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Fri, 03 Apr 2015 11:43:53 +0100Fri, 03 Apr 2015 11:43:53 +0100Modeling and Simulation of a Moving Rigid Body in a Rarefied Gas
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4012
We present a numerical scheme to simulate a moving rigid body with arbitrary shape suspended in a rarefied gas micro flows, in view of applications to complex computations of moving structures in micro or vacuum systems. The rarefied gas is simulated by solving the Boltzmann equation using a DSMC particle method. The motion of the rigid body is governed by the Newton-Euler equations, where the force and the torque on the rigid body is computed from the momentum transfer of the gas molecules colliding with the body. The resulting motion of the rigid body affects in turn again the gas flow in the surroundings. This means that a two-way coupling has been modeled. We validate the scheme by performing various numerical experiments in 1-, 2- and 3-dimensional computational domains. We have presented 1-dimensional actuator problem, 2-dimensional cavity driven flow problem, Brownian diffusion of a spherical particle both with translational and rotational motions, and finally thermophoresis on a spherical particles. We compare the numerical results obtained from the numerical simulations with the existing theories in each test examples. Samir Shresthadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4012Wed, 04 Mar 2015 11:43:53 +0100Testrig optimization by block loads: Remodelling of damage as Gaussian functions and their clustering method
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4003
In automotive testrigs we apply load time series to components such that the outcome is as close as possible to some reference data. The testing procedure should in general be less expensive and at the same time take less time for testing. In my thesis, I propose a testrig damage optimization problem (WSDP). This approach improves upon the testrig stress optimization problem (TSOP) used as a state of the art by industry experts.
In both (TSOP) and (WSDP), we optimize the load time series for a given testrig configuration. As the name suggests, in (TSOP) the reference data is the stress time series. The detailed behaviour of the stresses as functions of time are sometimes not the most important topic. Instead the damage potential of the stress signals are considered. Since damage is not part of the objectives in the (TSOP) the total damage computed from the optimized load time series is not optimal with respect to the reference damage. Additionally, the load time series obtained is as long as the reference stress time series and the total damage computation needs cycle counting algorithms and Goodmann corrections. The use of cycle counting algorithms makes the computation of damage from load time series non-differentiable.
To overcome the issues discussed in the previous paragraph this thesis uses block loads for the load time series. Using of block loads makes the damage differentiable with respect to the load time series. Additionally, in some special cases it is shown that damage is convex when block loads are used and no cycle counting algorithms are required. Using load time series with block loads enables us to use damage in the objective function of the (WSDP).
During every iteration of the (WSDP), we have to find the maximum total damage over all plane angles. The first attempt at solving the (WSDP) uses discretization of the interval for plane angle to find the maximum total damage at each iteration. This is shown to give unreliable results and makes maximum total damage function non-differentiable with respect to the plane angle. To overcome this, damage function for a given surface stress tensor due to a block load is remodelled by Gaussian functions. The parameters for the new model are derived.
When we model the damage by Gaussian function, the total damage is computed as a sum of Gaussian functions. The plane with the maximum damage is similar to the modes of the Gaussian Mixture Models (GMM), the difference being that the Gaussian functions used in GMM are probability density functions which is not the case in the damage approximation presented in this work. We derive conditions for a single maximum for Gaussian functions, similar to the ones given for the unimodality of GMM by Aprausheva et al. in [1].
By using the conditions for a single maximum we give a clustering algorithm that merges the Gaussian functions in the sum as clusters. Each cluster obtained through clustering is such that it has a single maximum in the absence of other Gaussian functions of the sum. The approximate point of the maximum of each cluster is used as the starting point for a fixed point equation on the original damage function to get the actual maximum total damage at each iteration.
We implement the method for the (TSOP) and the two methods (with discretization and with clustering) for (WSDP) on two example problems. The results obtained from the (WSDP) using discretization is shown to be better than the results obtained from the (TSOP). Furthermore we show that, (WSDP) using clustering approach to finding the maximum total damage, takes less number of iterations and is more reliable than using discretization.Chhitiz Buchasiadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4003Tue, 24 Feb 2015 11:08:29 +0100Combinations of Boolean Groebner Bases and SAT Solvers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3958
In this thesis, we combine Groebner basis with SAT Solver in different manners.
Both SAT solvers and Groebner basis techniques have their own strength and weakness.
Combining them could fix their weakness.
The first combination is using Groebner techniques to learn additional binary clauses for SAT solver from a selection of clauses. This combination is first proposed by Zengler and Kuechlin.
However, in our experiments, about 80 percent Groebner basis computations give no new binary clauses.
