  This work considers a weighted POD-greedy method to estimate statistical
outputs parabolic PDE problems with parametrized random data. The key idea of
weighted reduced basis methods is to weight the parameter-dependent error
estimate according to a probability measure in the set-up of the reduced space.
The error of stochastic finite element solutions is usually measured in a root
mean square sense regarding their dependence on the stochastic input
parameters. An orthogonal projection of a snapshot set onto a corresponding POD
basis defines an optimum reduced approximation in terms of a Monte Carlo
discretization of the root mean square error. The errors of a weighted
POD-greedy Galerkin solution are compared against an orthogonal projection of
the underlying snapshots onto a POD basis for a numerical example involving
thermal conduction. In particular, it is assessed whether a weighted POD-greedy
solutions is able to come significantly closer to the optimum than a
non-weighted equivalent. Additionally, the performance of a weighted POD-greedy
Galerkin solution is considered with respect to the mean absolute error of an
adjoint-corrected functional of the reduced solution.
