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.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.

Methods for data-driven multiscale model discovery for materials

“…The SINDy framework has been recently generalized by Loiseau and Brunton [39] to incorporate known physical constraints and symmetries in the equations by implementing a constrained sequentially thresholded leastsquares optimization. Such constraints are known to promote stability [46][47][48]. Loiseau et al [49] also demonstrated the ability of SINDy to identify dynamical systems models of high-dimensional systems, such as fluid flows, from a few physical sensor measurements.…”

Challenges of order reduction techniques for problems involving polymorphic uncertainty

“…A similar technique utilizing approximation instead of interpolation is the gappy POD . Other hyperreduction techniques are the energy conserving sampling and weighting (ECSW) and the Gauss‐Newton with approximated tensors (GNAT) …”

Towards a stable and fast dynamic skeletal muscle model

“…[1], [4]) of the first-order system (4) by setting u ≈ V u z u , v ≈ V v z v and w ≈ V w z w , i.e. by separately projecting the position, velocity and pressure DOFs.…”

“…The equations describing the muscle deformation over time are the balance of linear momentum subject to the incompressibility constraint ρ 0 (X ) ∂V ∂t (X , t) = ∇ · P(X , t) + B(X , t) , s.t. J(X , t) = 1 , ∀ X ∈ Ω 0 , (1) and a nonlinear constitutive equation of the form P(X , t) = P iso (X , t) + P aniso (X , t) + P act (X , t) + P visc (X , t)…”

“…Inserting equation (2) into equation (1), introducing a parameter µ ∈ P ⊂ R p for varying muscle properties and discretising the resulting equation using the finite element method (Taylor-Hood elements), one obtains the following second-order DAE system …”