By selecting smaller and more compact input for Groebner basis computations, we can significantly
reduce the number of inefficient Groebner basis computations, learn much more binary clauses. In addition,
the new strategy can reduce the solving time of a SAT Solver in general, especially for large and hard problems.
The second combination is using all-solution SAT solver and interpolation to compute Boolean Groebner bases of Boolean elimination ideals of a given ideal. Computing Boolean Groebner basis of the given ideal is an inefficient method in case we want to eliminate most of the variables from a big system of Boolean polynomials.
Therefore, we propose a more efficient approach to handle such cases.
In this approach, the given ideal is translated to the CNF formula. Then an all-solution SAT Solver is used to find the projection of all solutions of the given ideal. Finally, an algorithm, e.g. Buchberger-Moeller Algorithm, is used to associate the reduced Groebner basis to the projection.
We also optimize the Buchberger-Moeller Algorithm for lexicographical ordering and compare it with Brickenstein's interpolation algorithm.
Finally, we combine Groebner basis and abstraction techniques to the verification of some digital designs that contain complicated data paths.
For a given design, we construct an abstract model.
Then, we reformulate it as a system of polynomials in the ring \({\mathbb Z}_{2^k}[x_1,\dots,x_n]\).
The variables are ordered in a way such that the system has already been a Groebner basis w.r.t lexicographical monomial ordering.
Finally, the normal form is employed to prove the desired properties.
To evaluate our approach, we verify the global property of a multiplier and a FIR filter using the computer algebra system Singular. The result shows that our approach is much faster than the commercial verification tool from Onespin on these benchmarks.Thanh Hung Nguyendoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3958Thu, 18 Dec 2014 14:11:19 +0100Multilevel Constructions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3942
The thesis consists of the two chapters.
The first chapter is addressed to make a deep investigation of the MLMC method. In particular we take an optimisation view at the estimate. Rather than fixing the number of discretisation points \(n_i\) to be a geometric sequence, we are trying to find an optimal set up for \(n_i\) such that for a fixed error the estimate can be computed within a minimal time.
In the second chapter we propose to enhance the MLMC estimate with the weak extrapolation technique. This technique helps to improve order of a weak convergence of a scheme and as a result reduce CC of an estimate. In particular we study high order weak extrapolation approach, which is know not be inefficient in the standard settings. However, a combination of the MLMC and the weak extrapolation yields an improvement of the MLMC.Anton Kostiukdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3942Wed, 10 Dec 2014 08:29:03 +0100Zinsoptimiertes Schuldenmanagement
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3931
Das zinsoptimierte Schuldenmanagement hat zum Ziel, eine möglichst effiziente Abwägung zwischen den erwarteten Finanzierungskosten einerseits und den Risiken für den Staatshaushalt andererseits zu finden. Um sich diesem Spannungsfeld zu nähern, schlagen wir erstmals die Brücke zwischen den Problemstellungen des Schuldenmanagements und den Methoden der zeitkontinuierlichen, dynamischen Portfoliooptimierung.
Das Schlüsselelement ist dabei eine neue Metrik zur Messung der Finanzierungskosten, die Perpetualkosten. Diese spiegeln die durchschnittlichen zukünftigen Finanzierungskosten wider und beinhalten sowohl die bereits bekannten Zinszahlungen als auch die noch unbekannten Kosten für notwendige Anschlussfinanzierungen. Daher repräsentiert die Volatilität der Perpetualkosten auch das Risiko einer bestimmten Strategie; je langfristiger eine Finanzierung ist, desto kleiner ist die Schwankungsbreite der Perpetualkosten.
Die Perpetualkosten ergeben sich als Produkt aus dem Barwert eines Schuldenportfolios und aus der vom Portfolio unabhängigen Perpetualrate. Für die Modellierung des Barwertes greifen wir auf das aus der dynamischen Portfoliooptimierung bekannte Konzept eines selbstfinanzierenden Bondportfolios zurück, das hier auf einem mehrdimensionalen affin-linearen Zinsmodell basiert. Das Wachstum des Schuldenportfolios wird dabei durch die Einbeziehung des Primärüberschusses des Staates gebremst bzw. verhindert, indem wir diesen als externen Zufluss in das selbstfinanzierende Modell aufnehmen.
Wegen der Vielfältigkeit möglicher Finanzierungsinstrumente wählen wir nicht deren Wertanteile als Kontrollvariable, sondern kontrollieren die Sensitivitäten des Portfolios gegenüber verschiedenen Zinsbewegungen. Aus optimalen Sensitivitäten können in einem nachgelagerten Schritt dann optimale Wertanteile für verschiedenste Finanzierungsinstrumente abgeleitet werden. Beispielhaft demonstrieren wir dies mittels Rolling-Horizon-Bonds unterschiedlicher Laufzeit.
