Frontiers in Applied Mathematics and Statistics (Dec 2023)

Model order reduction for optimality systems through empirical gramians

  • Luca Mechelli,
  • Jan Rohleff,
  • Stefan Volkwein

DOI
https://doi.org/10.3389/fams.2023.1144142
Journal volume & issue
Vol. 9

Abstract

Read online

In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples.

Keywords