Windowed least-squares model reduction for dynamical systems

TL;DR

This paper introduces a windowed least-squares (WLS) method for model reduction in dynamical systems, improving accuracy and addressing limitations of existing techniques by optimizing residuals over time windows.

Contribution

The WLS approach generalizes existing methods and reduces dependence on time discretization, offering a flexible framework for spatial and spatiotemporal model reduction.

Findings

WLS yields more accurate solutions than Galerkin and LSPG.

WLS reduces space-time residuals effectively.

The approach addresses key deficiencies in existing model reduction techniques.

Abstract

This work proposes a windowed least-squares (WLS) approach for model-reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space-time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov-Galerkin (LSPG), and space-time LSPG projection. In addition, the approach addresses key deficiencies in existing model-reduction techniques, e.g., the dependence of LSPG and space-time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space-time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space-time residuals than Galerkin and LSPG projection.