A variational multiscale approach to PDE-constrained optimization problems arising in Data-Driven Computational Mechanics
Abstract
We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous setting, we establish well-posedness of such concomitant formulations. Then, we propose stable and consistent finite element approximations for these underlying primal and dual problems relying on the Variational MultiScale framework. For quasi-uniform finite element partitions, we investigate approximations' general properties and establish well-posedness for two canonical choices of the sub-grid scales, i.e., the Algebraic Sub-Grid Scale and Orthogonal Sub-Grid Scale. Moreover, for continuous finite element functions, we are able to move back and forth between the discrete primal and dual formulations only by changing the design of the stabilization parameters. To conclude, we stress-test the proposed approximations through a series of progressively sophisticated cases, providing both a comparative and qualitative assessment of their numerical performance.
Summary
This paper addresses the problem of solving PDE-constrained optimization problems arising in Data-Driven Computational Mechanics (DDCM), specifically in the context of reaction-diffusion. The authors focus on the "optimization subproblem" where continuous fields are sought to minimize their discrepancy with discrete fields derived from experimental data, subject to physical laws. They investigate both primal and dual formulations of the optimality conditions, establishing their well-posedness in the continuous setting. The core contribution lies in proposing stable and consistent finite element approximations for these primal and dual problems using the Variational Multiscale (VMS) framework. They analyze the properties of these approximations, establish well-posedness for Algebraic Sub-Grid Scale (ASGS) and Orthogonal Sub-Grid Scale (OSGS) methods, and demonstrate that the discrete primal and dual formulations can be interchanged by adjusting stabilization parameters. The paper concludes with numerical experiments to assess the performance of the proposed methods. This work matters to the field by providing a robust and well-analyzed finite element framework for tackling the "optimization subproblem" in DDCM, particularly for reaction-diffusion problems, offering a significant improvement over existing approaches. The authors' approach involves starting with the continuous primal and dual formulations of the PDE-constrained optimization problem and proving their well-posedness. They then discretize these formulations using finite elements and employ the VMS framework to derive necessary stabilization terms, ensuring stability and consistency. They analyze two specific sub-grid scale methods: ASGS and OSGS. Numerical experiments are conducted using the Firedrake finite element platform to stress-test the proposed approximations and compare their performance. The authors demonstrate that by carefully designing the stabilization parameters within the VMS framework, one can switch between the discrete primal and dual formulations, offering flexibility in solving these problems.
Key Insights
- •Novel Finite Element Formulations: The paper introduces novel stabilized finite element formulations for both primal and dual forms of the PDE-constrained optimization problem, achieving stability through the VMS framework.
- •Well-Posedness: The authors rigorously prove the well-posedness of both the continuous and discrete formulations (primal and dual) under certain conditions, providing a theoretical foundation for the proposed methods.
- •Equivalence via Stabilization: A key finding is that the discrete primal and dual formulations can be transformed into one another solely by modifying the stabilization parameters within the VMS framework.
- •Sub-Grid Scale Methods: The paper analyzes two specific sub-grid scale methods, ASGS and OSGS, establishing well-posedness for both choices with quasi-uniform finite element partitions.
- •Implementation with Firedrake: The implementation of the formulations in the Firedrake finite element platform provides a practical and accessible tool for researchers and engineers to utilize the proposed methods.
- •Second Law of Thermodynamics: While the paper simplifies the problem by omitting the explicit enforcement of the Second Law of Thermodynamics in the optimization problem, the authors verify a posteriori that the computed numerical solutions fulfill this law.
- •Relationship between ζ and κ: The paper highlights the dimensional relationship between the reaction coefficient ζ and the constant κ, showing that [ζ] = [κ] −1/2 [ℓ Ω ] −2 , where ℓ Ω represents the characteristic length of the problem.
Practical Implications
- •Data-Driven Computational Mechanics: The research provides a valuable tool for researchers and engineers working in DDCM, enabling them to solve the "optimization subproblem" for reaction-diffusion systems more effectively.
- •Reaction-Diffusion Modeling: The proposed methods can be applied to a wide range of real-world problems involving reaction-diffusion processes, such as chemical reactions, heat transfer, and biological systems.
- •Stabilized Finite Element Methods: The paper demonstrates the effectiveness of the VMS framework for stabilizing finite element approximations of PDE-constrained optimization problems, offering insights for developing similar methods for other types of PDEs.
- •Software Implementation: The availability of the code on GitHub (https://github.com/pedrobbazon/DDCM.git) allows practitioners to readily implement and test the proposed formulations in their own applications.
- •Future Research Directions: The authors suggest future research directions, including the explicit inclusion of the Second Law of Thermodynamics in the optimization problem to design more robust schemes.