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

Tobias Breiten, Shubhaditya Burela, Philipp Schulze

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.

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

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.

Paper Structure

This paper contains 13 sections, 4 theorems, 92 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model order reduction methods
  3. POD-Galerkin method
  4. sPOD-Galerkin method
  5. Optimal control with reduced-order models
  6. POD-based optimal control
  7. sPOD-based optimal control
  8. Numerical results
  9. Problem setup
  10. Algorithmic details
  11. Using ROMs in optimal control
  12. Timing analysis
  13. Conclusion and outlook

Key Result

Proposition 2.2

Consider an abstract control system of the form eq_def:PDE_continuous with $\mathcal{A}\colon\mathcal{D}(\mathcal{A})\subset H\to H$ being the infinitesimal generator of a strongly continuous group $S(t)\in\mathcal{L}(H)$, $H$ a Hilbert space, $y_0\in H$, $u\in L^2(0,T; U)$ with $U=\mathbb{R}^m$, an where $y$ denotes the mild solution of eq_def:PDE_continuous.

Figures (7)

  • Figure 1: Plots for the state and the target for the two example problems
  • Figure 2: Control shape functions
  • Figure 3: Number of controls vs. $\mathcal{J}$
  • Figure 4: Plots showing singular value trend over optimization steps for $m = 9$
  • Figure 5: Plots for $\mathcal{J}$ behavior for the single tilt problem
  • ...and 2 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 4.1