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In this work, we analyze Parametrized Advection-Dominated distributed OptimalControl Problems with random inputs in a Reduced Order Model (ROM) context. Allthe simulations are initially based on a finite element method (FEM)discretization; moreover, a space-time approach is considered when dealing withunsteady cases. To overcome numerical instabilities that can occur in theoptimality system for high values of the P\'eclet number, we consider aStreamline Upwind Petrov-Galerkin technique applied in anoptimize-then-discretize approach. We combine this method with the ROMframework in order to consider two possibilities of stabilization: Offline-Onlystabilization and Offline-Online stabilization. Moreover, we consider randomparameters and we use a weighted Proper Orthogonal Decomposition algorithm in apartitioned 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 compareall the approaches analyzing relative errors between the FEM and ROM solutionsand the computational efficiency based on the speedup-index.

Stabilized Weighted Reduced Order Methods for Parametrized  Advection-Dominated Optimal Control Problems governed by Partial Differential  Equations with Random Inputs