Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds

Patrick Buchfink; Silke Glas; Bernard Haasdonk

doi:10.5802/crmath.632

Article de recherche - Analyse numérique

Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds
[Bornes d’erreur pour la réduction de modèles sur des variétés construites par des applications polynomiales]

Patrick Buchfink ^1,² ; Silke Glas ² ; Bernard Haasdonk ¹

¹ Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
² University of Twente, Department of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The Netherlands

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1881-1891.

Résumé
Abstract

Pour la réduction de l’ordre des modèles basée sur la projection d’un sous-espace linéaire, il est bien connu que la largeur de Kolmogorov n décrit la meilleure erreur possible pour un modèle d’ordre réduit de taille $n$ . Dans cet article, nous fournissons des bornes d’erreur pour les modèles d’ordre réduit sur des variétés construites par des applications polynomiales. En particulier, nous montrons que les bornes d’erreur dépendent du degré polynomial $p$ de l’application ainsi que de la largeur linéaire de Kolmogorov $n$ pour le problème sous-jacent. Il en résulte une largeur de Kolmogorov $(n, p)$ , qui décrit une borne inférieure pour la meilleure erreur possible pour un modèle d’ordre réduit sur des variété construites par des applications polynomiales du degré polynomial $p$ et de taille réduite $n$ .

For projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov $n$ -width describes the best-possible error for a reduced order model (ROM) of size $n$ . In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. In particular, we show that the approximation bounds depend on the polynomial degree $p$ of the mapping function as well as on the linear Kolmogorov $n$ -width for the underlying problem. This results in a Kolmogorov $(n, p)$ -width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree $p$ and reduced size $n$ .

Reçu le : 2023-04-19
Révisé le : 2024-03-19
Accepté le : 2024-03-19
Publié le : 2024-12-03

DOI : 10.5802/crmath.632

Classification : 41A45, 41A46, 41A65, 65L70, 65M15, 65N15
Keywords: Model Order Reduction, nonlinear manifolds, polynomial mappings, polynomial $(n,p)$-widths
Mots-clés : Réduction de l’ordre des modèles, variétés non linéaires, transformations polynomiales, $(n,p)$-épaisseur polynomiales

Affiliations des auteurs :

Patrick Buchfink ^{1, 2} ; Silke Glas ² ; Bernard Haasdonk ¹

¹ Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
² University of Twente, Department of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The Netherlands

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Patrick Buchfink and Silke Glas and Bernard Haasdonk},
     title = {Approximation {Bounds} for {Model} {Reduction} on {Polynomially} {Mapped} {Manifolds}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1881--1891},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.632},
     language = {en},
}

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Patrick Buchfink; Silke Glas; Bernard Haasdonk. Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1881-1891. doi : 10.5802/crmath.632. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.632/

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