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

• Published in: SIAM Journal on Scientific Computing
• Main Authors: Strazzullo, Maria, Ballarin, Francesco, Mosetti, Renzo, Rozza, Gianluigi
• Format: Article in Journal/Newspaper
• Language: English
• Published: 2018
• Subjects: reduced order method; proper orthogonal decomposition; parametrized optimal control problem; PDE state equation; environmental marine application; quasi-geostrophic equation; Settore MAT/08 - Analisi Numerica; Saddle Point; North Atlantic
• Online Access: https://hdl.handle.net/20.500.11767/82665; https://doi.org/10.1137/17M1150591; https://arxiv.org/abs/1710.01640

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.