Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs

• Published in: Journal of numerical mathematics Vol. 33; no. 1; pp. 1 - 35
• Main Authors: Zoccolan, Fabio, Strazzullo, Maria, Rozza, Gianluigi
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 26.03.2025; Walter de Gruyter GmbH
• Subjects: 49J20; 49M41; 60H25; 60H35; 65M60; Advection; Algorithms; Finite element method; Mathematics; Monte Carlo simulation; Optimal control; Optimization; Parameterization; Partial differential equations; Peclet number; Proper Orthogonal Decomposition; Quadratures; random inputs; reduced order methods; Reduced order models; Stabilization; Tensors; time-dependent parametrized optimal control problem; uncertainty quantification; weighted proper orthogonal decomposition
• ISSN: 1570-2820; 1569-3953
• DOI: 10.1515/jnma-2023-0006

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 is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet 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 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.