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Mon, 01 Oct 2012 16:25:45 +0200Mon, 01 Oct 2012 16:25:45 +0200Intersection Theory of the Tropical Moduli Spaces of Curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3276
Tropical geometry is a very new mathematical domain. The appearance of
tropical geometry was motivated by its deep relations to other mathematical
branches. These include algebraic geometry, symplectic geometry, complex
analysis, combinatorics and mathematical biology.
In this work we see some more relations between algebraic geometry and
tropical geometry. Our aim is to prove a one-to-one correspondence between
the divisor classes on the moduli space of n-pointed rational stable curves
and the divisors of the moduli space of n-pointed abstract tropical curves.
Thus we state some results of the algebraic case first. In algebraic geometry
these moduli spaces are well understood. In particular, the group of divisor
classes is calculated by S. Keel. We recall the needed results in chapter one.
For the proof of the correspondence we use some results of toric geometry.
Further we want to show an equality of the Chow groups of a special toric
variety and the algebraic moduli space. Thus we state some results of the
toric geometry as well.
This thesis tries to discover some connection between algebraic and tropical
geometry. Thus we also need the corresponding tropical objects to the
algebraic objects. Therefore we give some necessary definitions such as fan,
tropical fan, morphisms between tropical fans, divisors or the topical moduli
space of all n-marked tropical curves. Since we need it, we show that the
tropical moduli space can be embedded as a tropical fan.
After this preparatory work we prove that the group of divisor classes in
v
classical algebraic geometry has it equivalence in tropical geometry. For this
it is useful to give a map from the group of divisor classes of the algebraic
moduli space to the group of divisors of the tropical moduli space. Our aim is
to prove the bijectivity of this map in chapter three. On the way we discover
a deep connection between the algebraic moduli space and the toric variety
given by the tropical fan of the tropical moduli space.Matthias Herolddiplomhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3276Mon, 01 Oct 2012 16:25:45 +0200Four Generations of Asset Pricing Models and Volatility Dynamics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2248
The scope of this diploma thesis is to examine the four generations of asset pricing models and the corresponding volatility dynamics which have been devepoled so far. We proceed as follows: In chapter 1 we give a short repetition of the Black-Scholes first generation model which assumes a constant volatility and we show that volatility should not be modeled as constant by examining statistical data and introducing the notion of implied volatility. In chapter 2, we examine the simplest models that are able to produce smiles or skews - local volatility models. These are called second generation models. Local volatility models model the volatility as a function of the stock price and time. We start with the work of Dupire, show how local volatility models can be calibrated and end with a detailed discussion of the constant elasticity of volatility model. Chapter 3 focuses on the Heston model which represents the class of the stochastic volatility models, which assume that the volatility itself is driven by a stochastic process. These are called third generation models. We introduce the model structure, derive a partial differential pricing equation, give a closed-form solution for European calls by solving this equation and explain how the model is calibrated. The last part of chapter 3 then deals with the limits and the mis-specifications of the Heston model, in particular for recent exotic options like reverse cliquets, Accumulators or Napoleons. In chapter 4 we then introduce the Bergomi forward variance model which is called fourth generation model as a consequence of the limits of the Heston model explained in chapter 3. The Bergomi model is a stochastic local volatility model - the spot price is modeled as a constant elasticity of volatility diffusion and its volatility parameters are functions of the so called forward variances which are specified as stochastic processes. We start with the model specification, derive a partial differential pricing equation, show how the model has to be calibrated and end with pricing examples and a concluding discussion.Sascha Desmettrediplomhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2248Mon, 24 Jan 2011 09:08:01 +0100Vector bundles on degenerations of elliptic curves of types II, III and IV
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2039
