  Reduced-order modeling is an efficient approach for solving parameterized
discrete partial differential equations when the solution is needed at many
parameter values. An offline step approximates the solution space and an online
step utilizes this approximation, the reduced basis, to solve a smaller reduced
problem at significantly lower cost, producing an accurate estimate of the
solution. For nonlinear problems, however, standard methods do not achieve the
desired cost savings. Empirical interpolation methods represent a modification
of this methodology used for cases of nonlinear operators or nonaffine
parameter dependence. These methods identify points in the discretization
necessary for representing the nonlinear component of the reduced model
accurately, and they incur online computational costs that are independent of
the spatial dimension $N$. We will show that empirical interpolation methods
can be used to significantly reduce the costs of solving parameterized versions
of the Navier-Stokes equations, and that iterative solution methods can be used
in place of direct methods to further reduce the costs of solving the algebraic
systems arising from reduced-order models.
