Mini-Kliegel Seminar: Set Oriented Numerics in Dynamics and Optimization

Dr. Michael Dellnitz

Over the last years so-called set oriented numerical methods have been developed in the context of the numerical treatment of dynamical systems. The basic idea is to cover the objects of interest -- for instance invariant manifolds or invariant measures -- by outer approximations which are created via adaptive multilevel subdivision techniques. These schemes allow for an extremely memory and time efficient discretization of the phase space and have the flexibility to be applied to several problem types.

In particular we will show that set oriented techniques can particularly be useful for the solution of multiobjective optimization problems. In these problems several objective functions have to be optimized at the same time. For instance, for a perfect economical production plan one wants to simultaneously minimize cost and maximize quality. As indicated by this example the different objectives typically contradict each other and therefore certainly do not have identical optima. Thus, the question arises how to approximate the "optimal compromises" which, in mathematical terms, define the so-called Pareto set. In order to make our set oriented numerical methods applicable we will first construct a dynamical system which possesses the Pareto set as an attractor. In a second step we will develop appropriate step size strategies. The corresponding techniques are applied to several real world applications.

In addition we will illustrate how to make use of set oriented numerical techniques for the approximation of transport processes which play an important role in many real world phenomena. We will mainly focus on two related applications: first we analyze the transport of asteroids in the solar system -- this work is particularly motivated by the explanation of the existence of the asteroid belt between Mars and Jupiter. Secondly we show how to analyze transport phenomena in ocean dynamics. Here the related mathematical models depend explicitly on time and this makes the numerical treatment inherently more difficult.

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