Buch, Englisch, 171 Seiten, PB, Format (B × H): 148 mm x 210 mm, Gewicht: 267 g
Reihe: Mathematik
Buch, Englisch, 171 Seiten, PB, Format (B × H): 148 mm x 210 mm, Gewicht: 267 g
Reihe: Mathematik
ISBN: 978-3-8439-2959-2
Verlag: Dr. Hut
In this thesis we study asymptotic limits applied to three optimization problems: optimal design of a semiconductor device and the zero space charge limit; control of a crowd, represented by either a particle system or a mean-field equation, with the help of external agents; consensus-based global optimization and its mean-field limit.
Our strategy can be summarized as follows: we analyze the models in the setting of optimal control or global optimization. Then we perform a limiting procedure to reduce the state information with the help of an appropriate limit. Finally, we investigate if the results obtained for both problems coincide in the limit, in order to verify that the chosen reduction approaches are reasonable. In addition to the analytical considerations we perform some numerical simulations.
In the first case we write the semiconductor device equation as solution of an additional optimization problem. Then, we face a bi-level optimization problem, which we analyze in detail. Due to the lack of an uniqueness results for the optimal controls, we show the convergence of the controls with the help of the concept of Gamma-convergence as the Debye length tends to zero. Further, we propose some asymptotic preserving numerical algorithms for the simulation and show their convergence. Finally, we show results obtained with the proposed algorithms.
In the second part we propose a huge system of ODEs to model the state of the optimal control problem. We proceed with the passage to the mean-field limit. In terms of a flow formulation, we are able to show some convergence rate for the controls as the number of particles tends to infinity. Again, we perform some numerical simulations to underline the theoretical results.
The third part is concerned with a consensus-based global optimization problem. We propose a particle game which is based on a weighted average. This average allows to pass the mean-field limit and is our candidate for the global minimizer. Under certain assumptions on the objective function, we show that the average approximates the minimizer arbitrary well. In the proof we rely on PDE methods. The numerical results in one and 20 dimensions indicate that the algorithm performs very well in high dimensions and requires few function evaluations.