Schließlich lösen wir zwei Optimierungsprobleme mit Methoden der stochastischen Kontrolltheorie. Dabei wird stets der erwartete Nutzen der Perpetualkosten maximiert. Die Nutzenfunktionen sind jeweils an das Schuldenmanagement angepasst und zeichnen sich insbesondere dadurch aus, dass höhere Kosten mit einem niedrigeren Nutzen einhergehen. Im ersten Problem betrachten wir eine Potenznutzenfunktion mit konstanter relativer Risikoaversion, im zweiten wählen wir eine Nutzenfunktion, welche die Einhaltung einer vorgegebenen Schulden- bzw. Kostenobergrenze garantiert.Christoph Petersdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3931Mon, 24 Nov 2014 09:09:39 +0100Variance Reduction Procedures for Market Risk Estimation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3885
Monte Carlo simulation is one of the commonly used methods for risk estimation on financial markets, especially for option portfolios, where any analytical approximation is usually too inaccurate. However, the usually high computational effort for complex portfolios with a large number of underlying assets motivates the application of variance reduction procedures. Variance reduction for estimating the probability of high portfolio losses has been extensively studied by Glasserman et al. A great variance reduction is achieved by applying an exponential twisting importance sampling algorithm together with stratification. The popular and much faster Delta-Gamma approximation replaces the portfolio loss function in order to guide the choice of the importance sampling density and it plays the role of the stratification variable. The main disadvantage of the proposed algorithm is that it is derived only in the case of Gaussian and some heavy-tailed changes in risk factors.
Hence, our main goal is to keep the main advantage of the Monte Carlo simulation, namely its ability to perform a simulation under alternative assumptions on the distribution of the changes in risk factors, also in the variance reduction algorithms. Step by step, we construct new variance reduction techniques for estimating the probability of high portfolio losses. They are based on the idea of the Cross-Entropy importance sampling procedure. More precisely, the importance sampling density is chosen as the closest one to the optimal importance sampling density (zero variance estimator) out of some parametric family of densities with respect to Kullback - Leibler cross-entropy. Our algorithms are based on the special choices of the parametric family and can now use any approximation of the portfolio loss function. A special stratification is developed, so that any approximation of the portfolio loss function under any assumption of the distribution of the risk factors can be used. The constructed algorithms can easily be applied for any distribution of risk factors, no matter if light- or heavy-tailed. The numerical study exhibits a greater variance reduction than of the algorithm from Glasserman et al. The use of a better approximation may improve the performance of our algorithms significantly, as it is shown in the numerical study.
The literature on the estimation of the popular market risk measures, namely VaR and CVaR, often refers to the algorithms for estimating the probability of high portfolio losses, describing the corresponding transition process only briefly. Hence, we give a consecutive discussion of this problem. Results necessary to construct confidence intervals for both measures under the mentioned variance reduction procedures are also given.
Mykhailo Pupashenkodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3885Wed, 01 Oct 2014 09:47:40 +0200New aspects of optimal investment in continuous time
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3867
This thesis focuses on dealing with some new aspects of continuous time portfolio optimization by using the stochastic control method.
First, we extend the Busch-Korn-Seifried model for a large investor by using the Vasicek model for the short rate, and that problem is solved explicitly for two types of intensity functions.
Next, we justify the existence of the constant proportion portfolio insurance (CPPI) strategy in a framework containing a stochastic short rate and a Markov switching parameter. The effect of Vasicek short rate on the CPPI strategy has been studied by Horsky (2012). This part of the thesis extends his research by including a Markov switching parameter, and the generalization is based on the B\"{a}uerle-Rieder investment problem. The explicit solutions are obtained for the portfolio problem without the Money Market Account as well as the portfolio problem with the Money Market Account.
Finally, we apply the method used in Busch-Korn-Seifried investment problem to explicitly solve the portfolio optimization with a stochastic benchmark.Nhat Thu Trandoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3867Tue, 09 Sep 2014 12:50:40 +0200Edgeworth Expansions for Binomial Trees
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3861
In the theory of option pricing one is usually concerned with evaluating expectations under the risk-neutral measure in a continuous-time model.
However, very often these values cannot be calculated explicitly and numerical methods need to be applied to approximate the desired quantity. Monte Carlo simulations, numerical methods for PDEs and the lattice approach are the methods typically employed. In this thesis we consider the latter approach, with the main focus on binomial trees.
