Kaiserslautern - Fachbereich Mathematik
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Fibre reinforced polymers(FRPs) are one the newest and modern materials. In FRPs a light polymer matrix holds but weak polymer matrix is strengthened by glass or carbon fibres. The result is a material that is light and compared to its weight, very strong.\par
The stiffness of the resulting material is governed by the direction and the length of the fibres. To better understand the behaviour of FRPs we need to know the fibre length distribution in the resulting material. The classic method for this is ashing, where a sample of the material is burned and destroyed. We look at CT images of the material. In the first part we assumed that we have a full fibre segmentation, we can fit an a cylinder to each individual fibre. In this setting we identified two problems, sampling bias and censoring.\par
Sampling bias occurs since a longer fibre has a higher probability to be visible in the observation window. To solve this problem we used a reweighed fibre length distribution. The weight depends on the used sampling rule.\par
For the censoring we used an EM algorithm. The EM algorithm is used to get a Maximum Likelihood estimator in cases of missing or censored data.\par
For this setting we deduced conditions such that the EM algorithm converges to at least a stationary point of the underlying likelihood function. We further found conditions such that if the EM converges to the correct ML estimator, the estimator is consistent and asymptotically normally distributed.\par
Since obtaining a full fibre segmentation is hard we further looked in the fibre endpoint process. The fibre end point process can be modelled as a Neymann-Scott cluster process. Using this model we can find a formula for the reduced second moment measure for this process. We use this formula to get an estimator for the fibre length distribution.\par
We investigated all estimators using simulation studies. We especially investigated their performance in the case of non overlapping fibres.
A classical conjecture in the representation theory of finite groups, the McKay conjecture, states that for any finite group and prime number p the number of complex irreducible characters of degree prime to p is equal to the number of complex irreducible characters of degree prime to p of the normalizer of a p-Sylow subgroup. Recently a reduction theorem was proved by Isaacs, Malle and Navarro: If all simple groups are “good”, then the McKay conjecture holds. In this work we are concerned with the problem of goodness for finite groups of Lie type in their defining characteristic. A simple group is called “good” if certain equivariant bijections between the involved character sets exist. We present a structural approach to the construction of such a bijection by utilizing the so-called “Steinberg-Map”. This yields very natural bijections and we prove most of the desired properties.
The aim of this thesis is to introduce an equilibrium insurance market model and study its properties and possible applications in risk class management.
First, an insurance market model based on an equilibrium approach is developed. Depending on the premium, the insured will choose the amount of coverage they buy in order to maximize their expected utility. The behavior of the insurer in different market regimes is then compared. While the premiums in markets with perfect competition are calculated in order to make no profit at all, insurers try to maximize their margins in a monopolistic market.
In markets modeled in this way several phenomena become evident. Perhaps the most important one is the so-called push-out effect. When customers with different attributes are insured together, insurance might become so expensive for one type of customers that those agents are better off with buying no insurance at all. The push-out effect was already shown for theoretical examples in the literature. We present a comprehensive analysis of the equilibrium insurance market model and the push-out effect for different insurance products such as life, health and disability insurance contracts using real-life data from different sources. In a concluding chapter we formulate indicators when a push-out can be expected and when not.
Machine learning regression approaches such as neural networks have gained vast popularity in recent years. The exponential growth of computing power has enabled larger and more evolved networks that can perform increasingly complex tasks. In our feasibility study about the use of neural networks in the regression of equilibrium insurance premiums it is shown that this regression is quite robust and the risk of overfitting can almost be excluded -- as long as the regression is performed on at least a few thousand data points.
Grouping customers of different risk types into contracts is important for the stability and the robustness of an insurance market. This motivates the study of the optimal assignment of risk classes into contracts, also known as rating classes. We provide a theoretical framework that makes use of techniques from different mathematical fields such as non-linear optimization, convex analysis, herding theory, game theory and combinatorics. In addition, we are able to show that the market specifications have a large impact on the optimal allocation of risk classes to contracts by the insurer. However, there does not need to be an optimal risk class assignment for each of these specifications.
To address this issue, we present two different approaches, one more theoretical and another that can easily be implemented in practice. An extension of our model to markets with capacity constraints rounds off the topic and extends the applicability of our approach.
The main objects of study in this thesis are abelian varieties and their endomorphism rings. Abelian varieties are not just interesting in their own right, they also have numerous applications in various areas such as in algebraic geometry, number theory and information security. In fact, they make up one of the best choices in public key cryptography and more recently in post-quantum cryptography. Endomorphism rings are objects attached to abelian varieties. Their computation plays an important role in explicit class field theory and in the security of some post-quantum cryptosystems.
