Projects

Joint project of Prof. Dr. Jens Lang and Prof. Dr. Stefan Ulbrich

DFG funded assistant: Debora Clever

Collaborating Assistant: Carsten Ziems

Description

To explore the fundamental scientific issues of high dimensional complex engineering applications such as optimal control problems with time-dependent partial differential algebraic equations (PDAEs) scalable numerical algorithms are requested. This means that the work necessary to solve increasingly larger problems should grow all but linearly – the optimal rate. In this joint project, we want to combine modern solution strategies to solve time-dependent systems of partial differential algebraic equations such as adaptive multilevel finite elements methods and error-controlled linearly implicit time integrators of higher order with novel adaptive multilevel inexact SQP methods, which provide an efficient handling of control and state constraints by interior-point or semismooth Newton strategies.

The optimization method controls the inexactness and accuracy of the PDAE-solver in an adaptive way by using appropriate error estimators. Adaptivity based on a posteriori error estimates enables us to judge the quality of the numerical approximations and used models to determine appropriate strategies to improve the accuracy of the overall optimization process. Successful adaptive methods lead to substantial savings in computer time and memory requirements. They can mean the difference between getting an answer or not to the optimization problem considered.

An optimal boundary control problem of the cooling down process of glass modelled by three-dimensional radiative heat transfer, and an optimal control of two-dimensional dopant's redistribution in silicon serve as showcase engineering applications where restrictions on state and control variables are essential.

DFG Schwerpunktprogramm 1253 “Optimization with Partial Differential Equations”