The binomial method is based on the concept of weak convergence. The discrete-time model is constructed so as to ensure convergence in distribution to the continuous process. This means that the expectations calculated in the binomial tree can be used as approximations of the option prices in the continuous model. The binomial method is easy to implement and can be adapted to options with different types of payout structures, including American options. This makes the approach very appealing. However, the problem is that in many cases, the convergence of the method is slow and highly irregular, and even a fine discretization does not guarantee accurate price approximations. Therefore, ways of improving the convergence properties are required.
We apply Edgeworth expansions to study the convergence behavior of the lattice approach. We propose a general framework, that allows to obtain asymptotic expansion for both multinomial and multidimensional trees. This information is then used to construct advanced models with superior convergence properties.
In binomial models we usually deal with triangular arrays of lattice random vectors. In this case the available results on Edgeworth expansions for lattices are not directly applicable. Therefore, we first present Edgeworth expansions, which are also valid for the binomial tree setting. We then apply these result to the one-dimensional and multidimensional Black-Scholes models. We obtain third order expansions
for general binomial and trinomial trees in the 1D setting, and construct advanced models for digital, vanilla and barrier options. Second order expansion are provided for the standard 2D binomial trees and advanced models are constructed for the two-asset digital and the two-asset correlation options. We also present advanced binomial models for a multidimensional setting.Alona Bockdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3861Tue, 02 Sep 2014 09:07:50 +0200Portfoliooptimierung im Binomialmodell
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3849
Die Dissertation "Portfoliooptimierung im Binomialmodell" befasst sich mit der Frage, inwieweit
das Problem der optimalen Portfolioauswahl im Binomialmodell lösbar ist bzw. inwieweit
die Ergebnisse auf das stetige Modell übertragbar sind. Dabei werden neben dem
klassischen Modell ohne Kosten und ohne Veränderung der Marktsituation auch Modellerweiterungen
untersucht.Henriette Krönerdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3849Thu, 14 Aug 2014 08:15:18 +0200Algorithms in SINGULAR: Parallelization, Syzygies, and Singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3840
This thesis, whose subject is located in the field of algorithmic commutative algebra and algebraic geometry, consists of three parts.
The first part is devoted to parallelization, a technique which allows us to take advantage of the computational power of modern multicore processors. First, we present parallel algorithms for the normalization of a reduced affine algebra A over a perfect field. Starting from the algorithm of Greuel, Laplagne, and Seelisch, we propose two approaches. For the local-to-global approach, we stratify the singular locus Sing(A) of A, compute the normalization locally at each stratum and finally reconstruct the normalization of A from the local results. For the second approach, we apply modular methods to both the global and the local-to-global normalization algorithm.
Second, we propose a parallel version of the algorithm of Gianni, Trager, and Zacharias for primary decomposition. For the parallelization of this algorithm, we use modular methods for the computationally hardest steps, such as for the computation of the associated prime ideals in the zero-dimensional case and for the standard bases computations. We then apply an innovative fast method to verify that the result is indeed a primary decomposition of the input ideal. This allows us to skip the verification step at each of the intermediate modular computations.
The proposed parallel algorithms are implemented in the open-source computer algebra system SINGULAR. The implementation is based on SINGULAR's new parallel framework which has been developed as part of this thesis and which is specifically designed for applications in mathematical research.
In the second part, we propose new algorithms for the computation of syzygies, based on an in-depth analysis of Schreyer's algorithm. Here, the main ideas are that we may leave out so-called "lower order terms" which do not contribute to the result of the algorithm, that we do not need to order the terms of certain module elements which occur at intermediate steps, and that some partial results can be cached and reused.
Finally, the third part deals with the algorithmic classification of singularities over the real numbers. First, we present a real version of the Splitting Lemma and, based on the classification theorems of Arnold, algorithms for the classification of the simple real singularities. In addition to the algorithms, we also provide insights into how real and complex singularities are related geometrically. Second, we explicitly describe the structure of the equivalence classes of the unimodal real singularities of corank 2. We prove that the equivalences are given by automorphisms of a certain shape. Based on this theorem, we explain in detail how the structure of the equivalence classes can be computed using SINGULAR and present the results in concise form. The probably most surprising outcome is that the real singularity type \(J_{10}^-\) is actually redundant.Andreas Steenpassdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3840Wed, 30 Jul 2014 10:37:00 +0200Algorithmic aspects of tropical intersection theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3823
In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.
