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

TL;DR

This paper develops stabilized reduced order models for parametrized advection-dominated optimal control problems with random inputs, combining FEM, space-time methods, and weighted POD to improve stability and efficiency in uncertainty quantification.

Contribution

It introduces a stabilized ROM framework with offline-online stabilization and weighted POD for uncertain advection-dominated PDE control problems.

Findings

Stabilized ROMs significantly reduce errors compared to non-stabilized models.

Offline-online stabilization improves computational efficiency.

Weighted POD effectively handles uncertainty quantification in the ROM context.

Abstract

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.