There are subexponential algorithms to compute the endomorphism rings of abelian varieties of dimension one and two. Prior to this work, all these subexponential algorithms came with a probability of failure and additional steps were required to unconditionally prove the output. In addition, these methods do not cover all abelian varieties of dimension two. The objective of this thesis is to analyse the subexponential methods and develop ways to deal with the exceptional cases.
We improve the existing methods by developing algorithms that always output the correct endomorphism ring. In addition to that, we develop a novel approach to compute endomorphism rings of some abelian varieties that could not be handled before. We also prove that the subexponential approaches are simply not good enough to cover all the cases. We use some of our results to construct a family of abelian surfaces with which we build post-quantum cryptosystems that are believed to resist subexponential quantum attacks - a desirable property for cryptosystems. This has the potential of providing an efficient non interactive isogeny based key exchange protocol, which is also capable of resisting subexponential quantum attacks and will be the first of its kind.
The German energy mix, which provides an overview of the sources of electricity available in Germany, is changing as a result of the expansion of renewable energy sources. With this shift towards sustainable energy sources such as wind and solar power, the electricity market situation is also in flux. Whereas in the past there were few uncertainties in electricity generation and only demand was subject to stochastic uncertainties, generation is now subject to stochastic fluctuations as well, especially due to weather dependency. To provide a supportive framework for this different situation, the electricity market has introduced, among other things, the intraday market, products with half-hourly and quarter-hourly time slices, and a modified balancing energy market design. As a result, both electricity price forecasting and optimization issues remain topical.
In this thesis, we first address intraday market modeling and intraday index forecasting. To do so, we move to the level of individual bids in the intraday market and use them to model the limit order books of intraday products. Based on statistics of the modeled limit order books, we present a novel estimator for the intraday indices. Especially for less liquid products, the order book statistics contain relevant information that allows for significantly more accurate predictions in comparison to the benchmark estimator.
Unlike the intraday market, the day ahead market allows smaller companies without their own trading department to participate since it is operated as a market with daily auctions. We optimize the flexibility offer of such a small company in the day ahead market and model the prices with a stochastic multi-factor model already used in the industry. To make this model accessible for stochastic optimization, we discretize it in time and space using scenario trees. Here we present existing algorithms for scenario tree generation as well as our own extensions and adaptations. These are based on the nested distance, which measures the distance between two distributions of stochastic processes. Based on the resulting scenario trees, we apply the stochastic optimization methods of stochastic programming, dynamic programming, and reinforcement learning to illustrate in which context the methods are appropriate.
Die vorliegende Dissertation besteht aus zwei Hauptteilen: Neue Ergebnisse aus der Gaußchen Analysis und ihre Anwendung auf die Theorie der Pfadintegrale. Das zentrale Resultat des ersten Teils ist die Charakterisierung aller regulären Distributionen die man mit Donsker's Delta multiplizieren kann. Dabei wird eine explizite Formel für solche Produkte, die sogenannte Wick-Formel, angegeben. Im Anwendungsteil dieser Arbeit wird zunächst eine komplex skalierte Feynman-Kac-Formel und ihre zugehörigen Kerne mit Hilfe dieser Wick-Formel gezeigt. Desweiteren werden Feynman Integranden für neue Klassen von Potentialen als White Noise Distributionen konstruiert.
Zusammenfassung. In dieser Arbeit werden Probleme der numerischen Lösung finiter Differenzenverfahren partieller Differentialgleichungen in einem algebraischen Ansatz behandelt. Es werden sowohl theoretische Ergebnisse präsentiert als auch die praktische Implementierung mithilfe der Systeme SINGULAR und QEPCAD vorgeführt. Dabei beziehen sich die algebraischen Methoden auf zwei unterschiedliche Aspekte bei finiten Differenzenverfahren: die Erzeugung von Schemata mithilfe von Gröbnerbasen und die darauf folgende Stabilitätsanalyse mittels Quantorenelimination durch algebraische zylindrische Dekomposition. Beim Aufbau der Arbeit werden in den ersten drei Kapiteln in einer Rückschau die nötigen Begriffe aus der Computeralgebra gelegt, die Grundzüge der numerischen Konvergenztheorie finiter Differenzenschemata erklärt sowie die Anwendung des CAD-Algorithmus zur Quantorenelimierung skizziert. Das Kapitel 4 entwickelt ausgehend vom zugrunde liegenden Kontext die Formulierung und die dafür nötigen Bedingungen an Differenzenschemata, die algebraisch nach Definition ein Ideal in einem Polynomring darstellen. Neben der praktischen Handhabbarkeit der Objekte liegt die Betonung auf größtmöglicher Allgemeinheit in den Definitionen der Begriffe. Es werden äquivalente Wege der Erzeugung sowie Eigenschaften der Eindeutigkeit unter sehr speziellen Bedingungen an die verwendeten Approximationen gezeigt. Die Anwendung des CAD-Algorithmus auf die Abschätzung des Symbols eines Schemas wird erläutert. Das fünfte Kapitel beschreibt die SINGULAR-Bibliothek findiff.lib, welche das Zusammenspiel von SINGULAR und QEPCAD garantiert und eine vollständige Automatisierung der Erzeugung und Stabilitätsanalyse eines finiten Differenzenverfahrens ermöglicht.