In the second part we apply the algorithmic toolkit developed in the first part to the study of tropical double Hurwitz cycles. Hurwitz cycles are a higher-dimensional generalization of Hurwitz numbers, which count covers of \(\mathbb{P}^1\) by smooth curves of a given genus with a certain fixed ramification behaviour. Double Hurwitz numbers provide a strong connection between various mathematical disciplines, including algebraic geometry, representation theory and combinatorics. The tropical cycles have a rather complex combinatorial nature, so it is very difficult to study them purely "by hand". Being able to compute examples has been very helpful
in coming up with theoretical results. Our main result states that all marked and unmarked Hurwitz cycles are connected in codimension one and that for a generic choice of simple ramification points the marked cycle is a multiple of an irreducible cycle. In addition we provide computational examples to show that this is the strongest possible statement.Simon Hampedoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3823Thu, 03 Jul 2014 09:26:06 +0200Efficient Algorithms for Flow Simulation related to Nuclear Reactor Safety
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3826
Safety analysis is of ultimate importance for operating Nuclear Power Plants (NPP). The overall
modeling and simulation of physical and chemical processes occuring in the course of an accident
is an interdisciplinary problem and has origins in fluid dynamics, numerical analysis, reactor tech-
nology and computer programming. The aim of the study is therefore to create the foundations
of a multi-dimensional non-isothermal fluid model for a NPP containment and software tool based
on it. The numerical simulations allow to analyze and predict the behavior of NPP systems under
different working and accident conditions, and to develop proper action plans for minimizing the
risks of accidents, and/or minimizing the consequences of possible accidents. A very large number
of scenarios have to be simulated, and at the same time acceptable accuracy for the critical param-
eters, such as radioactive pollution, temperature, etc., have to be achieved. The existing software
tools are either too slow, or not accurate enough. This thesis deals with developing customized al-
gorithm and software tools for simulation of isothermal and non-isothermal flows in a containment
pool of NPP. Requirements to such a software are formulated, and proper algorithms are presented.
The goal of the work is to achieve a balance between accuracy and speed of calculation, and to
develop customized algorithm for this special case. Different discretization and solution approaches
are studied and those which correspond best to the formulated goal are selected, adjusted, and when
possible, analysed. Fast directional splitting algorithm for Navier-Stokes equations in complicated
geometries, in presence of solid and porous obstales, is in the core of the algorithm. Developing
suitable pre-processor and customized domain decomposition algorithms are essential part of the
overall algorithm amd software. Results from numerical simulations in test geometries and in real
geometries are presented and discussed.Tatiana Gornakdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3826Thu, 03 Jul 2014 08:29:14 +0200Efficient algorithms for Asymmetric Flow Field Flow Fractionation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3811
This thesis is devoted to the modeling and simulation of Asymmetric Flow Field Flow Fractionation, which is a technique for separating particles of submicron scale. This process is a part of large family of Field Flow Fractionation techniques and has a very broad range of industrial applications, e. g. in microbiology, chemistry, pharmaceutics, environmental analysis.
Mathematical modeling is crucial for this process, as due to the own nature of the process, lab ex- periments are difficult and expensive to perform. On the other hand, there are several challenges for the mathematical modeling: huge dominance (up to 106 times) of the flow over the diffusion, highly stretched geometry of the device. This work is devoted to developing fast and efficient algorithms, which take into the account the challenges, posed by the application, and provide reliable approximations for the quantities of interest.
We present a new Multilevel Monte Carlo method for estimating the distribution functions on a compact interval, which are of the main interest for Asymmetric Flow Field Flow Fractionation. Error estimates for this method in terms of computational cost are also derived.
We optimize the flow control at the Focusing stage under the given constraints on the flow and present an important ingredients for the further optimization, such as two-grid Reduced Basis method, specially adapted for the Finite Volume discretization approach.Tigran Nagapetyandoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3811Wed, 04 Jun 2014 09:42:28 +0200Pedestrian Flow Models
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3803
There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries. Raghavender Etikyaladoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3803Fri, 16 May 2014 12:02:59 +0200Intersection theory with applications to the computation of Gromov-Witten invariants
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3750
This thesis is devoted to the computational aspects of intersection theory and enumerative geometry. The first results are a Sage package Schubert3 and a Singular library schubert.lib which both provide the key functionality necessary for computations in intersection theory and enumerative geometry. In particular, we describe an alternative method for computations in Schubert calculus via equivariant intersection theory. More concretely, we propose an explicit formula for computing the degree of Fano schemes of linear subspaces on hypersurfaces. As a special case, we also obtain an explicit formula for computing the number of linear subspaces on a general hypersurface when this number is finite. This leads to a much better performance than classical Schubert calculus.
Another result of this thesis is related to the computation of Gromov-Witten invariants. The most powerful method for computing Gromov-Witten invariants is the localization of moduli spaces of stable maps. This method was introduced by Kontsevich in 1995. It allows us to compute Gromov-Witten invariants via Bott's formula. As an insightful application, we computed the numbers of rational curves on general complete intersection Calabi-Yau threefolds in projective spaces up to degree six. The results are all in agreement with predictions made from mirror symmetry.
