Optimal control with the shifted proper orthogonal decomposition via a first-reduce-then-optimize framework

TL;DR

This paper develops a reduced-order modeling approach using shifted POD for efficient optimal control of transport-dominated PDEs, demonstrating improved computational performance over standard POD.

Contribution

It introduces a shifted POD-based reduced-order model for optimal control problems, with theoretical guarantees and performance comparison to standard POD.

Findings

The shifted POD model effectively captures transport phenomena with low-dimensional representations.

The reduced-order model guarantees existence and uniqueness of solutions and optimal control.

Shifted POD outperforms standard POD in computational efficiency.

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 low-dimensional representations of high-fidelity transport-dominated phenomena. In this work, a reduced-order model is constructed first, followed by the optimization of the reduced system. We consider a 1D linear advection equation problem and prove existence and uniqueness of solutions for the reduced-order model as well as the existence of an optimal control. Moreover, we compare the computational performance of the shifted POD method against the standard POD.