Mathematische Schriften Kassel
https://kobra.uni-kassel.de:443/handle/123456789/2010081834090
Sat, 10 Dec 2022 08:39:44 GMT2022-12-10T08:39:44ZIdentifying critical demand scenarios for the robust capacitated network design problem using principal component analysis
https://kobra.uni-kassel.de:443/handle/123456789/13584
In this paper, we consider the single-commodity robust network design problem. Given an undirected graph with capacity installation costs on its edges and a set S of scenarios with associated flow balance vectors that represent different scenarios of node supplies and demands, the goal is to find integer edge capacities that minimize the total installation cost and permit a feasible single commodity flow for each scenario. This problem arises, for example, in the design of power networks, which are dimensioned to accommodate many different load scenarios.
We propose a new method to identify a small subset S′ of the given scenarios, such that solving the robust network design problem for the smaller scenario set S′ leads to almost the same capacities as solving it for the full scenario set S. By considering only the scenarios in S′ , the size of the model that needs to be solved can be reduced substantially, while the error introduced by neglecting the remaining scenarios is kept very small. Our method only employs simple techniques from statistical data analysis, namely principal component analysis (PCA), and convex hull computations in low dimensions. Thus, its computational effort is very small and it is easily applicable to more complex network design problems.
We evaluate the effectiveness of the method in computational experiments for instances stemming from offshore power grid planning or telecommunication networks. Our results show that the proposed techniques are indeed well suited to identify small scenario subsets that lead to significantly reduced models with high quality solutions.
Tue, 30 Nov 2021 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/135842021-11-30T00:00:00ZBley, AndreasHahn, PhilippIn this paper, we consider the single-commodity robust network design problem. Given an undirected graph with capacity installation costs on its edges and a set S of scenarios with associated flow balance vectors that represent different scenarios of node supplies and demands, the goal is to find integer edge capacities that minimize the total installation cost and permit a feasible single commodity flow for each scenario. This problem arises, for example, in the design of power networks, which are dimensioned to accommodate many different load scenarios.
We propose a new method to identify a small subset S′ of the given scenarios, such that solving the robust network design problem for the smaller scenario set S′ leads to almost the same capacities as solving it for the full scenario set S. By considering only the scenarios in S′ , the size of the model that needs to be solved can be reduced substantially, while the error introduced by neglecting the remaining scenarios is kept very small. Our method only employs simple techniques from statistical data analysis, namely principal component analysis (PCA), and convex hull computations in low dimensions. Thus, its computational effort is very small and it is easily applicable to more complex network design problems.
We evaluate the effectiveness of the method in computational experiments for instances stemming from offshore power grid planning or telecommunication networks. Our results show that the proposed techniques are indeed well suited to identify small scenario subsets that lead to significantly reduced models with high quality solutions.No Chaos in Dixon's System
https://kobra.uni-kassel.de:443/handle/123456789/11470
The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.
Wed, 01 Jan 2020 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/114702020-01-01T00:00:00ZSeiler, Werner M.Seiß, MatthiasThe so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads
https://kobra.uni-kassel.de:443/handle/123456789/11318
We study a rate-independent system with non-convex energy and in the
case of a time-discontinuous loading. We prove existence of
the rate-dependent viscous regularization by time-incremental problems, while
the existence of the so called parameterized BV-solutions is obtained via
vanishing viscosity in a suitable parameterized setting. In addition, we prove
that the solution set is compact.
Wed, 25 Sep 2019 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/113182019-09-25T00:00:00ZKnees, DorotheeZanini, ChiaraWe study a rate-independent system with non-convex energy and in the
case of a time-discontinuous loading. We prove existence of
the rate-dependent viscous regularization by time-incremental problems, while
the existence of the so called parameterized BV-solutions is obtained via
vanishing viscosity in a suitable parameterized setting. In addition, we prove
that the solution set is compact.On the existence of symmetric minimizers
https://kobra.uni-kassel.de:443/handle/123456789/2018012354238
In this note we revisit a less known symmetrization method for functions with respect to a topological group, which we call G-averaging. We note that, although quite non-technical in nature, this method yields G-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstract theorem and show how it can be applied to the p-Laplace and polyharmonic Poisson problem in order to construct symmetric solutions. We also pose some open problems and explore further possibilities where the method of G-averaging could be applied to.
Tue, 23 Jan 2018 00:00:00 GMThttps://kobra.uni-kassel.de:443/handle/123456789/20180123542382018-01-23T00:00:00ZStylianou, AthanasiosIn this note we revisit a less known symmetrization method for functions with respect to a topological group, which we call G-averaging. We note that, although quite non-technical in nature, this method yields G-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstract theorem and show how it can be applied to the p-Laplace and polyharmonic Poisson problem in order to construct symmetric solutions. We also pose some open problems and explore further possibilities where the method of G-averaging could be applied to.