In this thesis we classify simple coherent sheaves on Kodaira fibers of types II, III and IV (cuspidal and tacnode cubic curves and a plane configuration of three concurrent lines). Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. In works of Burban, Drozd and Greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. It turns out, that all other degenerations of elliptic curves are vector-bundle-wild. Nevertheless, we prove that the category of coherent sheaves of an arbitrary reduced plane cubic curve, (including the mentioned Kodaira fibers) is brick-tame. The main technical tool of our approach is the representation theory of bocses. Although, this technique was mainly used for purely theoretical purposes, we illustrate its computational potential for investigating tame behavior in wild categories. In particular, it allows to prove that a simple vector bundle on a reduced cubic curve is determined by its rank, multidegree and determinant, generalizing Atiyah's classification. Our approach leads to an interesting class of bocses, which can be wild but are brick-tame.Lesya Bodnarchukdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2039Tue, 11 Nov 2008 11:01:01 +0100Soft Operators Decision Trees. Uncertainty and stability related issues
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2014
The nowadays increasing number of fields where large quantities of data are collected generates an emergent demand for methods for extracting relevant information from huge databases. Amongst the various existing data mining models, decision trees are widely used since they represent a good trade-off between accuracy and interpretability. However, one of their main problems is that they are very instable, which complicates the process of the knowledge discovery because the users are disturbed by the different decision trees generated from almost the same input learning samples. In the current work, binary tree classifiers are analyzed and partially improved. The analysis of tree classifiers goes from their topology from the graph theory point of view to the creation of a new tree classification model by means of combining decision trees and soft comparison operators (Mlynski, 2003) with the purpose to not only overcome the well known instability problem of decision trees, but also in order to confer the ability of dealing with uncertainty. In order to study and compare the structural stability of tree classifiers, we propose an instability coefficient which is based on the notion of Lipschitz continuity and offer a metric to measure the proximity between decision trees. This thesis converges towards its main part with the presentation of our model ``Soft Operators Decision Tree\'\' (SODT). Mainly, we describe its construction, application and the consistency of the mathematical formulation behind this. Finally we show the results of the implementation of SODT and compare numerically the stability and accuracy of a SODT and a crisp DT. The numerical simulations support the stability hypothesis and a smaller tendency to overfitting the training data with SODT than with crisp DT is observed. A further aspect of this inclusion of soft operators is that we choose them in a way so that the resulting goodness function (used by this method) is differentiable and thus allows to calculate the best split points by means of gradient descent methods. The main drawback of SODT is the incorporation of the unpreciseness factor, which increases the complexity of the algorithm.Eva Barrena Algaradoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2014Tue, 29 Jul 2008 16:11:54 +0200On the Local Multiscale Determination of the Earth`s Disturbing Potential From Discrete Deflections of the Vertical
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1947
As a first approximation the Earth is a sphere; as a second approximation it may be considered an ellipsoid of revolution. The deviations of the actual Earth's gravity field from the ellipsoidal 'normal' field are so small that they can be understood to be linear. The splitting of the Earth's gravity field into a 'normal' and a remaining small 'disturbing' field considerably simplifies the problem of its determination. Under the assumption of an ellipsoidal Earth model high observational accuracy is achievable only if the deviation (deflection of the vertical) of the physical plumb line, to which measurements refer, from the ellipsoidal normal is not ignored. Hence, the determination of the disturbing potential from known deflections of the vertical is a central problem of physical geodesy. In this paper we propose a new, well-promising method for modelling the disturbing potential locally from the deflections of the vertical. Essential tools are integral formulae on the sphere based on Green's function of the Beltrami operator. The determination of the disturbing potential from deflections of the vertical is formulated as a multiscale procedure involving scale-dependent regularized versions of the surface gradient of the Green function. The modelling process is based on a multiscale framework by use of locally supported surface curl-free vector wavelets.Willi Freeden; Thomas Fehlinger; Carsten Mayer; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1947Mon, 07 Apr 2008 17:08:11 +0200Nonlinear diffusion filtering of images using the topological gradient approach to edges detection
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1941
In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain \(\Omega \subset \mathbb{R}^2\). In order to generate a nonuniform grid, the adaptive coarsening technique is proposed.Monika Muszkietadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1941Mon, 17 Mar 2008 13:38:48 +0100Fast Wavelet Transform by Biorthogonal Locally Supported Radial Basis Functions on Fixed Spherical Grids
