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Verifiability of the Data-Driven Variational Multiscale Reduced Order Model

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Abstract

In this paper, we focus on the mathematical foundations of reduced order model (ROM) closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that the data-driven ROM closure studied here (i.e., the data-driven variational multiscale ROM) is verifiable. Finally, we investigate the verifiability of the data-driven variational multiscale ROM in the numerical simulation of the one-dimensional Burgers equation and a two-dimensional flow past a circular cylinder at Reynolds numbers \(Re=100\) and \(Re=1000\).

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Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Notes

  1. For the case \(r=20\), for about \(5\%\) of the total 400 possible k values, the constrained linear least squares solver lsqlin fails to converge. These k values are scattered around \(k=300\). We did not include the corresponding \(\mathcal {E} (L^2)\) and \(\eta (L^2)\) data in Fig. 1 and we also excluded them when computing the corresponding linear regression slope presented in Fig. 2.

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Acknowledgements

We thank the reviewers for the insightful comments and suggestions, which have significantly improved the paper. The work of the first, second, and sixth authors was supported by NSF through grants DMS-2012253 and CDS &E-MSS-1953113. The third author acknowledges the support by NSF through grant DMS-2108856. The fifth author acknowledges the support by European Union Funding for Research and Innovation – Horizon 2020 Program – in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics,” the PRIN 2017 “Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations” (NA-FROM-PDEs), and the INDAM-GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali.”

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Authors and Affiliations

  1. IFP Energies Nouvelles, 92500, Rueil-Malmaison, France

    Birgul Koc

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Changhong Mou

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Honghu Liu & Traian Iliescu

  4. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Zhu Wang

  5. MathLab, Mathematics Area, SISSA, 34136, Trieste, Italy

    Gianluigi Rozza

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  1. Birgul Koc
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  2. Changhong Mou
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Correspondence to Traian Iliescu.

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Koc, B., Mou, C., Liu, H. et al. Verifiability of the Data-Driven Variational Multiscale Reduced Order Model. J Sci Comput 93, 54 (2022). https://doi.org/10.1007/s10915-022-02019-y

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