  The definition of partial differential equation (PDE) models usually involves
a set of parameters whose values may vary over a wide range. The solution of
even a single set of parameter values may be quite expensive. In many cases,
e.g., optimization, control, uncertainty quantification, and other settings,
solutions are needed for many sets of parameter values. We consider the case of
the time-dependent Navier-Stokes equations for which a recently developed
ensemble-based method allows for the efficient determination of the multiple
solutions corresponding to many parameter sets. The method uses the average of
the multiple solutions at any time step to define a linear set of equations
that determines the solutions at the next time step. To significantly further
reduce the costs of determining multiple solutions of the Navier-Stokes
equations, we incorporate a proper orthogonal decomposition (POD) reduced-order
model into the ensemble-based method. The stability and convergence results for
the ensemble-based method are extended to the ensemble-POD approach. Numerical
experiments are provided that illustrate the accuracy and efficiency of
computations determined using the new approach.
