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Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

Michael Kartmann, Stefan Volkwein

TL;DR

This work develops a certified, adaptive model-order reduction framework for unconstrained linear-quadratic OCPs governed by time-varying parabolic PDEs. By leveraging variational discretization, a state Galerkin ROM naturally induces a reduced control structure, making the state-only reduced OCP equivalent to a state-and-control reduced problem and enabling an online POD-based reduction with rigorous a posteriori error bounds for the control and an error representation for the objective. The authors prove convergence of the adaptive ROM optimization algorithm and demonstrate substantial speedups over full-order solves, with Full-ROM outperforming state-only ROM in numerical experiments across varying problem conditioning. The results provide a practical, certified approach for efficiently solving PDE-constrained OCPs with online adaptivity and reliable error control, supported by numerical evidence and accessible code.

Abstract

We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimizer as the solely state-reduced problem. Lower and upper \emph{a posteriori} error bounds for the optimal control and a representation for the error in the optimal function value are provided. These bounds are used in an adaptive algorithm to solve the control problem. We prove its convergence and numerically demonstrate the advantage of combined control and state space reduction.

Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

TL;DR

This work develops a certified, adaptive model-order reduction framework for unconstrained linear-quadratic OCPs governed by time-varying parabolic PDEs. By leveraging variational discretization, a state Galerkin ROM naturally induces a reduced control structure, making the state-only reduced OCP equivalent to a state-and-control reduced problem and enabling an online POD-based reduction with rigorous a posteriori error bounds for the control and an error representation for the objective. The authors prove convergence of the adaptive ROM optimization algorithm and demonstrate substantial speedups over full-order solves, with Full-ROM outperforming state-only ROM in numerical experiments across varying problem conditioning. The results provide a practical, certified approach for efficiently solving PDE-constrained OCPs with online adaptivity and reliable error control, supported by numerical evidence and accessible code.

Abstract

We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimizer as the solely state-reduced problem. Lower and upper \emph{a posteriori} error bounds for the optimal control and a representation for the error in the optimal function value are provided. These bounds are used in an adaptive algorithm to solve the control problem. We prove its convergence and numerically demonstrate the advantage of combined control and state space reduction.
Paper Structure (17 sections, 10 theorems, 63 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 17 sections, 10 theorems, 63 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and notation
  3. Reduced-order models (ROMs) with control and state space reduction
  4. Full-order model (FOM)
  5. Reduced-order model (ROM)
  6. A posteriori error estimation
  7. Error estimator for the optimal control
  8. Optimal function value error representation
  9. Combined adaptive state and control reduction
  10. Construction of the reduced spaces with POD
  11. Convergence analysis
  12. Numerical Results
  13. Model problem
  14. Discretization and algorithmic setup
  15. Comparison of the state-reduced OCP and the control- and state-reduced OCP
  16. ...and 2 more sections

Key Result

Lemma 2

We have $\bar{u}^r = \hat{u}^r$. In particular, the associated optimal states and optimal adjoints are equal, that is, $\bar{y}^r\coloneqq\hat{y}^r$ and $\bar{p}^r \coloneqq \hat{p}^r$.

Figures (4)

  • Figure 1: Snapshots of the optimal controls generated by each algorithm
  • Figure 2: Snapshots of the target state $y_d$ and optimal states generated by each algorithm
  • Figure 3: The error quantities from \ref{['eq:error_est']}, and $|\nabla \hat{J}(u_k)|_{L^2(0,T;U)}$ plotted against iteration counter $k$ for different $\beta$, and both ROM (left column) and Full-ROM methods (right column)
  • Figure 4: Basis sizes plotted against iteration counter $k$ for different $\beta$

Theorems & Definitions (23)

  • Remark 1
  • Lemma 2
  • Remark 3: Optimality condition of the control- and state-reduced OCP
  • Lemma 4
  • proof
  • Corollary 5: Error estimator for the first-order optimality condition
  • proof
  • Remark 6
  • Theorem 7: Optimal value function error representation
  • proof
  • ...and 13 more