  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 $\ell^2$-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 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: (1) a reduction of both the spatial and temporal dimensions of the
dynamical system, (2) the removal of spurious temporal modes (e.g., unstable
growth) from the state space, and (3) error bounds that 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.
