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

Abstract

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.

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Funding

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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Correspondence to Alessandro Alla.

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Communicated by: Jan Hesthaven

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Alla, A., Hinze, M., Kolvenbach, P. et al. A certified model reduction approach for robust parameter optimization with PDE constraints. Adv Comput Math 45, 1221–1250 (2019). https://doi.org/10.1007/s10444-018-9653-1

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Keywords

  • Model order reduction
  • Parameter optimization
  • Robust optimization
  • Proper orthogonal decomposition

Mathematics Subject Classification (2010)

  • 35Q93
  • 49J20
  • 49K20