Oliver Lass

Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to illustrate and validate the optimal efficiency and robustness of the performance of the present multigrid strategy.

POD Galerkin schemes for nonlinear elliptic-parabolic systems

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 a priori error analysis 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. [C. R. Acad. Sci. Paris Ser. I, 339 (2004), pp. 667--672] and Chaturantabut and Sorensen [Application of POD and DEIM on a Dimension Reduction of Nonlinear Miscible Viscous Fingering in Porous Media, Technical Report, RICE University, 2009]. Numerical examples illustrate the efficiency of the proposed reduced-order modeling.

Adaptive POD basis computation for parametrized nonlinear systems using optimal snapshot location

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.

Parameter identification for nonlinear elliptic-parabolic systems with application in lithium-ion battery modeling

In this paper the authors consider a parameter estimation problem for a nonlinear systems, which consists of one parabolic equation for the concentration and two elliptic equations for the potentials. The measurements are given as boundary values for one of the potentials. For its numerical solution the Gauss Newton method is applied. To speed up the solution process, a reduced-order approach based on proper orthogonal decomposition is utilized, where the accuracy is controlled by error estimators. Parameters, which can not be identified from the measurements, are identified by the subset selection method with

Space mapping techniques for a structural optimization problem governed by the p-Laplace equation

QR

Model Order Reduction for Rotating Electrical Machines

QR pivoting. Numerical examples show the efficiency of the proposed approach.

Robust shape optimization of electric devices based on deterministic optimization methods and finite-element analysis with affine parametrization and design elements

We consider a nonlinear optimization problem governed by partial differential equations with uncertain parameters. It is addressed by a robust worst case formulation. The resulting optimization problem is of bilevel 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 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.

Uncertainty Quantification for a Permanent Magnet Synchronous Machine with Dynamic Rotor Eccentricity

Solving optimal control problems for real-world applications are hard to tackle numerically due to the large size and the complex underlying (partial differential equations based) models. In this paper, a structural optimization problem governed by the p-Laplace equation (fine model) is considered. A surrogate optimization is utilized to develop an efficient numerical optimization method. Here the p-Laplace equation is replaced by a simplified (coarse) model, a space mapping attempts to match, in the coarse model, the values of the p-Laplace equation. Numerical examples illustrate the presented approach.