  The majority of model reduction approaches use an efficient representation of
the state and then derive equations to temporally evolve the coefficients that
encode the state in the representation. In this paper, we instead employ an
efficient representation of the entire trajectory of the state over some time
interval and solve for the coefficients that define the trajectory on the
interval. We use spectral proper orthogonal decomposition (SPOD) modes, in
particular, which possess properties that make them suitable for model
reduction and are known to provide an accurate representation of trajectories.
In fact, with the same number of total coefficients, the SPOD representation is
substantially more accurate than any representation formed by specifying the
coefficients in a spatial (e.g., POD) basis for the many time steps that make
up the interval. We develop a method to solve for the SPOD coefficients that
encode the trajectories in forced linear dynamical systems given the forcing
and initial condition, thereby obtaining the accurate representation of the
trajectory. We apply the method to two examples, a linearized Ginzburg-Landau
problem and an advection-diffusion problem. In both, the error of the proposed
method is orders of magnitude lower than both POD-Galerkin and balanced
truncation applied to the same problem, as well as the most accurate solution
within the span of the POD modes. The method is also fast, with CPU time
comparable to or lower than both benchmarks in the examples we present.
