Galerkin v. leastsquares Petrov–Galerkin projection in nonlinear model reduction
Leastsquares Petrov–Galerkin (LSPG) modelreduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for largescale 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 timecontinuous level, while LSPG techniques do so at the timediscrete 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 timecontinuous representation, conditions under which the two techniques are equivalent, and timediscrete 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 compressibleflow problem with over one million unknowns, we show that increasing themore »
 Authors:

^{[1]};
^{[2]};
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 Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 George Mason Univ., Fairfax, VA (United States)
 Publication Date:
 Report Number(s):
 SAND20168178J; SAND201510809J
Journal ID: ISSN 00219991; 646813; TRN: US1700184
 Grant/Contract Number:
 AC0494AL85000
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Name: Journal of Computational Physics; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Research Org:
 Sandia National Lab. (SNLCA), Livermore, CA (United States); Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; model reduction; GNAT; leastsquares Petrov–Galerkin projection; Galerkin projection; CFD
 OSTI Identifier:
 1333617
 Alternate Identifier(s):
 OSTI ID: 1338307; OSTI ID: 1398568
Carlberg, Kevin Thomas, Barone, Matthew F., and Antil, Harbir. Galerkin v. leastsquares Petrov–Galerkin projection in nonlinear model reduction. United States: N. p.,
Web. doi:10.1016/j.jcp.2016.10.033.
Carlberg, Kevin Thomas, Barone, Matthew F., & Antil, Harbir. Galerkin v. leastsquares Petrov–Galerkin projection in nonlinear model reduction. United States. doi:10.1016/j.jcp.2016.10.033.
Carlberg, Kevin Thomas, Barone, Matthew F., and Antil, Harbir. 2016.
"Galerkin v. leastsquares Petrov–Galerkin projection in nonlinear model reduction". United States.
doi:10.1016/j.jcp.2016.10.033. https://www.osti.gov/servlets/purl/1333617.
@article{osti_1333617,
title = {Galerkin v. leastsquares Petrov–Galerkin projection in nonlinear model reduction},
author = {Carlberg, Kevin Thomas and Barone, Matthew F. and Antil, Harbir},
abstractNote = {Leastsquares Petrov–Galerkin (LSPG) modelreduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for largescale 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 timecontinuous level, while LSPG techniques do so at the timediscrete 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 timecontinuous representation, conditions under which the two techniques are equivalent, and timediscrete 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 compressibleflow 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 reducedorder model by an order of magnitude.},
doi = {10.1016/j.jcp.2016.10.033},
journal = {Journal of Computational Physics},
number = ,
volume = ,
place = {United States},
year = {2016},
month = {10}
}