  Understanding the linear growth of disturbances due to external forcing is
crucial for flow stability analysis, flow control, and uncertainty
quantification. These applications typically require a large number of forward
simulations of the forced linearized dynamics, often in a brute-force fashion.
When dealing with simple steady-state or periodic base flows, there exist
powerful and cost-effective solution operator techniques. Once these solution
operators are constructed, they can be used to determine the response to
various forcings with negligible computational cost. However, these methods do
not apply to problems with arbitrarily time-dependent base flows. This paper
develops and investigates reduced-order modeling with time-dependent bases
(TDBs) to build low-rank solution operators for forced linearized dynamics with
arbitrarily time-dependent base flows. In particular, we use forced optimally
time-dependent decomposition (f-OTD), which extracts the time-dependent
correlated structures of the flow response to various excitations. Several
demonstrations are included to illustrate the utility of the f-OTD low-rank
approximation for performing global transient stability analysis. Additionally,
we demonstrate the application of f-OTD in computing the post-transient
response of linearized Navier-Stokes equations to a large number of impulses,
which has applications in flow control.
