation for PDE’s. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 357-374. doi : 10.5802/crmeca.191. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.191/

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