KLUEDO RSS FeedNeueste Dokumente / Latest documents
https://kluedo.ub.uni-kl.de/index/index/
Mon, 31 Mar 2014 07:47:17 +0200Mon, 31 Mar 2014 07:47:17 +0200Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3770
The hypervolume subset selection problem consists of finding a subset, with a given cardinality, of a nondominated set of points that maximizes the hypervolume indicator. This problem arises in selection procedures of population-based heuristics for multiobjective optimization, and for which practically efficient algorithms are strongly required. In this article, we provide two new formulations for the two-dimensional variant of this problem.
The first is an integer programming formulation that can be solved by solving its linear relaxation. The second formulation is a \(k\)-link shortest path formulation on a special digraph with Monge property that can be solved by dynamic programming in \(\mathcal{O}(n^2)\) time complexity. This improves upon the existing result of \(O(n^3)\) in Bader.Tobias Kuhn; Carlos M. Fonseca; Luís Paquete; Stefan Ruzika; José Rui Figueirapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3770Mon, 31 Mar 2014 07:47:17 +0200Edgeworth expansions for lattice triangular arrays
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3765
Edgeworth expansions have been introduced as a generalization of the central limit theorem and allow to investigate the convergence properties of sums of i.i.d. random variables. We consider triangular arrays of lattice random vectors and obtain a valid Edgeworth expansion for this case. The presented results can be used, for example, to study the convergence behavior of lattice models.Alona Bockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3765Wed, 26 Mar 2014 12:14:40 +0100A multiscale model for pH-tactic invasion with time-varying carrying capacities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3757
We propose a model for acid-mediated tumor invasion involving two different scales: the microscopic one, for the dynamics of intracellular protons and their exchange with their extracellular counterparts, and the macroscopic scale of interactions between tumor cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behavior of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following [2,31] we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumor cells in response to the effects of acidity is considered as well. The global well posedness of the resulting multiscale model is proved with a regularization and fixed point argument. Numerical simulations are performed in order to illustrate the behavior of the model.Christina Surulescu; Gülnihal Meral; Christian Stinnerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3757Tue, 18 Mar 2014 14:06:43 +0100Intersection 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 +0100Isogeometric analysis of nonlinear Euler-Bernoulli beam vibrations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3718
In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle.
As application, the straight nonlinear Euler-Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.Oliver Weeger; Utz Wever; Bernd Simeonpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3718Mon, 10 Feb 2014 11:22:39 +0100An Inexact Interior Point Method for the Large-Scale Simulation of Granular Material
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3694
Non-smooth contact dynamics provides an increasingly popular simulation framework for granular material. In contrast to classical discrete element methods, this approach is stable for arbitrary time steps and produces visually acceptable results in very short computing time. Yet when it comes to the prediction of draft forces, non-smooth contact dynamics is typically not accurate enough. We therefore propose to combine the method class with an interior point algorithm for higher accuracy. Our specific algorithm is based on so-called Jordan algebras and exploits the relation to symmetric cones in order to tackle the conical constraints that are intrinsic to frictional contact problems. In every interior point iteration a linear system has to be solved. We analyze how the interior point method behaves when it is combined with Krylov subspace solvers and incomplete factorizations. We show that efficient preconditioners and efficient linear solvers are essential for the method to be applicable to large-scale problems. Using BiCGstab as a linear solver and incomplete Cholesky factorizations, we substantially improve the accuracy in comparison to the projected Gauss-Jacobi solver.Jan Kleinert; Bernd Simeon; Martin Obermayrpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3694Wed, 29 Jan 2014 10:40:35 +0100Monitoring time series based on estimating functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3693
A large class of estimators including maximum likelihood, least squares and M-estimators are based on estimating functions. In sequential change point detection related monitoring functions can be used to monitor new incoming observations based on an initial estimator, which is computationally efficient because possible numeric optimization is restricted to the initial estimation. In this work, we give general regularity conditions under which we derive the asymptotic null behavior of the corresponding tests in addition to their behavior under alternatives, where conditions become particularly simple for sufficiently smooth estimating and monitoring functions. These regularity conditions unify and even extend a large amount of existing procedures in the literature, while they also allow us to derive monitoring schemes in time series that have not yet been considered in the literature including non-linear autoregressive time series and certain count time series such as binary or Poisson autoregressive models. We do not assume that the estimating and monitoring function are equal or even of the same dimension, allowing for example to combine a non-robust but more precise initial estimator with a robust monitoring scheme. Some simulations and data examples illustrate the usefulness of the described procedures.Claudia Kirch; Joseph Tadjuidje Kamgaingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3693Wed, 29 Jan 2014 10:20:27 +0100ADER schemes and high order coupling on networks of hyperbolic conservation laws
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3674
In this article we present a method to extend high order finite volume schemes
to networks of hyperbolic conservation laws with algebraic coupling conditions. This method is based on an ADER approach in time to solve the
generalized Riemann problem at the junction. Additionally to the high order accuracy, this approach maintains an exact conservation of quantities if
stated by the coupling conditions. Several numerical examples confirm the
benefits of a high order coupling procedure for high order accuracy and stable
shock capturing.
