Model Reduction For Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering

TL;DR

This paper develops reduced order methods, specifically POD-Galerkin techniques, to efficiently solve parametrized optimal control problems governed by PDEs in environmental marine sciences, demonstrated through two real-world applications.

Contribution

It introduces a POD-Galerkin reduction approach for parametrized linear quadratic optimal control problems with saddle-point structures, applied to environmental marine scenarios.

Findings

Reduced order methods significantly decrease computational time.

The approach is effective across different models and geographical scales.

Validated on pollutant control and ocean dynamic tracking applications.

Abstract

We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations, in its linear and nonlinear version, describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.