Set Oriented Numerical Methods for Dynamical Systems and Optimization
Set Oriented Numerical Methods for Dynamical Systems and Optimization
Over the last two decades so-calledÌýset orientedÌýnumerical methodsÌýhaveÌýbeenÌýdeveloped in the context of the numerical treatmentÌýof dynamical systems.ÌýThe basic idea is to cover the objectsÌýofÌýinterest - for instanceÌýinvariantÌýsetsÌýorÌýinvariant measuresÌý-Ìýby outer approximations which are createdÌývia multilevelÌýsubdivision techniques.ÌýAt the beginning of this centuryÌýtheseÌýmethods have been modified in such a way thatÌýthey are also applicable toÌýtheÌýnumerical treatment ofÌýmultiobjective optimizationÌýproblems. Due to theÌýfactÌýthat they are set oriented in nature these techniques allow for theÌýdirectÌýcomputation of theÌýentire so-calledÌýPareto set. ÌýIn this talk recent developments in the area of set oriented numerics will beÌýpresented both for dynamical systems and optimization problems. TheÌýreliability of these methods will be demonstrated by several applicationsÌýsuch as the approximation of transport processes in ocean dynamics, or theÌýoptimization of a cruise control with respect to energy consumption and travelÌýdistance. Moreover a new algorithmic idea will be described which allowsÌýto compute invariant sets directly by Newton's method.