Hiep Dangdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3750Fri, 14 Mar 2014 08:54:11 +0100On the distribution of eigenspaces in classical groups over finite rings and the Cohen-Lenstra heuristic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3732
In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other. Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups. Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\). Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet. In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case. Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements.Michael Adamdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3732Tue, 18 Feb 2014 13:17:02 +0100Multi-Class Image Segmentation via Convex and Biconvex Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3656
This thesis is divided into two parts. Both cope with multi-class image segmentation and utilize
non-smooth optimization algorithms.
The topic of the first part, namely unsupervised segmentation, is the application of clustering
to image pixels. Therefore, we start with an introduction of the biconvex center-based clustering
algorithms c-means and fuzzy c-means, where c denotes the number of classes. We show that
fuzzy c-means can be seen as an approximation of c-means in terms of power means.
Since noise is omnipresent in our image data, these simple clustering models are not suitable
for its segmentation. To this end, we introduce a general and finite dimensional segmentation
model that consists of a data term stemming from the aforementioned clustering models plus a
continuous regularization term. We tackle this optimization model via an alternating minimiza-
tion approach called regularized c-centers (RcC). Thereby, we fix the centers and optimize the
segment membership of the pixels and vice versa. In this general setting, we prove convergence
in the sense of set-valued algorithms using Zangwill’s Theory [172].
Further, we present a segmentation model with a total variation regularizer. While updating
the cluster centers is straightforward for fixed segment memberships of the pixels, updating the
segment membership can be solved iteratively via non-smooth, convex optimization. Thereby,
we do not iterate a convex optimization algorithm until convergence. Instead, we stop as soon as
we have a certain amount of decrease in the objective functional to increase the efficiency. This
algorithm is a particular implementation of RcC providing also the corresponding convergence
theory. Moreover, we show the good performance of our method in various examples such as
simulated 2d images of brain tissue and 3d volumes of two materials, namely a multi-filament
composite superconductor and a carbon fiber reinforced silicon carbide ceramics. Thereby, we
exploit the property of the latter material that two components have no common boundary in
our adapted model.
The second part of the thesis is concerned with supervised segmentation. We leave the area
of center based models and investigate convex approaches related to graph p-Laplacians and
reproducing kernel Hilbert spaces (RKHSs). We study the effect of different weights used to
construct the graph. In practical experiments we show on the one hand image types that
are better segmented by the p-Laplacian model and on the other hand images that are better
segmented by the RKHS-based approach. This is due to the fact that the p-Laplacian approach
provides smoother results, while the RKHS approach provides often more accurate and detailed
segmentations. Finally, we propose a novel combination of both approaches to benefit from the
advantages of both models and study the performance on challenging medical image data.
Behrang Shafeidoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3656Mon, 25 Nov 2013 08:30:52 +0100Curve interactions in R^2: An analytical and stochastical approach
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3646
In the last few years a lot of work has been done in the investigation of Brownian motion with point interaction(s) in one and higher dimensions. Roughly speaking a Brownian motion with point interaction is nothing else than a Brownian motion whose generator is disturbed by a measure supported in just one point.
The purpose of the present work is the introducing of curve interactions of the two dimensional Brownian motion for a closed curve \(\mathcal{C}\). We will understand a curve interaction as a self-adjoint extension of the restriction of the Laplacian to the set of infinitely often continuously differentiable functions with compact support in \(\mathbb{R}^{2}\) which are constantly 0 at the closed curve. We will give a full description of all these self-adjoint extensions.
In the second chapter we will prove a generalization of Tanaka's formula to \(\mathbb{R}^{2}\). We define \(g\) to be a so-called harmonic single layer with continuous layer function \(\eta\) in \(\mathbb{R}^{2}\). For such a function \(g\) we prove
\begin{align}
g\left(B_{t}\right)=g\left(B_{0}\right)+\int\limits_{0}^{t}{\nabla g\left(B_{s}\right)\mathrm{d}B_{s}}+\int\limits_{0}^{t}\eta\left(B_{s}\right)\mathrm{d}L\left(s,\mathcal{C}\right)
\end{align}
where \(B_{t}\) is just the usual Brownian motion in \(\mathbb{R}^{2}\) and \(L\left(t,\mathcal{C}\right)\) is the connected unique local time process of \(B_{t}\) on the closed curve \(\mathcal{C}\).
We will use the generalized Tanaka formula in the following chapter to construct classes of processes related to curve interactions. In a first step we get the generalization of point interactions in a second step we get processes which behaves like a Brownian motion in the complement of \(\mathcal{C}\) and has an additional movement along the curve in the time- scale of \(L\left(t,\mathcal{C}\right)\). Such processes do not exist in the one point case since there we cannot move when the Brownian motion is in the point.
