  In this work, we analyze Parametrized Advection-Dominated distributed Optimal
Control Problems with random inputs in a Reduced Order Model (ROM) context. All
the simulations are initially based on a finite element method (FEM)
discretization; moreover, a space-time approach is considered when dealing with
unsteady cases. To overcome numerical instabilities that can occur in the
optimality system for high values of the P\'eclet number, we consider a
Streamline Upwind Petrov-Galerkin technique applied in an
optimize-then-discretize approach. We combine this method with the ROM
framework in order to consider two possibilities of stabilization: Offline-Only
stabilization and Offline-Online stabilization. Moreover, we consider random
parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a
partitioned approach to deal with the issue of uncertainty quantification.
Several quadrature techniques are used to derive weighted ROMs: tensor rules,
isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare
all the approaches analyzing relative errors between the FEM and ROM solutions
and the computational efficiency based on the speedup-index.
