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

Fabio Zoccolan; Maria Strazzullo; Gianluigi Rozza

doi:10.1515/jnma-2023-0006

Article

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

Fabio Zoccolan , Maria Strazzullo and Gianluigi Rozza

Published/Copyright: October 7, 2024

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From the journal Journal of Numerical Mathematics Volume 33 Issue 1

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 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 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.

Keywords: reduced order methods; time-dependent parametrized optimal control problem; stabilization; weighted proper orthogonal decomposition; random inputs; uncertainty quantification

MSC 2010 Classification: 65M60; 49J20; 49M41; 60H25; 60H35

Corresponding author: Gianluigi Rozza, mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy, E-mail: gianluigi.rozza@sissa.it

Acknowledgment

The computations in this work have been performed with RBniCS [54] library, developed at SISSA mathLab, which is an implementation in FEniCS [56] of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.

Research ethics: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Competing interests: The authors state no conflict of interest.
Research funding: We acknowledge the support by European Union Funding for Research and Innovation – Horizon 2020 Program – in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics”. We also acknowledge the PRIN 2017 “Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM-GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”.
Data availability: Not applicable.

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Received: 2023-01-04

Accepted: 2024-08-09

Published Online: 2024-10-07

Published in Print: 2025-03-26

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Articles in the same Issue

https://doi.org/10.1515/jnma-2023-0006

Keywords for this article

reduced order methods; time-dependent parametrized optimal control problem; stabilization; weighted proper orthogonal decomposition; random inputs; uncertainty quantification