Was man simulieren kann, das kann man auch optimieren... : Zur mathematischen Optimierung komplexer technologischer Prozesse

Der Wunsch nach Optimierung spielt in vielen Lebensbereichen eine wichtige Rolle. Moderne mathematische Verfahren können helfen, auch komplizierte industrielle Prozesse zu optimieren.
Optimization of processes described by differential equations is very important for industrial applications. Moreover, physical restrictions on the process leads to additional pointwise inequality constraints. The first part of this article is devoted to historical aspects. The famous problem of Dido from Karthago 2000 years ago was the first optimization problem for a function. Calculus of variation was founded 1696 by the work of Johann Bernoulli and others. However, the theory of optimization with differential equations is very young: The first paper of Boltyanskij, Gamkrelidze und Pontryagin was published in 1956. In the beginning, the theory of optimal controls was mainly devoted to systems of ordinary differential equations with applications in robotics or aeronautics. Meanwhile, optimization with differential equations is very important in a lot of different fields, for instance medicine, nanotechnology, or new materials. Discretization of the differential equations leads to large nonlinear optimization problems. The DFGpriority programme 1253 “Optimization with partial differential equations“ specifies the average number of optimization variables in such an optimization problem by 107 unknowns. Clearly, the solving of such huge problems is mathematically challenging. Whole classes of methods like genetic algorithms or gradient methods cannot be used for such problems. Highly specified active set strategies and interior point methods are suitable tools to attack discretized optimal control problems. Very important for the practical realization are discretization aspects. The ultimative goal is to solve such optimal control problems with a given accuracy at low costs. In the last years different strategies are developed to solve optimal control problems with a discretization of moderate size and high accuracy. Since optimal controls have usually kinks and other singularities, completely new strategies have to be developed.
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