---
_id: '8753'
abstract:
- lang: eng
  text: In a wide range of applications it is desirable to optimally control a dynamical
    system with respect to concurrent, potentially competing goals. This gives rise
    to a multiobjective optimal control problem where, instead of computing a single
    optimal solution, the set of optimal compromises, the so-called Pareto set, has
    to be approximated. When the problem under consideration is described by a partial
    differential equation (PDE), as is the case for fluid flow, the computational
    cost rapidly increases and makes its direct treatment infeasible. Reduced order
    modeling is a very popular method to reduce the computational cost, in particular
    in a multi query context such as uncertainty quantification, parameter estimation
    or optimization. In this article, we show how to combine reduced order modeling
    and multiobjective optimal control techniques in order to efficiently solve multiobjective
    optimal control problems constrained by PDEs. We consider a global, derivative
    free optimization method as well as a local, gradient-based approach for which
    the optimality system is derived in two different ways. The methods are compared
    with regard to the solution quality as well as the computational effort and they
    are illustrated using the example of the flow around a cylinder and a backward-facing-step
    channel flow.
author:
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  id: '16494'
  last_name: Ober-Blöbaum
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: Peitz S, Ober-Blöbaum S, Dellnitz M. Multiobjective Optimal Control Methods
    for the Navier-Stokes Equations Using Reduced Order Modeling. <i>Acta Applicandae
    Mathematicae</i>. 2018;161(1):171–199. doi:<a href="https://doi.org/10.1007/s10440-018-0209-7">10.1007/s10440-018-0209-7</a>
  apa: Peitz, S., Ober-Blöbaum, S., &#38; Dellnitz, M. (2018). Multiobjective Optimal
    Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling.
    <i>Acta Applicandae Mathematicae</i>, <i>161</i>(1), 171–199. <a href="https://doi.org/10.1007/s10440-018-0209-7">https://doi.org/10.1007/s10440-018-0209-7</a>
  bibtex: '@article{Peitz_Ober-Blöbaum_Dellnitz_2018, title={Multiobjective Optimal
    Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling},
    volume={161}, DOI={<a href="https://doi.org/10.1007/s10440-018-0209-7">10.1007/s10440-018-0209-7</a>},
    number={1}, journal={Acta Applicandae Mathematicae}, author={Peitz, Sebastian
    and Ober-Blöbaum, Sina and Dellnitz, Michael}, year={2018}, pages={171–199} }'
  chicago: 'Peitz, Sebastian, Sina Ober-Blöbaum, and Michael Dellnitz. “Multiobjective
    Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling.”
    <i>Acta Applicandae Mathematicae</i> 161, no. 1 (2018): 171–199. <a href="https://doi.org/10.1007/s10440-018-0209-7">https://doi.org/10.1007/s10440-018-0209-7</a>.'
  ieee: 'S. Peitz, S. Ober-Blöbaum, and M. Dellnitz, “Multiobjective Optimal Control
    Methods for the Navier-Stokes Equations Using Reduced Order Modeling,” <i>Acta
    Applicandae Mathematicae</i>, vol. 161, no. 1, pp. 171–199, 2018, doi: <a href="https://doi.org/10.1007/s10440-018-0209-7">10.1007/s10440-018-0209-7</a>.'
  mla: Peitz, Sebastian, et al. “Multiobjective Optimal Control Methods for the Navier-Stokes
    Equations Using Reduced Order Modeling.” <i>Acta Applicandae Mathematicae</i>,
    vol. 161, no. 1, 2018, pp. 171–199, doi:<a href="https://doi.org/10.1007/s10440-018-0209-7">10.1007/s10440-018-0209-7</a>.
  short: S. Peitz, S. Ober-Blöbaum, M. Dellnitz, Acta Applicandae Mathematicae 161
    (2018) 171–199.
date_created: 2019-03-29T13:30:41Z
date_updated: 2022-01-21T10:01:41Z
department:
- _id: '101'
doi: 10.1007/s10440-018-0209-7
intvolume: '       161'
issue: '1'
language:
- iso: eng
page: 171–199
project:
- _id: '52'
  name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication: Acta Applicandae Mathematicae
publication_identifier:
  issn:
  - 0167-8019
  - 1572-9036
publication_status: published
status: public
title: Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using
  Reduced Order Modeling
type: journal_article
user_id: '15694'
volume: 161
year: '2018'
...