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

Abstract

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 increasingmore » 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.« less

Authors:

Carlberg, Kevin Thomas ^[1]; Barone, Matthew F. ^[2]; Antil, Harbir ^[3]

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+ Show Author Affiliations

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)

Publication Date:

Thu Oct 20 00:00:00 EDT 2016

Research Org.:

Sandia National Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:

1333617

Alternate Identifier(s):

OSTI ID: 1338307; OSTI ID: 1398568

Report Number(s):

SAND-2016-8178J; SAND-2015-10809J
Journal ID: ISSN 0021-9991; 646813; TRN: US1700184

Grant/Contract Number:  

AC04-94AL85000

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Name: Journal of Computational Physics; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

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