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

TL;DR

This paper compares Galerkin and least-squares Petrov-Galerkin (LSPG) projection methods for nonlinear model reduction, revealing conditions for equivalence, error bounds, and the importance of matching time step to spectral content for improved accuracy and efficiency.

Contribution

It provides a detailed theoretical and computational comparison of Galerkin and LSPG methods, including new conditions for equivalence and error bounds, and insights on time step selection.

Findings

Decreasing time step does not always reduce error in LSPG ROM.

Matching time step to spectral content improves accuracy and reduces computational cost.

LSPG can be equivalent to Galerkin under certain conditions.

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. 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.