Invited Session Tue.3.MA 415

Tuesday, 15:15 - 16:45 h, Room: MA 415

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

Optimization applications in industry II


Chair: Dietmar Hömberg



Tuesday, 15:15 - 15:40 h, Room: MA 415, Talk 1

Stefan Ulbrich
Multilevel optimization based on adaptive discretizations and reduced order models for engineering applications

Coauthor: J. Carsten Ziems


We consider optimization problems governed by partial differential
equations. Multilevel techniques use a hierarchy of approximations to
this infinite dimensional problem and offer the potential to carry out
most optimization iterations on comparably coarse discretizations.
Motivated by engineering applications we discuss the efficient interplay between the optimization method, adaptive discretizations of the PDE, reduced order models derived from these discretizations, and error estimators.
To this end, we describe an adaptive multilevel SQP method that generates a hierarchy of adaptive discretizations during the optimization iteration using adaptive finite-element approximations and reduced order models such as POD. The adaptive refinement strategy is based on a posteriori error estimators for the PDE-constraint, the adjoint equation and the criticality measure. The resulting optimization methods allows to use existing adaptive PDE-solvers and error estimators in a modular way.
We demonstrate the efficiency of the approach by numerical examples for engineering applications.



Tuesday, 15:45 - 16:10 h, Room: MA 415, Talk 2

Martin Grepl
A certified reduced basis approach for parametrized linear-quadratic optimal control problems

Coauthor: Mark Kärcher


The solution of optimal control problems governed by partial differential equations (PDEs) using classical discretization techniques such as finite elements or finite volumes is computationally very expensive and time-consuming since the PDE must be solved many times. One way of decreasing the computational burden is the surrogate model based approach, where the original high-dimensional model is replaced by its reduced order approximation. However, the solution of the reduced order optimal control problem is suboptimal and reliable error estimation is therefore crucial.
In this talk, we present error estimation procedures for linear-quadratic optimal control problems governed by parametrized parabolic PDEs. To this end, employ the reduced basis method as a surrogate model for the solution of the optimal control problem and develop rigorous and efficiently evaluable a posteriori error bounds for the optimal control and the associated cost functional. Besides serving as a certificate of fidelity for the suboptimal solution, our a posteriori error bounds are also a crucial ingredient in generating the reduced basis with greedy algorithms.



Tuesday, 16:15 - 16:40 h, Room: MA 415, Talk 3

Irwin Yousept
PDE-constrained optimization involving eddy current equations


Eddy current equations consist of a coupled system of first-order PDEs arising from Maxwell's equations by neglecting the displacement current. Applications of such equations can be found in many modern technologies such as in induction heating, magnetic levitation, optimal design of electromagnetic meta-materials, and many others. In this talk, we discuss several PDE-constrained optimization problems involving time-harmonic eddy current equations as equality constraints. Recent theoretical and numerical results are presented.


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