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Title: 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. However, 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. Here, 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 Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), 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; SAND-2015-10809J; 646813; TRN: US1700184
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
Citation Metrics:
Cited by: 107 works
Citation information provided by
Web of Science

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Cited By (12)

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

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Related Subjects

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