  This work presents the windowed space-time least-squares Petrov-Galerkin
method (WST-LSPG) for model reduction of nonlinear parameterized dynamical
systems. WST-LSPG is a generalization of the space-time least-squares
Petrov-Galerkin method (ST-LSPG). The main drawback of ST-LSPG is that it
requires solving a dense space-time system with a space-time basis that is
calculated over the entire global time domain, which can be unfeasible for
large-scale applications. Instead of using a temporally-global space-time trial
subspace and minimizing the discrete-in-time full-order model (FOM) residual
over an entire time domain, the proposed WST-LSPG approach addresses this
weakness by (1) dividing the time simulation into time windows, (2) devising a
unique low-dimensional space-time trial subspace for each window, and (3)
minimizing the discrete-in-time space-time residual of the dynamical system
over each window. This formulation yields a problem with coupling confined
within each window, but sequential across the windows. To enable high-fidelity
trial subspaces characterized by a relatively minimal number of basis vectors,
this work proposes constructing space-time bases using tensor decompositions
for each window. WST-LSPG is equipped with hyper-reduction techniques to
further reduce the computational cost. Numerical experiments for the
one-dimensional Burgers' equation and the two-dimensional compressible
Navier-Stokes equations for flow over a NACA 0012 airfoil demonstrate that
WST-LSPG is superior to ST-LSPG in terms of accuracy and computational gain.
