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Two-stage model reduction approaches for the efficient and certified solution of parametrized optimal control problems

Hendrik Kleikamp, Lukas Renelt

TL;DR

This contribution develops an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system and proposes different strategies for building the involved reduced order models by separate reduction of the dynamical systems and the final time adjoint states.

Abstract

In this contribution we develop an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system. The fully reduced model combines reduced basis approximations of the system dynamics and of the manifold of optimal final time adjoint states to achieve a computational complexity independent of the original state space. Such a combination is particularly beneficial in the case where a deviation in a low-dimensional output is penalized. In addition, an offline-online decomposed a posteriori error estimator bounding the error between the approximate final time adjoint with respect to the optimal one is derived and its reliability proven. We propose different strategies for building the involved reduced order models, for instance by separate reduction of the dynamical systems and the final time adjoint states or via greedy procedures yielding a combined and fully reduced model. These algorithms are evaluated and compared for a two-dimensional heat equation problem. The numerical results show the desired accuracy of the reduced models and highlight the speedup obtained by the newly combined reduced order model in comparison to an exact computation of the optimal control or other reduction approaches.

Two-stage model reduction approaches for the efficient and certified solution of parametrized optimal control problems

TL;DR

This contribution develops an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system and proposes different strategies for building the involved reduced order models by separate reduction of the dynamical systems and the final time adjoint states.

Abstract

In this contribution we develop an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system. The fully reduced model combines reduced basis approximations of the system dynamics and of the manifold of optimal final time adjoint states to achieve a computational complexity independent of the original state space. Such a combination is particularly beneficial in the case where a deviation in a low-dimensional output is penalized. In addition, an offline-online decomposed a posteriori error estimator bounding the error between the approximate final time adjoint with respect to the optimal one is derived and its reliability proven. We propose different strategies for building the involved reduced order models, for instance by separate reduction of the dynamical systems and the final time adjoint states or via greedy procedures yielding a combined and fully reduced model. These algorithms are evaluated and compared for a two-dimensional heat equation problem. The numerical results show the desired accuracy of the reduced models and highlight the speedup obtained by the newly combined reduced order model in comparison to an exact computation of the optimal control or other reduction approaches.
Paper Structure (21 sections, 6 theorems, 58 equations, 1 algorithm)

This paper contains 21 sections, 6 theorems, 58 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Parametrized linear time-varying systems and optimal control
  3. Linear time-varying control systems
  4. Parametrized linear-quadratic optimal control problems
  5. Optimality system and optimal final time adjoint states
  6. Reduced order modeling for system dynamics
  7. Projection-based reduced basis model order reduction
  8. A posteriori error estimation
  9. Model order reduction for final time optimal adjoint states
  10. Projection-based reduced basis model order reduction
  11. A posteriori error estimation
  12. Efficient and fully reduced model for parametrized optimal control problems
  13. Fully reduced model by combining system and final time adjoint reduction
  14. Efficiently computable a posteriori error estimate
  15. Definition and reliability of the error estimator
  16. ...and 6 more sections

Key Result

Theorem 1

Given a parameter $\mu\in\mathcal{P}$, the optimal control of the problem in equ:optimal-control-problem can equivalently be expressed as the solution to the optimality system where $M\coloneqq C^*\mathcal{R}_{Y}C\in\mathcal{L}\left(X,X'\right)$. The adjoint mass operator $E_{\mathrm{ad}}\in\mathcal{L}\left(X',X'\right)$ is given as $E_{\mathrm{ad}}=\mathcal{R}_{X}E^*\mathcal{R}_{X}^{-1}$, i.e. w

Theorems & Definitions (15)

  • Theorem 1: Optimality system for the linear-quadratic optimal control problem
  • proof
  • Lemma 1: Linear system of equations for the optimal final time adjoint state
  • proof
  • Remark 1: Counterparts to the reduced operators in the matrix case
  • Lemma 2: A posteriori error estimator for the reduced order model of the primal system; see haasdonk2011efficient
  • proof
  • Lemma 3: A posteriori error estimator for the reduced order model of the adjoint system
  • proof
  • Remark 2: Numerical approximation of the integrals in the error estimators
  • ...and 5 more