Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds
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...
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Académie des sciences
2024-12-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.632/ |
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| author | Buchfink, Patrick Glas, Silke Haasdonk, Bernard |
| author_facet | Buchfink, Patrick Glas, Silke Haasdonk, Bernard |
| author_sort | Buchfink, Patrick |
| collection | DOAJ |
| description | 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 $n$. |
| format | Article |
| id | doaj-art-f09aa9efd6ea49c6a9e07db702060c8f |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-f09aa9efd6ea49c6a9e07db702060c8f2025-02-07T11:27:00ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-12-01362G131881189110.5802/crmath.63210.5802/crmath.632Approximation Bounds for Model Reduction on Polynomially Mapped ManifoldsBuchfink, Patrick0Glas, Silke1https://orcid.org/0000-0003-3274-1615Haasdonk, Bernard2Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany; University of Twente, Department of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The NetherlandsUniversity of Twente, Department of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The NetherlandsInstitute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, GermanyFor 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 $n$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.632/Model Order Reductionnonlinear manifoldspolynomial mappingspolynomial $(n,p)$-widths |
| spellingShingle | Buchfink, Patrick Glas, Silke Haasdonk, Bernard Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds Comptes Rendus. Mathématique Model Order Reduction nonlinear manifolds polynomial mappings polynomial $(n,p)$-widths |
| title | Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds |
| title_full | Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds |
| title_fullStr | Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds |
| title_full_unstemmed | Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds |
| title_short | Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds |
| title_sort | approximation bounds for model reduction on polynomially mapped manifolds |
| topic | Model Order Reduction nonlinear manifolds polynomial mappings polynomial $(n,p)$-widths |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.632/ |
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