We consider a nonlinear optimization problem governed by partial differential equations (PDE) with uncertain parameters. It is addressed by a robust worst case formulation. The resulting optimization problem is of bi-level structure and is difficult to treat numerically. We propose an approximate robust formulation that employs linear and quadratic approximations. To speed up the computation, reduced order models based on proper orthogonal decomposition (POD) in combination with a posteriori error estimators are developed. The proposed strategy is then applied to the optimal placement of a permanent magnet in the rotor of a synchronous machine with moving rotor. Numerical results are presented to validate the presented approach.

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Abstract. In this paper the authors study a nonlinear elliptic-parabolic system, which is motivated by mathematical models for lithium ion batteries. For the reliable and fast numerical solution of the system a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The strategy is justified by an a-priori error estimate for the error between the solution to the coupled system and its POD approximation. The nonlinear coupling is realized by variants of the empirical interpolation introduced by Barrault et al. [3] and Chaturantabut et al. [4]. Numerical examples illustrate the efficiency of the proposed reduced-order modeling.

The construction of reduced-order models for parametrized partial differential systems using proper orthogonal decomposition (POD) is based on the information of the so-called snapshots. These provide the spatial distribution of the nonlinear system at discrete parameter and/or time instances. In this work a strategy is used, where the POD reduced-order model is improved by choosing additional snapshot locations in an optimal way; see Kunisch and Volkwein (ESAIM: M2AN, 44:509-529, 2010). These optimal snapshot locations influences the POD basis functions and therefore the POD reduced-order model. This strategy is used to build up a POD basis on a parameter set in an adaptive way. The approach is illustrated by the construction of the POD reduced-order model for the complex-valued Helmholtz equation.

We investigate an optimal control problem governed by a parametric elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model and optimization parameters. The resulting optimization problem, then, has a bi-level structure for the solution of this problem which leads a non-linear optimization problem with a min-max formulation. The idea is to utilize a suitable approximation of the robust counterpart. However, this approach turns out to be very expensive, therefore we propose a goal-oriented model order reduction approach which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which then is utilized in the numerical optimization. Numerical results are presented to validate the presented approach.