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1932
The thesis is concerned with multiscale approximation by means of radial basis functions on hierarchically structured spherical grids. A new approach is proposed to construct a biorthogonal system of locally supported zonal functions. By use of this biorthogonal system of locally supported zonal functions, a spherical fast wavelet transform (SFWT) is established. Finally, based on the wavelet analysis, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators are shown to be efficiently and economically solvable.Ali A. Moghisehdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1932Mon, 25 Feb 2008 10:55:30 +0100Time-Space Multiscale Analysis by Use of Tensor Product Wavelets and its Application to Hydrology and GRACE Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1924
This paper presents a wavelet analysis of temporal and spatial variations of the Earth's gravitational potential based on tensor product wavelets. The time--space wavelet concept is realized by combining Legendre wavelets for the time domain and spherical wavelets for the space domain. In consequence, a multiresolution analysis for both, temporal and spatial resolution, is formulated within a unified concept. The method is then numerically realized by using first synthetically generated data and, finally, several real data sets.Helga Nutz; Kerstin Wolfpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1924Fri, 11 Jan 2008 11:11:17 +0100Capacity Inverse Minimum Cost Flow Problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1914
Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.Cigdem Güler; Horst Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1914Mon, 03 Dec 2007 12:01:57 +0100A-infinity-bimodules and Serre A-infinity-functors
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1910
This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which form a framework in which the theory of A-infinity-categories is developed. We prove that (unital) A-infinity-categories constitute a closed symmetric multicategory. We define the notion of A-infinity-bimodule similarly to Tradler and show that it is equivalent to an A-infinity-functor of two arguments which takes values in the differential graded category of complexes of k-modules, where k is a commutative ground ring. Serre A-infinity-functors are defined via A-infinity-bimodules following ideas of Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category over a field admits a Serre A-infinity-functor if and only if its homotopy category admits an ordinary Serre functor. The proof uses categories and Serre functors enriched in the homotopy category of complexes of k-modules. Another important ingredient is an A-infinity-version of the Yoneda Lemma.Oleksandr Manzyukdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1910Tue, 06 Nov 2007 14:25:46 +0100Vector Field Approximation on Regular Surfaces in Terms of Outer Harmonic Representations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1904
The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.Anna Lutherdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1904Thu, 25 Oct 2007 14:03:47 +0200Some asymptotics for local least-squares regression with regularization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1902
We derive some asymptotics for a new approach to curve estimation proposed by Mr'{a}zek et al. cite{MWB06} which combines localization and regularization. This methodology has been considered as the basis of a unified framework covering various different smoothing methods in the analogous two-dimensional problem of image denoising. As a first step for understanding this approach theoretically, we restrict our discussion here to the least-squares distance where we have explicit formulas for the function estimates and where we can derive a rather complete asymptotic theory from known results for the Priestley-Chao curve estimate. In this paper, we consider only the case where the bias dominates the mean-square error. Other situations are dealt with in subsequent papers.Jürgen Franke; Joseph Tadjuidje; Stefan Didas; Joachim Weickertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1902Thu, 11 Oct 2007 12:37:44 +0200The difference of the solutions of the elastic and elastoplastic boundary value problem and an approach to multiaxial stress-strain correction
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1900
