Evolve Then Filter Regularization for Stochastic Reduced Order Modeling

Evolve Then Filter Regularization for Stochastic Reduced Order Modeling Xuping Xie 1,†*, Feng Bao 2,‡ and Clayton Webster 3,† 1 Oak Ridge National Lab; xiex@ornl.gov 2 Florida State University; bao@math.fsu.edu 3 Oak Ridge National Lab; webstercg@ornl.gov * Correspondence: xiex@ornl.gov † Address: One Beth Valley Road, Oak Ridge National Lab, Oak Ridge, TN, 37831 ‡ Address: 1017 Academic Way, Tallahassee, FL 32306

[1]  Honghu Liu,et al.  Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs , 2014, 1403.4198.

[2]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[3]  Monika Neda,et al.  Numerical Analysis of Filter-Based Stabilization for Evolution Equations , 2012, SIAM J. Numer. Anal..

[4]  Leo G. Rebholz,et al.  Approximate Deconvolution Models of Turbulence , 2012 .

[5]  Steven L. Brunton,et al.  Sparsity enabled cluster reduced-order models for control , 2016, J. Comput. Phys..

[6]  Gianluigi Rozza,et al.  Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations , 2015 .

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

[8]  A. Quarteroni,et al.  A reduced computational and geometrical framework for inverse problems in hemodynamics , 2013, International journal for numerical methods in biomedical engineering.

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

[10]  Darryl D. Holm,et al.  Regularization modeling for large-eddy simulation , 2002, nlin/0206026.

[11]  Traian Iliescu,et al.  An evolve‐then‐filter regularized reduced order model for convection‐dominated flows , 2015, 1506.07555.

[12]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[13]  Matthew F. Barone,et al.  On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment , 2010 .

[14]  Gianluigi Rozza,et al.  Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation , 2009 .

[15]  Traian Iliescu,et al.  Proper orthogonal decomposition closure models for fluid flows: Burgers equation , 2013, 1308.3276.

[16]  Gilead Tadmor,et al.  Reduced-Order Modelling for Flow Control , 2013 .

[17]  Omer San,et al.  Neural network closures for nonlinear model order reduction , 2017, Adv. Comput. Math..

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

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

[20]  Karsten Urban,et al.  Reduced Basis Methods for Parameterized Partial Differential Equations with Stochastic Influences Using the Karhunen-Loève Expansion , 2013, SIAM/ASA J. Uncertain. Quantification.

[21]  Earl H. Dowell,et al.  Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation , 2013, Journal of Fluid Mechanics.

[22]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

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

[24]  Juan Du,et al.  Non-linear model reduction for the Navier-Stokes equations using residual DEIM method , 2014, J. Comput. Phys..

[25]  Laurent Cordier,et al.  Calibration of POD reduced‐order models using Tikhonov regularization , 2009 .

[26]  M. Germano.  Differential filters for the large eddy numerical simulation of turbulent flows , 1986 .

[27]  Traian Iliescu,et al.  SUPG reduced order models for convection-dominated convection–diffusion–reaction equations , 2014 .

[28]  Arnulf Jentzen,et al.  Galerkin Approximations for the Stochastic Burgers Equation , 2013, SIAM J. Numer. Anal..

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

[30]  P. Kloeden,et al.  Taylor Approximations for Stochastic Partial Differential Equations , 2011 .

[31]  S. Aachen.  Stochastic Differential Equations An Introduction With Applications , 2016 .

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

[33]  Matthew F. Barone,et al.  Stable Galerkin reduced order models for linearized compressible flow , 2009, J. Comput. Phys..

[34]  Steven L. Brunton,et al.  Dynamic Mode Decomposition with Control , 2014, SIAM J. Appl. Dyn. Syst..

[35]  J. Burkardt,et al.  REDUCED ORDER MODELING OF SOME NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[36]  M. Germano.  Differential filters of elliptic type , 1986 .

[37]  Omer San,et al.  Machine learning closures for model order reduction of thermal fluids , 2018, Applied Mathematical Modelling.

[38]  G. Lord,et al.  A numerical scheme for stochastic PDEs with Gevrey regularity , 2004 .

[39]  Gianluigi Rozza,et al.  Certified reduced basis approximation for parametrized partial differential equations and applications , 2011 .

[40]  Steven L. Brunton,et al.  Compressive sampling and dynamic mode decomposition , 2013, 1312.5186.

[41]  Y. Maday,et al.  Reduced Basis Techniques for Stochastic Problems , 2010, 1004.0357.

[42]  Feriedoun Sabetghadam,et al.  α Regularization of the POD-Galerkin dynamical systems of the Kuramoto-Sivashinsky equation , 2012, Appl. Math. Comput..

[43]  Gianluigi Rozza,et al.  A Weighted Reduced Basis Method for Elliptic Partial Differential Equations with Random Input Data , 2013, SIAM J. Numer. Anal..

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

[45]  I. Gyongy,et al.  On Numerical Approximation of Stochastic Burgers' Equation , 2006 .

[46]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[47]  Charbel Farhat,et al.  The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows , 2012, J. Comput. Phys..

[48]  Dirk Blömker.  Amplitude Equations for Stochastic Partial Differential Equations , 2007, Interdisciplinary Mathematical Sciences.

[49]  David Galbally,et al.  Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems , 2009 .

[50]  Steven L. Brunton,et al.  Sparse Identification of Nonlinear Dynamics (SINDy) , 2016 .

[51]  M. A. Muñoz,et al.  Multiplicative Noise in Non-equilibrium Phase Transitions: A tutorial , 2003, cond-mat/0303650.

[52]  Thomas Y. Hou,et al.  Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics , 2006, J. Comput. Phys..

[53]  G. Rozza,et al.  Stabilized reduced basis method for parametrized advection-diffusion PDEs , 2014 .

[54]  Traian Iliescu,et al.  Regularized Reduced Order Models for a Stochastic Burgers Equation , 2017, 1701.01155.

[55]  Charbel Farhat,et al.  Stabilization of projection‐based reduced‐order models , 2012 .