A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

Eileen Kammann; Fredi Tröltzsch; Stefan Volkwein

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 2, page 555-581
  • ISSN: 0764-583X

Abstract

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We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.

How to cite

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Kammann, Eileen, Tröltzsch, Fredi, and Volkwein, Stefan. "A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 555-581. <http://eudml.org/doc/273256>.

@article{Kammann2013,
abstract = {We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.},
author = {Kammann, Eileen, Tröltzsch, Fredi, Volkwein, Stefan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {optimal control; semilinear partial differential equations; error estimation; proper orthogonal decomposition; semilinear parabolic equations},
language = {eng},
number = {2},
pages = {555-581},
publisher = {EDP-Sciences},
title = {A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD},
url = {http://eudml.org/doc/273256},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Kammann, Eileen
AU - Tröltzsch, Fredi
AU - Volkwein, Stefan
TI - A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 2
SP - 555
EP - 581
AB - We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.
LA - eng
KW - optimal control; semilinear partial differential equations; error estimation; proper orthogonal decomposition; semilinear parabolic equations
UR - http://eudml.org/doc/273256
ER -

References

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