In the theoretical part of this thesis, the difference of the solutions of the elastic and the elastoplastic boundary value problem is analysed, both for linear kinematic and combined linear kinematic and isotropic hardening material. We consider both models in their quasistatic, rate-independent formulation with linearised geometry. The main result of the thesis is, that the differences of the physical obervables (the stresses, strains and displacements) can be expressed as composition of some linear operators and play operators with respect to the exterior forces. Explicit homotopies between both solutions are presented. The main analytical devices are Lipschitz estimates for the stop and the play operator. We present some generalisations of the standard estimates. They allow different input functions, different initial memories and different scalar products. Thereby, the underlying time involving function spaces are the Sobolov spaces of first order with arbitrary integrability exponent between one and infinity. The main results can easily be generalised for the class of continuous functions with bounded total variation. In the practical part of this work, a method to correct the elastic stress tensor over a long time interval at some chosen points of the body is presented and analysed. In contrast to widespread uniaxial corrections (Neuber or ESED), our method takes multiaxiality phenomena like cyclic hardening/softening, ratchetting and non-masing behaviour into account using Jiang's model of elastoplasticity. It can be easily adapted to other constitutive elastoplastic material laws. The theory for our correction model is developped for linear kinematic hardening material, for which error estimated are derived. Our numerical algorithm is very fast and designed for the case that the elastic stress is piecewise linear. The results for the stresses can be significantly improved with Seeger's empirical strain constraint. For the improved model, a simple predictor-correcor algorithm for smooth input loading is established.Holger Langdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1900Thu, 11 Oct 2007 11:20:18 +0200A piecewise analytical solution for Jiangs model of elastoplasticity
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1898
In this article, we present an analytic solution for Jiang's constitutive model of elastoplasticity. It is considered in its stress controlled form for proportional stress loading under the assumptions that the one-to-one coupling of the yield surface radius and the memory surface radius is switched off, that the transient hardening is neglected and that the ratchetting exponents are constant.Holger Lang; Klaus Dressler; Rene Pinnaureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1898Tue, 02 Oct 2007 12:27:59 +0200New heuristics for the minimum fundamental cut basis problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1887
Given an undirected connected network and a weight function finding a basis of the cut space with minimum sum of the cut weights is termed Minimum Cut Basis Problem. This problem can be solved, e.g., by the algorithm of Gomory and Hu [GH61]. If, however, fundamentality is required, i.e., the basis is induced by a spanning tree T in G, the problem becomes NP-hard. Theoretical and numerical results on that topic can be found in Bunke et al. [BHMM07] and in Bunke [Bun06]. In the following we present heuristics with complexity O(m log n) and O(mn), where n and m are the numbers of vertices and edges respectively, which obtain upper bounds on the aforementioned problem and in several cases outperform the heuristics of Schwahn [Sch05].Alexander J. Perez Tchernov; Anne M. Schwahnpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1887Tue, 24 Jul 2007 14:31:16 +0200Local Smoothing Methods in Image Processing
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1878
In this article a new data-adaptive method for smoothing of bivariate functions is developed. The smoothing is done by kernel regression with rotational invariant bivariate kernels. Two or three local bandwidth parameters are chosen automatically by a two-step plug-in approach. The algorithm starts with small global bandwidth parameters, which adapt during a few iterations to the noisy image. In the next step local bandwidths are estimated. Some general asymptotic results about Gasser-Müller-estimators and optimal bandwidth selection are given. The derived local bandwidth estimators converge and are asymptotically normal.Vera Friederichspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1878Tue, 03 Jul 2007 13:00:09 +0200The Application of Reproducing Kernel Based Spline Approximation to Seismic Surface and Body Wave Tomography: Theoretical Aspects and Numerical Results
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1872
The main aim of this work was to obtain an approximate solution of the seismic traveltime tomography problems with the help of splines based on reproducing kernel Sobolev spaces. In order to be able to apply the spline approximation concept to surface wave as well as to body wave tomography problems, the spherical spline approximation concept was extended for the case where the domain of the function to be approximated is an arbitrary compact set in R^n and a finite number of discontinuity points is allowed. We present applications of such spline method to seismic surface wave as well as body wave tomography, and discuss the theoretical and numerical aspects of such applications. Moreover, we run numerous numerical tests that justify the theoretical considerations.Abel Amirbekyandoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1872Tue, 26 Jun 2007 14:43:44 +0200Neue Aspekte der Portfolio-Optimierung und der Modellierung von Bondindizes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1875
