Efficient Reduced Order Modeling for Advection-Dominated Partial Differential Equations under Optimal Control

Core Concepts

This paper presents a Streamline Upwind Petrov-Galerkin (SUPG) reduced order method for efficiently solving advection-dominated partial differential equations under optimal control.

Abstract

The paper focuses on developing an efficient reduced order modeling (ROM) approach for solving advection-dominated partial differential equations (PDEs) under optimal control.
Key highlights:

The authors consider linear-quadratic optimal control problems for advection-diffusion PDEs with high Péclet numbers, where numerical instabilities can occur.
A Streamline Upwind Petrov-Galerkin (SUPG) technique is used in the optimality system to overcome these instabilities, following an optimize-then-discretize approach.
For parabolic problems, a stabilized space-time framework is considered, with stabilization applied to the bilinear forms involving time derivatives.
Two ROM settings are analyzed: one where stabilization is only applied in the offline phase, and another where it is also used in the online phase.
Proper Orthogonal Decomposition (POD) is used to construct the reduced bases for the state, control, and adjoint variables in a partitioned approach.
Computational experiments are presented for two test cases - the Graetz-Poiseuille problem and the Propagating Front in a Square problem - demonstrating the effectiveness of the SUPG-based ROM approach.

Stats

The Péclet number, defined as |b|h/(2ε), is used as an indicator of the dominance of the advection term over the diffusion term.
The stabilization parameter δ_K is chosen based on local assumptions to ensure coercivity of the SUPG bilinear form.

Quotes

"It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting."
"To the best of our knowledge, SUPG for Parabolic OCPs in an optimize-then-discretized approach is still a novelty element in literature from a theoretical perspective."

Key Insights Distilled From

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

by Fabio Zoccol... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2301.01973.pdf

Deeper Inquiries

How can the proposed SUPG-based ROM approach be extended to handle more complex optimal control problems, such as those with nonlinear or time-dependent PDEs

The proposed SUPG-based ROM approach can be extended to handle more complex optimal control problems by incorporating techniques to address nonlinear or time-dependent PDEs. For nonlinear problems, the ROM methodology can be adapted to include nonlinear terms in the discretized equations. This may involve using techniques like nonlinear Galerkin projection or nonlinear model reduction methods to capture the nonlinear behavior of the system. Additionally, the ROM basis functions can be enriched to better represent the nonlinear features of the problem.
For time-dependent PDEs, the ROM approach can be extended by incorporating time discretization schemes that are suitable for the specific problem. This may involve using implicit or semi-implicit time integration methods to handle the time evolution of the system. The stabilization techniques used in the SUPG method can also be adapted to ensure stability and accuracy in the time-dependent ROM formulation.

What are the potential challenges in applying this methodology to real-world engineering applications with high-dimensional parameter spaces

Applying the proposed methodology to real-world engineering applications with high-dimensional parameter spaces may pose several challenges. Some potential challenges include:

Computational Complexity: Handling high-dimensional parameter spaces can significantly increase the computational cost of generating and storing snapshots for the ROM. This can lead to scalability issues, especially when dealing with a large number of parameters.

Curse of Dimensionality: As the dimensionality of the parameter space increases, the number of snapshots required to accurately represent the system also increases exponentially. This can lead to difficulties in constructing an efficient reduced basis.

Modeling Errors: In real-world applications, there may be uncertainties and modeling errors that can affect the accuracy of the ROM. Dealing with these uncertainties and ensuring robustness in the reduced model can be a challenge.

Adaptability: Adapting the ROM methodology to handle varying or uncertain parameter spaces in real-time applications can be complex. Developing adaptive strategies to update the reduced basis and handle changes in the parameter space is crucial.

Can the insights gained from this work be leveraged to develop adaptive ROM strategies that can automatically detect and handle instabilities during the online phase

The insights gained from this work can be leveraged to develop adaptive ROM strategies that can automatically detect and handle instabilities during the online phase. Some approaches to achieve this include:

Online Stabilization: Implementing online stabilization techniques that can dynamically adjust the stabilization parameters based on the solution behavior during the online phase. This can help in mitigating instabilities as they arise.

Error Estimation: Incorporating error estimation techniques to monitor the accuracy of the ROM solution during the online phase. By detecting deviations from the high-fidelity solution, adaptive strategies can be triggered to improve the stability and accuracy of the reduced model.

Dynamic Basis Adaptation: Developing algorithms that can adaptively update the reduced basis during the online phase to capture changes in the system dynamics or parameter space. This can help in maintaining stability and accuracy in varying conditions.

By integrating these adaptive strategies into the ROM framework, it is possible to enhance the robustness and reliability of reduced order models in handling instabilities in real-time applications.