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A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

Fabio Zoccolan, Maria Strazzullo, Gianluigi Rozza

TL;DR

This work addresses efficient reduced-order modeling for advection-dominated Linear-Quadratic OCPs with high $Pe$. It introduces SUPG-stabilized truth discretization within an optimize-then-discretize framework and a space–time formulation for parabolic problems, followed by POD-based ROMs built with an offline-online strategy. Through Graetz-Poiseuille and Propagating Front in a Square benchmarks, the authors demonstrate that Online-Offline stabilization yields high-accuracy reduced solutions and large speedups, while Only-Offline stabilization can fail to achieve adequate accuracy. The study advances robust ROM techniques for stabilized parabolic OCPs in advection-dominated regimes and suggests avenues for further improvements, including uncertainty quantification and online stabilization enhancements.

Abstract

In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

TL;DR

This work addresses efficient reduced-order modeling for advection-dominated Linear-Quadratic OCPs with high . It introduces SUPG-stabilized truth discretization within an optimize-then-discretize framework and a space–time formulation for parabolic problems, followed by POD-based ROMs built with an offline-online strategy. Through Graetz-Poiseuille and Propagating Front in a Square benchmarks, the authors demonstrate that Online-Offline stabilization yields high-accuracy reduced solutions and large speedups, while Only-Offline stabilization can fail to achieve adequate accuracy. The study advances robust ROM techniques for stabilized parabolic OCPs in advection-dominated regimes and suggests avenues for further improvements, including uncertainty quantification and online stabilization enhancements.

Abstract

In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.
Paper Structure (15 sections, 1 theorem, 64 equations, 12 figures, 4 tables, 1 algorithm)

This paper contains 15 sections, 1 theorem, 64 equations, 12 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Steady Problems
  4. Unsteady Problems
  5. Truth Discretization
  6. SUPG stabilization for Advection-Dominated OCP($\boldsymbol{\mu}$)s
  7. SUPG for Time-Dependent Advection-Dominated OCP($\boldsymbol{\mu}$)s
  8. ROMs for advection-dominated OCP($\boldsymbol \mu$)s
  9. Offline-Online Procedure for ROMs
  10. Proper Orthogonal Decomposition
  11. Only-Offline vs Online-Offline stabilization
  12. Numerical Results
  13. Numerical Experiments for the Graetz-Poiseuille Problem
  14. Numerical Experiments for Propagating Front in a Square Problem
  15. Conclusions and Perspectives

Key Result

Theorem 3.1.3

Let $m, r \geq 1$ and $(y, u, p)$ be the solution of Opt-lin_system with $y \in H^{m+1}(\Omega), p \in H^{r+1}(\Omega) .$ Furthermore, let $y^{\mathcal{N}}, u^{\mathcal{N}}, p^{\mathcal{N}}$ be the numerical solution of supg-system. If $\delta_K$ satisfies where $\delta_1, \delta_2 >0$ are chosen constant, and $\eta_{\text{inv}}$ is defined as the following inverse constant with $|\cdot|_{1,K}$

Figures (12)

  • Figure 1: Geometry of the Graetz-Poiseuille Problem.
  • Figure 2: (Top) Only-Offline stabilized state (left) and its error with respect to the FEM solution (right); (Bottom) Only-Offline stabilized adjoint (left) and its error with respect to the FEM solution (right); for the Graetz-Poiseuille Problem; $N=1$, $\mathcal{P} = [10^4,10^6]$, $\mu_1=10^5$, $h=0.029$, $\delta_K =1.0$, $\alpha=0.01$.
  • Figure 3: (Top) Online-Offline stabilized state (left) and its error with respect to the FEM solution (right); (Bottom) Online-Offline stabilized adjoint (left) and its error with respect to the FEM solution (right); for the Graetz-Poiseuille Problem; $N=6$, $\mathcal{P} = [10^4,10^6]$, $\mu_1=10^5$, $h=0.029$, $\delta_K =1.0$, $\alpha=0.01$.
  • Figure 4: Relative errors between FEM and reduced solution for state (left), control (center) and adjoint (right), for Online-Offline and Only-Offline stabilization, $\alpha=0.01$, $N_{\text{test}}=100$, $h=0.029$, $\mathcal{P}=[10^4,10^6]$. Graetz-Poiseuille Problem.
  • Figure 5: (Top) SUPG FEM solution for the state and (Bottom) for the adjoint at $t=0.1$, $t=1.5$, $t=3.0$. Parabolic Graetz-Poiseuille Problem, $\mu_1=10^{5}$, $N_t=30$, $h=0.038$, $\delta_K =1.0$, $\alpha=0.01$.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Remark 2.1.2: Notation
  • Remark 2.2.2
  • Definition 3.1.1: Advection-Diffusion Equations
  • Definition 3.1.2: Advection-Dominated problem
  • Theorem 3.1.3: Error for state and adjoint variables
  • Remark 4.2.1: Time-dependent problems
  • Remark 5.0.1