Efficient time integration and nonlinear model reduction for incompressible hyperelastic materials
(2013)
This thesis deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements. We analyze the structure of the equations of motion and show that a differential-algebraic system of index 1 with a singular perturbation term needs to be solved. In the limit case the index may jump to index 3 and thus renders the time integration into a difficult problem. For the time integration we apply Rosenbrock methods and study their convergence behavior for a test problem, which highlights the importance of the well-known Scholz conditions for this problem class. Numerical tests demonstrate that such linear-implicit methods are an attractive alternative to established time integration methods in structural dynamics. In the second part we combine the simulation of nonlinear materials with a model reduction step. We use the method of proper orthogonal decomposition and apply it to the discretized system of second order. For a nonlinear model reduction to be efficient we approximate the nonlinearity by following the lookup approach. In a practical example we show that large CPU time savings can achieved. This work is in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications.
We present a new efficient and robust algorithm for topology optimization of 3D cast parts. Special constraints are fulfilled to make possible the incorporation of a simulation of the casting process into the optimization: In order to keep track of the exact position of the boundary and to provide a full finite element model of the structure in each iteration, we use a twofold approach for the structural update. A level set function technique for boundary representation is combined with a new tetrahedral mesh generator for geometries specified by implicit boundary descriptions. Boundary conditions are mapped automatically onto the updated mesh. For sensitivity analysis, we employ the concept of the topological gradient. Modification of the level set function is reduced to efficient summation of several level set functions, and the finite element mesh is adapted to the modified structure in each iteration of the optimization process. We show that the resulting meshes are of high quality. A domain decomposition technique is used to keep the computational costs of remeshing low. The capabilities of our algorithm are demonstrated by industrial-scale optimization examples.
In this thesis, the quasi-static Biot poroelasticity system in bounded multilayered domains in one and three dimensions is studied. In more detail, in the one-dimensional case, a finite volume discretization for the Biot system with discontinuous coefficients is derived. The discretization results in a difference scheme with harmonic averaging of the coefficients. Detailed theoretical analysis of the obtained discrete model is performed. Error estimates, which establish convergence rates for both primary as well as flux unknowns are derived. Besides, modified and more accurate discretizations, which can be applied when the interface position coincides with a grid node, are obtained. These discretizations yield second order convergence of the fluxes of the problem. Finally, the solver for the solution of the produced system of linear equations is developed and extensively tested. A number of numerical experiments, which confirm the theoretical considerations are performed. In the three-dimensional case, the finite volume discretization of the system involves construction of special interpolating polynomials in the dual volumes. These polynomials are derived so that they satisfy the same continuity conditions across the interface, as the original system of PDEs. This technique allows to obtain such a difference scheme, which provides accurate computation of the primary as well as of the flux unknowns, including the points adjacent to the interface. Numerical experiments, based on the obtained discretization, show second order convergence for auxiliary problems with known analytical solutions. A multigrid solver, which incorporates the features of the discrete model, is developed in order to solve efficiently the linear system, produced by the finite volume discretization of the three-dimensional problem. The crucial point is to derive problem-dependent restriction and prolongation operators. Such operators are a well-known remedy for the scalar PDEs with discontinuous coefficients. Here, these operators are derived for the system of PDEs, taking into account interdependence of different unknowns within the system. In the derivation, the interpolating polynomials from the finite volume discretization are employed again, linking thus the discretization and the solution processes. The developed multigrid solver is tested on several model problems. Numerical experiments show that, due to the proper problem-dependent intergrid transfer, the multigrid solver is robust with respect to the discontinuities of the coefficients of the system. In the end, the poroelasticity system with discontinuous coefficients is used to model a real problem. The Biot model, describing this problem, is treated numerically, i.e., discretized by the developed finite volume techniques and then solved by the constructed multigrid solver. Physical characteristics of the process, such as displacement of the skeleton, pressure of the fluid, components of the stress tensor, are calculated and then presented at certain cross-sections.