Zwei zentrale Probleme der modernen Finanzmathematik sind die Portfolio-Optimierung und die Optionsbewertung. Während es bei der Portfolio-Optimierung darum geht, das Vermögen optimal auf verschiedene Anlagemöglichkeiten zu verteilen, versucht die Optionsbewertung faire Preise von derivativen Finanzinstrumenten zu bestimmen. In dieser Arbeit werden Fragestellungen aus beiden dieser Themenbereiche bearbeitet. Die Arbeit beginnt mit einem Kapitel über Grundlagen, in dem zum Beispiel das Portfolio-Problem von Merton dargestellt und die Black/Scholes-Formel zur Optionsbewertung hergeleitet wird. In Kapitel 2 wird das Portfolio-Problem von Morton und Pliska betrachtet, die in das Merton-Modell fixe Transaktionskosten eingeführt haben. Dabei muß der Investor bei jeder Transaktion einen fixen Anteil vom derzeitigen Vermögen als Kosten abführen. Es wird die asymptotische Approximation dieses Modells von Atkinson und Wilmott vorgestellt und die optimale Portfoliostrategie aus den Marktparametern hergeleitet. Danach werden die tatsächlichen Transaktionskosten abgeschätzt und ein User Guide zur praktischen Anwendung dieses Transaktionskostenmodells angegeben. Zum Schluß wird das Modell numerisch analysiert, indem unter anderem die erwartete Handelszeit und die Güte der Abschätzung der tatsächlichen Transaktionskosten berechnet werden. Ein Portfolio-Problem mit internationalen Märkten wird in Kapitel 3 vorgestellt. Dem Investor steht zusätzlich zu seinem Heimatland noch ein weiteres Land für seine Vermögensanlagen zur Verfügung. Dabei werden die Preisprozesse für die ausländischen Wertpapiere mit einem stochastischen Wechselkurs in die Heimatwährung umgerechnet. In einer statischen Analyse wird unter anderem berechnet, wieviel weniger Vermögen der Investor benötigt, um das gleiche erwartete Endvermögen zu erhalten wie in dem Fall, wenn ihm keine Auslandsanlagen zur Verfügung stehen. Kapitel 4 behandelt drei verschiedene Portfolio-Probleme mit Sprung-Diffusions-Prozessen. Nach der Herleitung eines Verifikationssatzes wird das Problem bei Anlagemöglichkeit in eine Aktie und in ein Geldmarktkonto jeweils für eine konstante und eine stochastische Zinsrate untersucht. Im ersten Fall wird eine implizite Darstellung für den optimalen Portfolioprozeß und eine Bedingung angegeben, unter der diese Darstellung eindeutig lösbar ist. Außerdem wird der optimale Portfolioprozeß für verschiedene Verteilungen für die Sprunghöhe untersucht. Im Falle einer stochastischen Zinsrate kann nur ein Kandidat für den optimalen Lösungsprozeß angeben werden. Dieser hat wieder eine implizite Darstellung. Das letzte Portfolio-Problem ist eine Abwandlung des Modells aus Kapitel 3. Wird dort der Wechselkurs durch eine geometrisch Brownsche Bewegung modelliert, ist er hier ein reiner Sprungprozeß. Es wird wieder der optimale Portfolioprozeß hergeleitet, wobei ein Anteil davon unter Umständen nur numerisch lösbar ist. Eine hinreichende Bedingung für die Lösbarkeit wird angegeben. In Kapitel 5 werden verschiedene Bewertungsansätze für Optionen auf Bondindizes präsentiert. Es wird eine Methode vorgestellt, mit der die Optionen anhand von Marktpreisen bewertet werden können. Für den Fall, daß es nicht genug Marktpreise gibt, wird ein Verfahren angegeben, um den Bondindex realitätsnah zu simulieren und künstliche Marktpreise zu erzeugen. Diese Preise können dann für eine Kalibrierung verwendet werden.Tin-Kwai Mandoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1875Tue, 26 Jun 2007 14:41:22 +0200A condition that a continuously deformed, simply connected body does not penetrate itself
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1874
In this article we give a sufficient condition that a simply connected flexible body does not penetrate itself, if it is subjected to a continuous deformation. It is shown that the deformation map is automatically injective, if it is just locally injective and injective on the boundary of the body. Thereby, it is very remarkable that no higher regularity assumption than continuity for the deformation map is required. The proof exclusively relies on homotopy methods and the Jordan-Brouwer separation theorem.Holger Langreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1874Sat, 23 Jun 2007 13:31:37 +0200Weight-Constrained Minimum Spanning Tree Problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1873
In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.Sebastian Tobias Henndiplomhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1873Thu, 21 Jun 2007 22:56:38 +0200Ovoide 8-dimensionaler quadratischer Räume
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1867
Diese Arbeit beschäftigt sich mit Methoden zur Klassifikation von Ovoiden in quadratischen Räumen. Die Anwendung der dazu entwickelten Algorithmen erfolgt hauptsächlich in achtdimensionalen Räumen speziell über den Körpern GF(7), GF(8) und GF(9). Zu verschiedenen, zumeist kleinen, zyklischen Gruppen werden hier die unter diesen Gruppen invarianten Ovoide bestimmt. Die bei dieser Suche auftretenden Ovoide sind alle bereits bekannt. Es ergeben sich jedoch Restriktionen an die Stabilisatoren gegebenenfalls existierender, unbekannter Ovoide.Andrea Mindnichdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1867Sat, 02 Jun 2007 13:21:12 +0200Efficient numerical methods for the Biot poroelasticity system in multilayered domains
