Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction

Journal Article · · Journal of Computational Physics
 [1];  [2];  [3]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  3. George Mason Univ., Fairfax, VA (United States)
+ Show Author Affiliations

Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.

Research Organization:
Sandia National Laboratories (SNL-CA), Livermore, CA (United States); Sandia National Laboratories, Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC04-94AL85000
OSTI ID:
1333617
Alternate ID(s):
OSTI ID: 1338307
OSTI ID: 1398568
Report Number(s):
SAND--2016-8178J; 646813
Journal Information:
Journal of Computational Physics, Journal Name: Journal of Computational Physics; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (42)

Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basisa posteriori error bounds journal January 2005
On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment journal August 2010
Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations
  • Carlberg, Kevin; Bou-Mosleh, Charbel; Farhat, Charbel
  • International Journal for Numerical Methods in Engineering, Vol. 86, Issue 2 https://doi.org/10.1002/nme.3050
journal October 2010
Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency: NONLINEAR MODEL REDUCTION AND ENERGY-CONSERVING SAMPLING AND WEIGHTING journal April 2014
Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation journal April 1991
Stability Properties of POD-Galerkin Approximations for the Compressible Navier-Stokes Equations journal March 2000
On Low-Dimensional Galerkin Models for Fluid Flow journal June 2000
Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction journal February 2015
Two-Step Greedy Algorithm for Reduced Order Quadratures journal May 2013
Stabilization of projection-based reduced order models of the Navier–Stokes journal August 2012
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations: Application to Transport and Continuum Mechanics journal May 2008
Application of a three-field nonlinear fluid–structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter journal January 2003
Structure-preserving model reduction for mechanical systems journal October 2003
Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison journal September 2012
A spectral viscosity method for correcting the long-term behavior of POD models journal February 2004
A priori hyperreduction method: an adaptive approach journal January 2005
Enablers for robust POD models journal February 2009
Stable Galerkin reduced order models for linearized compressible flow journal April 2009
Non-linear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods journal February 2013
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows journal June 2013
Model reduction for compressible flows using POD and Galerkin projection journal February 2004
The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows journal January 1999
Low-order modelling of laminar flow regimes past a confined square cylinder journal January 1999
The dynamics of coherent structures in the wall region of a turbulent boundary layer journal July 1988
Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation journal July 2013
Long time stability and convergence for fully discrete nonlinear galerkin methods journal January 1990
Dissipativity of numerical schemes journal August 1991
Turbulence and the dynamics of coherent structures. III. Dynamics and scaling journal January 1987
Missing Point Estimation in Models Described by Proper Orthogonal Decomposition journal November 2008
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods journal November 2001
Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space journal January 2008
Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s journal December 1977
Nonlinear Galerkin Methods journal October 1989
Nonlinear Model Reduction via Discrete Empirical Interpolation journal January 2010
Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation journal January 2012
Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems journal January 2012
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems journal January 2015
Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics journal January 2015
A New Look at Proper Orthogonal Decomposition journal January 2003
Optimizing cubature for efficient integration of subspace deformations journal December 2008
Karhunen–Loève procedure for gappy data journal January 1995
Aeroelastic Dynamic Analysis of a Full F-16 Configuration for Various Flight Conditions journal March 2003

Cited By (13)

Randomized methods to characterize large-scale vortical flow network text January 2019
Randomized low‐rank approximation methods for projection‐based model order reduction of large nonlinear dynamical problems journal January 2019
Commutation error in reduced order modeling of fluid flows journal December 2019
Data-Driven Science and Engineering book February 2019
Constrained sparse Galerkin regression journal January 2018
Sparse reduced-order modelling: sensor-based dynamics to full-state estimation journal April 2018
Assessment of reduced-order modeling strategies for convective heat transfer journal January 2020
Randomized methods to characterize large-scale vortical flow networks journal November 2019
Investigations and Improvement of Robustness of Reduced-Order Models of Reacting Flow journal December 2019
Exploration of Reduced-Order Models for Rocket Combustion Applications conference January 2018
Challenges in Reduced Order Modeling of Reacting Flows conference July 2018
Investigations and Improvement of Robustness of Reduced-Order Models of Reacting Flow conference January 2019
Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation text January 2017

Similar Records

Galerkin v. discrete-optimal projection in nonlinear model reduction
Journal Article · Wed Apr 01 00:00:00 EDT 2015 · Sandia journal manuscript; Not yet accepted for publication · OSTI ID:1427252
Space–Time Least-Squares Petrov–Galerkin Projection for Nonlinear Model Reduction
Journal Article · Tue Jan 01 23:00:00 EST 2019 · SIAM Journal on Scientific Computing · OSTI ID:1525743

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CFD
GNAT
Galerkin projection
least-squares Petrov–Galerkin projection
model reduction