  Partial differential equations (PDEs) are often dependent on input quantities
which are inherently uncertain. To quantify this uncertainty, these PDEs must
be solved over a large ensemble of parameters. Even for a single realization
this can a computationally intensive process. In the case of flows governed by
the Navier-Stokes equations, an efficient method has been devised for computing
an ensemble of solutions. To further reduce the computational cost of this
method, an ensemble proper orthogonal decomposition (POD) method was recently
proposed.
  The main contribution of this work is the introduction of POD spatial
filtering for ensemble-POD methods. The POD spatial filter makes possible the
construction of the Leray ensemble-POD model, which is a regularized reduced
order model for the numerical simulation of convection-dominated flows. The
Leray ensemble-POD model employs the POD spatial filter to smooth (regularize)
the convection term in the Navier-Stokes equations and greatly diminishes the
numerical inaccuracies produced by the ensemble-POD method in the numerical
simulation of convection-dominated flows. Specifically, for the numerical
simulation of a convection-dominated two-dimensional flow between two offset
cylinders, we show that the Leray ensemble-POD method yields accurate results,
whereas the ensemble-POD is highly inaccurate.
  The second contribution of this work is a new numerical discretization of the
variable viscosity ensemble algorithm in which the average viscosity is
replaced with the maximum viscosity. It is shown that this new numerical
discretization is significantly more stable than those in current use.
Furthermore, error estimates for the novel Leray ensemble-POD algorithm with
this new numerical discretization are also proven.
