Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

Abstract

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted 2 -norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.

SNS: A Solution-Based Nonlinear Subspace Method for Time-Dependent Model Order Reduction

“…This enables the ST-GNAT to achieve a speed-up when it is applied to nonlinear problems. Section 5.3 in [13] discusses three different options to determine the sampling indices (i.e.,Z). However, all these three options are simple variations of Algorithm 3 in [11] and Algorithm 5 in [10].…”

“…The original ST-GNAT paper introduces three different ways of collecting space-time residual snapshots, that are in turn used for the space-time residual basis construction (see Section 5.2 in [13]). Below is a list of the approaches introduced in […”

“…The SNS method is useful for model order reduction of a large-scale nonlinear dynamical problems to reduce the offline cost. It is especially useful for ST-GNAT that has shown promising results, such as a good accuracy with a considerable online speedup for hyperbolic problems in a recent paper [13], because ST-GNAT involves an expensive offline cost related to collecting nonlinear term snapshots. Error analysis shows that the oblique projection error bound of the SNS method depends on the condition number of the matrix M (e.g., a volume matrix generated from a discretization of a specific numerical scheme).…”

A long short-term memory embedding for hybrid uplifted reduced order models

“…In our study, this basis is computed using the POD method presented in Section 2. Then, the full-order operators are projected onto the same subspace S. In other words, the residual of the governing ODE is enforced to be orthogonal to S. Galerkin projection can be viewed as a special case of Petrov-Galerkin method [80][81][82][83] considering the same trial subspace as a test subspace. In the following, we present the governing equations for our test cases, namely 1D Burgers equation and 2D Navier-Stokes equations as well as their low-order approximations.…”

Commutation error in reduced order modeling of fluid flows

“…Reduced order models (ROMs) [4,14,18,19,27,34] have been used for decades in the efficient numerical simulation of fluid flows [2,3,6,8,11,12,17,19,24,27,32,38,40,39,46]. However, when the ROM dimension is too low to capture the relevant flow features, ROMs are generally supplemented with a Correction term [1,5,15,16,19,21,27,29,36,42].…”