A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

Mark Kärcher; Martin A. Grepl

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 6, page 1615-1638
  • ISSN: 0764-583X

Abstract

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We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.

How to cite

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Kärcher, Mark, and Grepl, Martin A.. "A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.6 (2014): 1615-1638. <http://eudml.org/doc/273129>.

@article{Kärcher2014,
abstract = {We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.},
author = {Kärcher, Mark, Grepl, Martin A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {optimal control; reduced basis method; a posteriori error estimation; model order reduction; parameter-dependent systems; partial differential equations; parabolic problems; linear-quadratic problems; parabolic PDEs; a-posteriori error estimates},
language = {eng},
number = {6},
pages = {1615-1638},
publisher = {EDP-Sciences},
title = {A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems},
url = {http://eudml.org/doc/273129},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Kärcher, Mark
AU - Grepl, Martin A.
TI - A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 6
SP - 1615
EP - 1638
AB - We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.
LA - eng
KW - optimal control; reduced basis method; a posteriori error estimation; model order reduction; parameter-dependent systems; partial differential equations; parabolic problems; linear-quadratic problems; parabolic PDEs; a-posteriori error estimates
UR - http://eudml.org/doc/273129
ER -

References

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