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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

TL;DR

The paper addresses fast, reliable evaluation of parametrized, advection-dominated optimal control problems with random inputs by developing stabilized reduced-order models that combine SUPG stabilization with weighted POD. It compares offline-only and offline-online stabilization strategies within a space-time FEM framework and employs diverse quadrature rules to handle uncertainty quantification. The results show that Offline-Online stabilization using weighted POD—particularly the Weighted Monte-Carlo variant—delivers near-projection accuracy with substantial computational speedups, and sparse-grid sampling often outperforms tensor-based approaches. The work demonstrates the effectiveness of wROMs for real-time or many-query contexts in stochastic PDE-constrained optimization and outlines directions for extending to boundary controls and nonlinear dynamics.

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.

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

TL;DR

The paper addresses fast, reliable evaluation of parametrized, advection-dominated optimal control problems with random inputs by developing stabilized reduced-order models that combine SUPG stabilization with weighted POD. It compares offline-only and offline-online stabilization strategies within a space-time FEM framework and employs diverse quadrature rules to handle uncertainty quantification. The results show that Offline-Online stabilization using weighted POD—particularly the Weighted Monte-Carlo variant—delivers near-projection accuracy with substantial computational speedups, and sparse-grid sampling often outperforms tensor-based approaches. The work demonstrates the effectiveness of wROMs for real-time or many-query contexts in stochastic PDE-constrained optimization and outlines directions for extending to boundary controls and nonlinear dynamics.

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.
Paper Structure (15 sections, 52 equations, 29 figures, 4 tables, 1 algorithm)

This paper contains 15 sections, 52 equations, 29 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation for Random Input Optimal Control Problems
  3. Mathematical Setting
  4. High-Fidelity Discretization
  5. SUPG stabilization for Advection-Dominated OCP($\boldsymbol{\mu}$)s
  6. Setting for Stabilized Advection-Dominated OCP($\boldsymbol{\mu}$)s - Steady case
  7. Setting for Stabilized Advection-Dominated OCP($\boldsymbol{\mu}$)s - Unsteady case
  8. Weighted ROMs for random inputs advection-dominated OCP($\boldsymbol \mu$)s
  9. Offline phase
  10. Weighted Proper Orthogonal Decomposition
  11. Online phase
  12. Numerical Results
  13. Numerical Tests for the Graetz-Poiseuille Problem
  14. Numerical Tests for Propagating Front in a Square Problem
  15. Conclusions and Perspectives

Figures (29)

  • Figure 1: Geometry of the Graetz-Poiseuille Problem.
  • Figure 2: (top) FEM not stabilized and (bottom) FEM stabilized solution, $y$ (right) and $u$ (left), $\boldsymbol{\mu} = (10^{5},1.5)$, $h=0.034$, $\alpha=0.01$, $\delta_K=1.0$.
  • Figure 3: Grid points for the quadrature formulae of the weighted POD regarding the Graetz-Poiseuille Problem; (top) Monte-Carlo method with $\boldsymbol{\mu}$ following distribution \ref{['beta-graetz']} on the parameter space $\mathcal{P}$ (left), Smolyak grid based on a Gauss-Jacobi rule (center), Tensor grid based on a Gauss-Jacobi rule (right); (bottom) Pseudo-Random method based on a Halton sequence (left), Smolyak grid based on a Clenshaw-Curtis rule (center), Tensor grid based on a Clenshaw-Curtis rule (right).
  • Figure 4: Singular values decay for the snapshot matrices for the Graetz-Poiseuille Problem with $\boldsymbol{\mu}$ following distribution \ref{['beta-graetz']} on the parameter space $\mathcal{P}$; State (left), Control (center), Adjoint (right); Standard POD (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-Random based on Halton numbers (pink).
  • Figure 5: Projection Errors onto the POD space for the Graetz-Poiseuille Problem with $\boldsymbol{\mu}$ following distribution \ref{['beta-graetz']} on the parameter space $\mathcal{P}$; State (left), Control (center), Adjoint (right); Standard POD (blue), wPOD Monte-Carlo (orange), Gauss-Jacobi tensor rule (green), Gauss-Jacobi Smolyak grid (red), Clenshaw-Curtis tensor rule (cyan), Clenshaw-Curtis Smolyak grid (dark green), Pseudo-Random based on Halton numbers (pink).
  • ...and 24 more figures

Theorems & Definitions (5)

  • Definition 2.1.2: Linear-Quadratic OCP($\boldsymbol{\mu}$
  • Remark 2.1.3: Notation
  • Remark 2.1.4: Parabolic Problems
  • Definition 3.0.1: Advection-Diffusion Equations
  • Definition 3.0.2: Péclet number and Advection-Dominated problem