---
_id: '8754'
abstract:
- lang: eng
  text: In this chapter, we combine a global, derivative-free subdivision algorithm
    for multiobjective optimization problems with a posteriori error estimates for
    reduced-order models based on Proper Orthogonal Decomposition in order to efficiently
    solve multiobjective optimization problems governed by partial differential equations.
    An error bound for a semilinear heat equation is developed in such a way that
    the errors in the conflicting objectives can be estimated individually. The resulting
    algorithm constructs a library of locally valid reduced-order models online using
    a Greedy (worst-first) search. Using this approach, the number of evaluations
    of the full-order model can be reduced by a factor of more than 1000.
author:
- first_name: Dennis
  full_name: Beermann, Dennis
  last_name: Beermann
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Stefan
  full_name: Volkwein, Stefan
  last_name: Volkwein
citation:
  ama: 'Beermann D, Dellnitz M, Peitz S, Volkwein S. Set-Oriented Multiobjective Optimal
    Control of PDEs Using Proper Orthogonal Decomposition. In: <i>Reduced-Order Modeling
    (ROM) for Simulation and Optimization</i>. Cham; 2018:47-72. doi:<a href="https://doi.org/10.1007/978-3-319-75319-5_3">10.1007/978-3-319-75319-5_3</a>'
  apa: Beermann, D., Dellnitz, M., Peitz, S., &#38; Volkwein, S. (2018). Set-Oriented
    Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition.
    In <i>Reduced-Order Modeling (ROM) for Simulation and Optimization</i> (pp. 47–72).
    Cham. <a href="https://doi.org/10.1007/978-3-319-75319-5_3">https://doi.org/10.1007/978-3-319-75319-5_3</a>
  bibtex: '@inbook{Beermann_Dellnitz_Peitz_Volkwein_2018, place={Cham}, title={Set-Oriented
    Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition},
    DOI={<a href="https://doi.org/10.1007/978-3-319-75319-5_3">10.1007/978-3-319-75319-5_3</a>},
    booktitle={Reduced-Order Modeling (ROM) for Simulation and Optimization}, author={Beermann,
    Dennis and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan}, year={2018},
    pages={47–72} }'
  chicago: Beermann, Dennis, Michael Dellnitz, Sebastian Peitz, and Stefan Volkwein.
    “Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition.”
    In <i>Reduced-Order Modeling (ROM) for Simulation and Optimization</i>, 47–72.
    Cham, 2018. <a href="https://doi.org/10.1007/978-3-319-75319-5_3">https://doi.org/10.1007/978-3-319-75319-5_3</a>.
  ieee: D. Beermann, M. Dellnitz, S. Peitz, and S. Volkwein, “Set-Oriented Multiobjective
    Optimal Control of PDEs Using Proper Orthogonal Decomposition,” in <i>Reduced-Order
    Modeling (ROM) for Simulation and Optimization</i>, Cham, 2018, pp. 47–72.
  mla: Beermann, Dennis, et al. “Set-Oriented Multiobjective Optimal Control of PDEs
    Using Proper Orthogonal Decomposition.” <i>Reduced-Order Modeling (ROM) for Simulation
    and Optimization</i>, 2018, pp. 47–72, doi:<a href="https://doi.org/10.1007/978-3-319-75319-5_3">10.1007/978-3-319-75319-5_3</a>.
  short: 'D. Beermann, M. Dellnitz, S. Peitz, S. Volkwein, in: Reduced-Order Modeling
    (ROM) for Simulation and Optimization, Cham, 2018, pp. 47–72.'
date_created: 2019-03-29T13:31:20Z
date_updated: 2022-01-06T07:04:00Z
department:
- _id: '101'
doi: 10.1007/978-3-319-75319-5_3
language:
- iso: eng
page: 47-72
place: Cham
project:
- _id: '52'
  name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication: Reduced-Order Modeling (ROM) for Simulation and Optimization
publication_identifier:
  isbn:
  - '9783319753188'
  - '9783319753195'
publication_status: published
status: public
title: Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal
  Decomposition
type: book_chapter
user_id: '47427'
year: '2018'
...