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1865
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.Anna Naumovichdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1865Wed, 16 May 2007 21:04:13 +0200Convergent Finite Element Discretizations of the Density Gradient Equation for Quantum Semiconductors
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1864
We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. Especially, we give a finite element description of the so--called nonlinear scheme introduced by {it Ancona}. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, especially with respect to the influence of approximate vacuum boundary conditions.Rene Pinnau; Jorge Mauricio Ruizpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1864Fri, 11 May 2007 20:46:05 +0200The Optimal Shape of the Reflex Tube of a Bass Loudspeaker
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1862
In the thesis the author presents a mathematical model which describes the behaviour of the acoustical pressure (sound), produced by a bass loudspeaker. The underlying physical propagation of sound is described by the non--linear isentropic Euler system in a Lagrangian description. This system is expanded via asymptotical analysis up to third order in the displacement of the membrane of the loudspeaker. The differential equations which describe the behaviour of the key note and the first order harmonic are compared to classical results. The boundary conditions, which are derived up to third order, are based on the principle that the small control volume sticks to the boundary and is allowed to move only along it. Using classical results of the theory of elliptic partial differential equations, the author shows that under appropriate conditions on the input data the appropriate mathematical problems admit, by the Fredholm alternative, unique solutions. Moreover, certain regularity results are shown. Further, a novel Wave Based Method is applied to solve appropriate mathematical problems. However, the known theory of the Wave Based Method, which can be found in the literature, so far, allowed to apply WBM only in the cases of convex domains. The author finds the criterion which allows to apply the WBM in the cases of non--convex domains. In the case of 2D problems we represent this criterion as a small proposition. With the aid of this proposition one is able to subdivide arbitrary 2D domains such that the number of subdomains is minimal, WBM may be applied in each subdomain and the geometry is not altered, e.g. via polygonal approximation. Further, the same principles are used in the case of 3D problem. However, the formulation of a similar proposition in cases of 3D problems has still to be done. Next, we show a simple procedure to solve an inhomogeneous Helmholtz equation using WBM. This procedure, however, is rather computationally expensive and can probably be improved. Several examples are also presented. We present the possibility to apply the Wave Based Technique to solve steady--state acoustic problems in the case of an unbounded 3D domain. The main principle of the classical WBM is extended to the case of an external domain. Two numerical examples are also presented. In order to apply the WBM to our problems we subdivide the computational domain into three subdomains. Therefore, on the interfaces certain coupling conditions are defined. The description of the optimization procedure, based on the principles of the shape gradient method and level set method, and the results of the optimization finalize the thesis.Jevgenijs Jegorovsdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1862Mon, 07 May 2007 18:44:35 +0200Mathematical Analysis of Macroscopic Models for Slow Dense Granular Flow
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1859
In this dissertation we present analysis of macroscopic models for slow dense granular flow. Models are derived from plasticity theory with yield condition and flow rule. Corner stone equations are conservation of mass and conservation of momentum with special constitutive law. Such models are considered in the class of generalised Newtonian fluids, where viscosity depends on the pressure and modulo of the strain-rate tensor. We showed the hyperbolic nature for the evolutionary model in 1D and ill-posed behaviour for 2D and 3D. The steady state equations are always hyperbolic. In the 2D problem we derived a prototype nonlinear backward parabolic equation for the velocity and the similar equation for the shear-rate. Analysis of derived PDE showed the finite blow up time. Blow up time depends on the initial condition. Full 2D and antiplane 3D model were investigated numerically with finite element method. For 2D model we showed the presence of boundary layers. Antiplane 3D model was investigated with the Runge Kutta Discontinuous Galerkin method with mesh addoption. Numerical results confirmed that such a numerical method can be a good choice for the simulations of the slow dense granular flow.Aleksander Grmdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1859Mon, 30 Apr 2007 10:53:36 +0200