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978-3-8439-0878-8, Reihe Mathematik

Debora Clever
Adaptive Multilevel Methods for PDAE-Constrained Optimal Control Problems

218 Seiten, Dissertation Technische Universität Darmstadt (2013), Softcover, A5

Zusammenfassung / Abstract

Optimal control theory for real-world applications is a challenging task of high interdisciplinary degree. It combines engineering science, computer science and mathematics, in particular the numerical analysis and the optimization. The main goal of this work is to develop a fully adaptive optimization environment, suitable to solve real-world optimal control problems restricted by partial differential algebraic equations (PDAEs). To this end, we embed the optimization problem into function spaces, reduce the problem to the control component and rely on continuous adjoint calculus. The environment is build up in a modular way, including the following three modules:

The first module includes the fully space-time adaptive solution of the involved PDAEs. We build on the software package KARDOS, which we augment to serve optimization requirements, such as autonomous grid and data management between up to four correlated PDAEs and the online evolution of functionals like objective, reduced gradient and applications of the reduced Hessian.

The second module includes several optimization techniques, ranging from a simple gradient method to Newton-type methods with good convergence properties. It provides suitable projection strategies to handle point-wise constraints on the control.

The third module includes a multilevel strategy, which tailors the grid refinement to the optimization progress. It controls the inconsistencies caused by inexact reduction to the control component and ensures global convergence of the finite dimensional control iterates to a stationary point of the infinite dimensional problem.

In a second step the environment is used to solve an optimal boundary control problem arising in glass manufacturing during the cooling process. The physical behavior of the cooling process is modeled by radiative heat transfer and simplified by spherical harmonics resulting in systems of partial differential algebraic equations. The performance of the environment and the results of the optimization are studied at basis of two models of different complexity in two and three spatial dimensions.