A certified model reduction approach for robust parameter optimization with PDE constraints

  • Alessandro AllaEmail author
  • Michael Hinze
  • Philip Kolvenbach
  • Oliver Lass
  • Stefan Ulbrich


We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting nonlinear optimization problem has a bilevel structure due to the min-max formulation. To approximate the worst case in the optimization problem, we propose linear and quadratic approximations. However, this approach still turns out to be very expensive; therefore, we propose an adaptive model order reduction technique which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which is then utilized in the numerical optimization. Numerical results are presented to validate the presented approach.


Model order reduction Parameter optimization Robust optimization Proper orthogonal decomposition 

Mathematics Subject Classification (2010)

35Q93 49J20 49K20 


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Funding information

This study is supported by the German BMBF in the context of the SIMUROM project (grant no. 05M2013) and the support of the German Research Foundation in the context of SFB 805.


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Authors and Affiliations

  1. 1.Department of MathematicsPUC-RioRio de JaneiroBrazil
  2. 2.Department of MathematicsUniversität HamburgHamburgGermany
  3. 3.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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