By establishing an approximation of a curve interaction by operators of the form Laplacian \(+V_{n}\) with "nice" potentials \(V_{n}\) we are able to deduce the existence of superprocesses related to curve interactions.
The last step is to give an approximation of these superprocesses by a sytem of branching particles. This approximation gives a better understanding of the related mass creation. Benedikt Heinrichdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3646Wed, 13 Nov 2013 15:30:37 +0100Time Domain Full Waveform Inversion Using ADI Modeling
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3599
Constructing accurate earth models from seismic data is a challenging task. Traditional methods rely on ray based approximations of the wave equation and reach their limit in geologically complex areas. Full waveform inversion (FWI) on the other side seeks to minimize the misﬁt between modeled and observed data without such approximation.
While superior in accuracy, FWI uses a gradient based iterative scheme that makes it also very computationally expensive. In this thesis we analyse and test an Alternating Direction Implicit (ADI) scheme in order to reduce the costs of the two dimensional time domain algorithm for solving the acoustic wave equation. The ADI scheme can be seen as an intermediate between explicit and implicit ﬁnite diﬀerence modeling schemes. Compared to full implicit schemes the ADI scheme only requires the solution of much smaller matrices and is thus less computationally demanding. Using ADI we can handle coarser discretization compared to an explicit method. Although order of convergence and CFL conditions for the examined explicit method and ADI scheme are comparable, we observe that the ADI scheme is less prone to dispersion. Furhter, our algorithm is eﬃciently parallelized with vectorization and threading techniques. In a numerical comparison, we can demonstrate a runtime advantage of the ADI scheme over an explicit method of the same accuracy.
With the modeling in place, we test and compare several inverse schemes in the second part of the thesis. With the goal of avoiding local minima and improving speed of convergence, we use diﬀerent minimization functions and hierarchical approaches. In several tests, we demonstrate superior results of the L1 norm compared to the L2 norm – especially in the presence of noise. Furthermore we show positive eﬀects for applying three diﬀerent multiscale approaches to the inverse problem. These methods focus on low frequency, early recording, or far oﬀset during early iterations of the minimization and then proceed iteratively towards the full problem. We achieve best results with the frequency based multiscale scheme, for which we also provide a heuristical method of choosing iteratively increasing frequency bands.
Finally, we demonstrate the eﬀectiveness of the diﬀerent methods ﬁrst on the Marmousi model and then on an extract of the 2004 BP model, where we are able to recover both high contrast top salt structures and lower contrast inclusions accurately.Bernd Klimmdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3599Wed, 28 Aug 2013 08:50:38 +0200Trading to stops: The investigation of state-based stopping rules
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3578
The use of trading stops is a common practice in financial markets for a variety of reasons: it provides a simple way to control losses on a given trade, while also ensuring that profit-taking is not deferred indefinitely; and it allows opportunities to consider reallocating resources to other investments. In this thesis, it is explained why the use of stops may be desirable in certain cases.
This is done by proposing a simple objective to be optimized. Some simple and commonly-used rules for the placing and use of stops are investigated; consisting of fixed or moving barriers, with fixed transaction costs. It is shown how to identify optimal levels at which to set stops, and the performances of different rules and strategies are compared. Thereby, uncertainty and altering of the drift parameter of the investment are incorporated.Nora Imkellerdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3578Tue, 13 Aug 2013 08:11:31 +0200Multivariate Polynomial Interpolation and the Lifting Scheme with an Application to Scattered Data Approximation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3566
This thesis deals with generalized inverses, multivariate polynomial interpolation and approximation of scattered data. Moreover, it covers the lifting scheme, which basically links the aforementioned topics. For instance, determining filters for the lifting scheme is connected to multivariate polynomial interpolation. More precisely, sets of interpolation sites are required that can be interpolated by a unique polynomial of a certain degree. In this thesis a new class of such sets is introduced and elements from this class are used to construct new and computationally more efficient filters for the lifting scheme.
Furthermore, a method to approximate multidimensional scattered data is introduced which is based on the lifting scheme. A major task in this method is to solve an ordinary linear least squares problem which possesses a special structure. Exploiting this structure yields better approximations and therefore this particular least squares problem is analyzed in detail. This leads to a characterization of special generalized inverses with partially prescribed image spaces.Dominik Stahldoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3566Wed, 10 Jul 2013 11:14:17 +0200Overlapping Domain Decomposition Preconditioners for Multi-Phase Elastic Composites
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3565
The application behind the subject of this thesis are multiscale simulations on highly heterogeneous particle-reinforced composites with large jumps in their material coefficients. Such simulations are used, e.g., for the prediction of elastic properties. As the underlying microstructures have very complex geometries, a discretization by means of finite elements typically involves very fine resolved meshes. The latter results in discretized linear systems of more than \(10^8\) unknowns which need to be solved efficiently. However, the variation of the material coefficients even on very small scales reveals the failure of most available methods when solving the arising linear systems. While for scalar elliptic problems of multiscale character, robust domain decomposition methods are developed, their extension and application to 3D elasticity problems needs to be further established.
