Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition

TL;DR

This paper develops reduced-order models using shifted proper orthogonal decomposition to efficiently solve optimal control problems for transport-dominated PDEs, demonstrating improved computational performance over traditional methods.

Contribution

It introduces a shifted POD-based reduced-order modeling framework for transport PDEs and compares two approaches for integrating model reduction with optimal control.

Findings

Shifted POD effectively captures transport phenomena with low-dimensional models.

The proposed methods outperform standard POD in computational efficiency.

Two frameworks for combining model reduction and optimization are validated.

Abstract

Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against the conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search.