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

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.