For approximating parametric problem solutions, Reduced Basis Methods (RBMs) are frequently proposed. They intend to reduce the computational costs of High Fidelity (HF) codes while maintaining the HF accuracy. They generally require an offline/online decomposition and a significant modification of the HF code in order for the online computation to be performed in short (or even real) time. We focus on the Non-Intrusive Reduced Basis (NIRB) two-grid method in this paper. Its main feature is the use of the HF code as a “black-box” on a coarse mesh for a new parameter during the online process, followed by an accuracy improvement based on the reduced basis paradigm that is realized in a very short time. Unlike other more intrusive methods, this approach does not necessitate code modification. As a result, it costs significantly less than an HF evaluation. This method was developed for elliptic equations with finite elements and has since then been extended to finite volume schemes.
In this paper, we extend the NIRB two-grid method to parabolic equations. We recover optimal estimates in natural norms, using the heat equation as a model problem and present several numerical results on the heat equation and on the Brusselator problem.
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Mots-clés : Non-intrusive reduced basis, parabolic equations, finite elements method
Elise Grosjean 1; Yvon Maday 2
CC-BY 4.0
@article{SMAI-JCM_2023__9__227_0,
author = {Elise Grosjean and Yvon Maday},
title = {Error estimate of the {Non-Intrusive} {Reduced} {Basis} {(NIRB)} two-grid method with parabolic equations},
journal = {The SMAI Journal of computational mathematics},
pages = {227--256},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {9},
year = {2023},
doi = {10.5802/smai-jcm.100},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/}
}
TY - JOUR AU - Elise Grosjean AU - Yvon Maday TI - Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations JO - The SMAI Journal of computational mathematics PY - 2023 SP - 227 EP - 256 VL - 9 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/ DO - 10.5802/smai-jcm.100 LA - en ID - SMAI-JCM_2023__9__227_0 ER -
%0 Journal Article %A Elise Grosjean %A Yvon Maday %T Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations %J The SMAI Journal of computational mathematics %D 2023 %P 227-256 %V 9 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/ %R 10.5802/smai-jcm.100 %G en %F SMAI-JCM_2023__9__227_0
Elise Grosjean; Yvon Maday. Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 227-256. doi : 10.5802/smai-jcm.100. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.100/
[1] An ’empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 339-9 (2004), pp. 667-672 | DOI | Numdam | MR | Zbl
[2] The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., Volume 25 (1993) no. 1, pp. 539-575 | DOI
[3] The mathematical theory of finite element methods, 15, Springer, 2007
[4] A priori convergence of the Greedy algorithm for the parametrized reduced basis method, ESAIM, Math. Model. Numer. Anal., Volume 46 (2012) no. 3, pp. 595 - 603 | DOI | Numdam | MR | Zbl
[5] Model order reduction for problems with large convection effects, Contributions to partial differential equations and applications (Computational Methods in Applied Sciences), Volume 47, Springer, 2019, pp. 131-150 | DOI | MR | Zbl
[7] Contribution à l’analyse numérique de quelques problèmes en chimie quantique et mécanique, Ph. D. Thesis, Université Pierre et Marie Curie-Paris VI (2009) | Numdam
[8] A two-grid finite-element/reduced basis scheme for the approximation of the solution of parametric dependent P.D.E, Actes du 9e Colloque national en calcul des structures, Giens (2009)
[9] A non-intrusive reduced basis approach for parametrized heat transfer problems, J. Comput. Phys., Volume 376 (2019), pp. pp.617-633 | DOI | MR | Zbl
[10] A Non-Intrusive Reduced Basis Method for Urban Flows Simulation, WCCM-ECCOMAS’20, 14th World Congress of Computational Mechanics and 8th European Congress on Computational Methods in Applied Sciences and Engineering (2021), p. 12p
[11] Greedy nonintrusive reduced order model for fluid dynamics, AIAA J., Volume 56 (2018) no. 12, pp. 4927-4943 | DOI
[12] Non-intrusive data-driven reduced-order modeling for time-dependent parametrized problems (2023) | arXiv
[13] Partial differential equations, American Mathematical Society, 2010
[14] POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition, Comput. Methods Appl. Mech. Eng., Volume 388 (2022), 114181, 27 pages | DOI | MR | Zbl
[15] Operator inference for non-intrusive model reduction with nonlinear manifolds (2022) | arXiv
[16] Variations and further developments on the Non-Intrusive Reduced Basis two-grid method, Ph. D. Thesis, Mathématiques Appliquées Sorbonne Université (2022) http://www.theses.fr/2022sorus019
[17] Error estimate of the non-intrusive reduced basis method with finite volume schemes, ESAIM, Math. Model. Numer. Anal., Volume 55 (2021) no. 5, pp. 1941-1961 | DOI | MR | Zbl
[18] The non-intrusive reduced basis two-grid method applied to sensitivity analysis (2023) | arXiv
[19] Data-driven reduced order modeling for time-dependent problems, Comput. Methods Appl. Mech. Eng., Volume 345 (2019), pp. 75-99 | DOI | MR | Zbl
[20] Convergence rates of the pod–greedy method, ESAIM, Math. Model. Numer. Anal., Volume 47 (2013) no. 3, pp. 859-873 | DOI | Numdam | MR | Zbl
[21] Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM, Math. Model. Numer. Anal., Volume 42 (2008) no. 2, pp. 277-302 | DOI | Numdam | MR | Zbl
[22] A real-time variational data assimilation method with data-driven model enrichment for time-dependent problems, Comput. Methods Appl. Mech. Eng., Volume 405 (2023), 115868, 37 pages | DOI | MR | Zbl
[23] New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3-4, pp. 251-266 | DOI | MR | Zbl
[24] Reduced basis methods for time-dependent problems, Acta Numer., Volume 31 (2022), pp. 265-345 | DOI | MR | Zbl
[25] Certified reduced basis methods for parametrized partial differential equations, Springer, 2016 | DOI
[26] A certified reduced basis method for the Fokker–Planck equation of dilute polymeric fluids: FENE dumbbells in extensional flow, SIAM J. Sci. Comput., Volume 32 (2010) no. 2, pp. 793-817 | DOI | MR | Zbl
[27] Über die beste Annaherung von Funktionen einer gegebenen Funktionenklasse, Ann. Math. (1936), pp. 107-110 | DOI | MR | Zbl
[28] Problemes aux limites non homogenes. II, Ann. Inst. Fourier, Volume 11 (1961), pp. 137-178 | DOI | Numdam | MR
[29] Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces, SIAM J. Sci. Comput., Volume 35 (2013) no. 6, p. A2417-A2441 | DOI | MR | Zbl
[30] Numerical solution of two-dimensional reaction–diffusion Brusselator system, Appl. Math. Comput., Volume 217 (2011) no. 12, pp. 5404-5415 | DOI | MR | Zbl
[31] Feel++: A computational framework for galerkin methods and advanced numerical methods, ESAIM: Proceedings, Volume 38, EDP Sciences (2012), pp. 429-455 | DOI | MR
[32] Reduced Basis Methods for Partial Differential Equations: an introduction, 92, Springer, 2015
[33] Approches multi-fidélités pour la simulation rapide d’écoulements d’air en milieu urbain, Ph. D. Thesis, Université Gustave Eiffel (2021)
[34] Galerkin finite element methods for parabolic problems, 25, Springer, 2007
[35] Non-intrusive reduced-order modelling of the Navier–Stokes equations based on RBF interpolation, Int. J. Numer. Methods Fluids, Volume 79 (2015) no. 11, pp. 580-595 | DOI | MR | Zbl
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