Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics
Abstract
We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal fields (gradient and flux) that are closest to some predefined discrete fields taken from a material data set. The optimization is performed over primal fields that satisfy the physical conservation law and the geometrical compatibility. We consider a reaction term in the conservation law, which has the effect of coupling all the optimality conditions. We first establish the well-posedness in the continuous setting. Then, we propose stable finite element discretizations that consistently approximate the continuous formulation, preserving its saddle-point structure and allowing for equal-order interpolation of all fields. Finally, we demonstrate the effectiveness of the proposed methods through a set of numerical examples.
Cite
@article{arxiv.2506.10894,
title = {Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics},
author = {Pedro B. Bazon and Cristian G. Gebhardt and Gustavo C. Buscaglia and Roberto F. Ausas},
journal= {arXiv preprint arXiv:2506.10894},
year = {2025}
}