The focus of the thesis lies in the development and analysis of robust overlapping domain decomposition methods for multiscale problems in linear elasticity. The method combines corrections on local subdomains with a global correction on a coarser grid. As the robustness of the overall method is mainly determined by how well small scale features of the solution can be captured on the coarser grid levels, robust multiscale coarsening strategies need to be developed which properly transfer information between fine and coarse grids.
We carry out a detailed and novel analysis of two-level overlapping domain decomposition methods for the elasticity problems. The study also provides a concept for the construction of multiscale coarsening strategies to robustly solve the discretized linear systems, i.e. with iteration numbers independent of variations in the Young's modulus and the Poisson ratio of the underlying composite. The theory also captures anisotropic elasticity problems and allows applications to multi-phase elastic materials with non-isotropic constituents in two and three spatial dimensions.
Moreover, we develop and construct new multiscale coarsening strategies and show why they should be preferred over standard ones on several model problems. In a parallel implementation (MPI) of the developed methods, we present applications to real composites and robustly solve discretized systems of more than \(200\) million unknowns.Marco Buckdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3565Tue, 09 Jul 2013 11:22:50 +0200Factorization of multivariate polynomials
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3555
Factorization of multivariate polynomials is a cornerstone of many applications in computer algebra. To compute it, one uses an algorithm by Zassenhaus who used it in 1969 to factorize univariate polynomials over \(\mathbb{Z}\). Later Musser generalized it to the multivariate case. Subsequently, the algorithm was refined and improved.
In this work every step of the algorithm is described as well as the problems that arise in these steps.
In doing so, we restrict to the coefficient domains \(\mathbb{F}_{q}\), \(\mathbb{Z}\), and \(\mathbb{Q}(\alpha)\) while focussing on a fast implementation. The author has implemented almost all algorithms mentioned in this work in the C++ library factory which is part of the computer algebra system Singular.
Besides, a new bound on the coefficients of a factor of a multivariate polynomial over \(\mathbb{Q}(\alpha)\) is proven which does not require \(\alpha\) to be an algebraic integer. This bound is used to compute Hensel lifting and recombination of factors in a modular fashion. Furthermore, several sub-steps are improved.
Finally, an overview on the capability of the implementation is given which includes benchmark examples as well as random generated input which is supposed to give an impression of the average performance.Martin Mok-Don Leedoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3555Thu, 27 Jun 2013 15:12:00 +0200Moduli spaces of rational tropical stable maps into smooth tropical varieties
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3528
This thesis is concerned with tropical moduli spaces, which are an important tool in tropical enumerative geometry. The main result is a construction of tropical moduli spaces of rational tropical covers of smooth tropical curves and of tropical lines in smooth tropical surfaces. The construction of a moduli space of tropical curves in a smooth tropical variety is reduced to the case of smooth fans. Furthermore, we point out relations to intersection theory on suitable moduli spaces on algebraic curves.Dennis Ochsedoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3528Wed, 05 Jun 2013 16:14:25 +0200Fibre Processes and their Applications
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3515
The main purpose of the study was to improve the physical properties of the modelling of compressed materials, especially fibrous materials. Fibrous materials are finding increasing application in the industries. And most of the materials are compressed for different applications. For such situation, we are interested in how the fibre arranged, e.g. with which distribution. For given materials it is possible to obtain a three-dimensional image via micro computed tomography. Since some physical parameters, e.g. the fibre lengths or the directions for points in the fibre, can be checked under some other methods from image, it is beneficial to improve the physical properties by changing the parameters in the image.
In this thesis, we present a new maximum-likelihood approach for the estimation of parameters of a parametric distribution on the unit sphere, which is various as some well known distributions, e.g. the von-Mises Fisher distribution or the Watson distribution, and for some models better fit. The consistency and asymptotic normality of the maximum-likelihood estimator are proven. As the second main part of this thesis, a general model of mixtures of these distributions on a hypersphere is discussed. We derive numerical approximations of the parameters in an Expectation Maximization setting. Furthermore we introduce a non-parametric estimation of the EM algorithm for the mixture model. Finally, we present some applications to the statistical analysis of fibre composites. Na Zhangdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3515Mon, 27 May 2013 09:32:01 +0200