POD-based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control

TL;DR

This paper develops and compares tailored POD-based reduced order methods for efficient simulation of parametric PDE optimal control problems with varying boundary controls, addressing challenges posed by non-affine and transport phenomena behaviors.

Contribution

It introduces two novel reduced order strategies—geometric recasting and local POD—for boundary control problems with parametric PDEs, overcoming limitations of classical methods.

Findings

Geometric recasting simplifies the problem by working in a reference domain.

Local POD improves accuracy in complex geometries.

The proposed methods outperform classical POD in accuracy and efficiency.

Abstract

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.