Verifiability of the Data-Driven Variational Multiscale Reduced Order Model

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 a data-driven ROM closure (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. Reduced order model and variational multiscale and data-driven model and verifiability

[1]  Traian Iliescu,et al.  Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison , 2011, 1106.3585.

[2]  Traian Iliescu,et al.  Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network , 2019, Fluids.

[3]  Fei Lu,et al.  Data-Driven Model Reduction for Stochastic Burgers Equations , 2020, Entropy.

[4]  Annalisa Quaini,et al.  Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature , 2019, International Journal of Computational Fluid Dynamics.

[5]  Traian Iliescu,et al.  Variational multiscale proper orthogonal decomposition: Navier‐stokes equations , 2012, 1210.7389.

[6]  Santiago Badia,et al.  Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows , 2015 .

[7]  Gianluigi Rozza,et al.  A reduced order variational multiscale approach for turbulent flows , 2018, Advances in Computational Mathematics.

[8]  Changhong Mou,et al.  Commutation error in reduced order modeling of fluid flows , 2018, Advances in Computational Mathematics.

[9]  K. Willcox,et al.  Data-driven operator inference for nonintrusive projection-based model reduction , 2016 .

[10]  Zhu Wang,et al.  Artificial viscosity proper orthogonal decomposition , 2011, Math. Comput. Model..

[11]  Leo G. Rebholz,et al.  Improved Accuracy in Algebraic Splitting Methods for Navier-Stokes Equations , 2017, SIAM J. Sci. Comput..

[12]  L. Berselli,et al.  Mathematics of Large Eddy Simulation of Turbulent Flows , 2005 .

[13]  Samuele Rubino,et al.  A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations , 2021, J. Comput. Phys..

[14]  Gianluigi Rozza,et al.  Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations , 2020, Comput. Math. Appl..

[15]  Pierre Sagaut,et al.  Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow , 2003, Journal of Fluid Mechanics.

[16]  O. San,et al.  Data-Driven Variational Multiscale Reduced Order Models , 2020, ArXiv.

[17]  Gianluigi Rozza,et al.  Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations , 2015 .

[18]  Traian Iliescu,et al.  A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flows , 2017, IMA Journal of Numerical Analysis.

[19]  R. Codina,et al.  Projection-based reduced order models for flow problems: A variational multiscale approach , 2020, Computer Methods in Applied Mechanics and Engineering.

[20]  Annalisa Quaini,et al.  Pressure stabilization strategies for a LES filtering Reduced Order Model , 2021, Fluids.

[21]  Assad A. Oberai,et al.  Approximate optimal projection for reduced‐order models , 2016 .

[22]  Annalisa Quaini,et al.  A POD-Galerkin reduced order model for a LES filtering approach , 2020, J. Comput. Phys..

[23]  Zhu Wang,et al.  Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations , 2013, Math. Comput..

[24]  Zhu Wang,et al.  Approximate Deconvolution Reduced Order Modeling , 2015, 1510.02726.

[25]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[27]  Gianluigi Rozza,et al.  Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height , 2019, Comput. Math. Appl..

[28]  Changhong Mou,et al.  Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation , 2019, International Journal of Computational Fluid Dynamics.

[29]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[30]  Roger Lewandowski,et al.  Mathematical and Numerical Foundations of Turbulence Models and Applications , 2014 .

[31]  Gianluigi Rozza,et al.  Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models , 2018, J. Sci. Comput..

[32]  Omer San,et al.  Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks , 2018, Computational and Applied Mathematics.

[33]  Traian Iliescu,et al.  Data-Driven Filtered Reduced Order Modeling of Fluid Flows , 2017, SIAM J. Sci. Comput..

[35]  M. Mohebujjaman,et al.  Physically constrained data‐driven correction for reduced‐order modeling of fluid flows , 2018, International Journal for Numerical Methods in Fluids.

[36]  Mickaël D. Chekroun,et al.  Stochastic parameterizing manifolds and non-markovian reduced equations : stochastic manifolds for nonlinear SPDEs II/ Mickaël D. Chekroun, Honghu Liu, Shouhong Wang , 2014 .

[37]  Charles-Henri Bruneau,et al.  Enablers for robust POD models , 2009, J. Comput. Phys..

[38]  Volker John,et al.  Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder , 2004 .

[39]  Peter Benner,et al.  Data‐Driven Learning of Reduced‐Order Dynamics for a Parametrized Shallow Water Equation , 2020, PAMM.

[40]  Gianluigi Rozza,et al.  On a Certified Smagorinsky Reduced Basis Turbulence Model , 2017, SIAM J. Numer. Anal..

[41]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[42]  Zhu Wang,et al.  Variational Multiscale Proper Orthogonal Decomposition: Convection-Dominated Convection-Diffusion Equations , 2011, 1103.2729.

[43]  B. R. Noack.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 2013 .

[44]  Volker John,et al.  Large Eddy Simulation of Turbulent Incompressible Flows - Analytical and Numerical Results for a Class of LES Models , 2003, Lecture Notes in Computational Science and Engineering.

[45]  K. Duraisamy,et al.  A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method , 2017, 1712.09669.

[46]  William Layton,et al.  Introduction to the Numerical Analysis of Incompressible Viscous Flows , 2008 .

[47]  Zhu Wang,et al.  Numerical analysis of the Leray reduced order model , 2017, J. Comput. Appl. Math..

[48]  Volker John,et al.  On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows , 2015, SIAM Rev..