Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences

TL;DR

This paper presents a reduced order modeling approach using POD-Galerkin methods to efficiently solve complex parametrized non-linear and time-dependent optimal flow control problems governed by PDEs, with applications in biomedical and environmental sciences.

Contribution

It introduces a POD-Galerkin reduction framework applied to parametrized optimality systems for flow control, tested on Stokes and Navier-Stokes equations.

Findings

Efficient reduced order models achieve rapid solutions.

Method accurately approximates solutions in biomedical and environmental contexts.

Applicable to both time-dependent and steady non-linear flow problems.

Abstract

We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.