Raul Borsche; Jochen Kallpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3674Mon, 16 Dec 2013 11:49:28 +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 +0100Geometric Ergodicity of Binary Autoregressive Models with Exogenous Variables
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3647
In this paper we introduce a binary autoregressive model. In contrast to the typical autoregression framework, we allow the conditional distribution of the observed process to depend on past values of the time series and some exogenous variables. Such processes have
potential applications in econometrics, medicine and environmental sciences. In this
paper, we establish stationarity and geometric ergodicity of these
processes under suitable conditions on the parameters of the model. Such properties are
important for understanding the stability properties of the model as well as for deriving the
asymptotic behavior of the parameter estimators.Claudia Kirch; Joseph Tadjuidje Kamgaingworkingpaperhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3647Wed, 13 Nov 2013 15:43:19 +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 +0100A Two-Stage Robustness Approach to Evacuation Planning with Buses
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3623
We consider the problem of scheduling a bus fleet to evacuate persons from an endangered region. As most of the planning data is subject to uncertainty, we develop a two-stage bicriteria robust formulation, which considers both the evacuation time, and the vulnerability of the schedule to changing evacuation circumstances.
As the resulting integer program is too large to solve it directly using an off-the-shelf solver, we develop an iterative algorithm that successively adds new scenarios to the currently considered subproblem. In computational experiments, we show that this approach is fast enough to deal with an instance modeling an evacuation case within the city of Kaiserslautern, Germany.Marc Goerigk; Kaouthar Deghdak; Vincent T'Kindtpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3623Wed, 09 Oct 2013 15:22:46 +0200Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3621
We prove the global existence, along with some basic boundedness properties, of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. The model has been proposed in [22] and accounts on the macroscopic level for the evolution of cell and tissue densities, along with the concentration of a chemoattractant, while on the subcellular level it involves the binding of integrins to soluble and insoluble components of the peritumoral region. The connection between the two scales is realized with the aid of a contractivity function characterizing the ability of the tumor cells to adapt their motility behavior
to their subcellular dynamics.
The resulting system, consisting of three partial and three ordinary differential equations including a temporal delay, in particular involves chemotactic and haptotactic cross-diffusion. In order to overcome technical obstacles stemming from the corresponding highest-order interaction terms, we base our analysis on a certain functional, inter alia involving the cell and tissue densities in the diffusion and haptotaxis terms respectively, which is shown to enjoy a quasi-dissipative property. This will be used as a starting point for the derivation of a series of integral estimates finally allowing for the construction of a generalized solution as the limit of solutions to suitably regularized problems.Christian Stinner; Christina Surulescu; Michael Winklerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3621Wed, 09 Oct 2013 09:02:12 +0200Time 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 +0200Linearized Riesz Transform and Quasi-Monogenic Shearlets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3596
The only quadrature operator of order two on \(L_2 (\mathbb{R}^2)\) which covaries with orthogonal
transforms, in particular rotations is (up to the sign) the Riesz transform. This property
was used for the construction of monogenic wavelets and curvelets. Recently, shearlets
were applied for various signal processing tasks. Unfortunately, the Riesz transform does
not correspond with the shear operation. In this paper we propose a novel quadrature operator called linearized Riesz transform which is related to the shear operator. We prove
properties of this transform and analyze its performance versus the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding
optical filters. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets and prove that they form a tight frame. Numerical experiments show
the good fit of the directional information given by the shearlets and the orientation ob-
tained from the quasi-monogenic shearlet coefficients. Finally we provide experiments on
the directional analysis of textures using our quasi-monogenic shearlets.Sören Häuser; Bettina Heise; Gabriele Steidlpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3596Thu, 22 Aug 2013 11:00:31 +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 +0200On a multiscale model involving cell contractivity and its effects on tumor invasion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3571
Cancer cell migration is an essential feature in the process of tumor spread and establishing of metastasis. It characterizes the invasion observed on the level of the cell population, but it is also tightly connected to the events taking place on the subcellular level. These are conditioning the motile and proliferative behavior of the cells, but are also influenced by it. In this work we propose a multiscale model linking these two levels and aiming to assess their interdependence. On the subcellular, microscopic scale it accounts for integrin binding to soluble and insoluble components present in the peritumoral environment, which is seen as the onset of biochemical events leading to changes in the cell's ability to contract and modify its shape. On the macroscale of the cell population this leads to modifications in the diffusion and haptotaxis performed by the tumor cells and implicitly to changes in the tumor environment. We prove the (local) well posedness of our model and perform numerical simulations in order to illustrate the model predictions.Gülnihal Meral; Christian Stinner; Christina Surulescupreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3571Thu, 25 Jul 2013 15:04:23 +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 +0200On the Generality of the Greedy Algorithm for Solving Matroid Base Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3535
It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions. Lara Turner; Matthias Ehrgott; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3535Wed, 19 Jun 2013 08:27:31 +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 +0200A Branch-Cut-and-Price Approach to the Bus Evacuation Problem with Integrated Collection Point and Shelter Decisions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3527
We consider the problem of evacuating a region with the help of buses. For a given set of possible collection points where evacuees gather, and possible shelter locations where evacuees are brought to, we need to determine both collection points and shelters we would like to use, and bus routes that evacuate the region in minimum time.
We model this integrated problem using an integer linear program, and present a branch-cut-and-price algorithm that generates bus tours in its pricing step. In computational experiments we show that our approach is able to solve instances of realistic size in sufficient time for practical application, and considerably outperforms the usage of a generic ILP solver. Marc Goerigk; Bob Grün; Philipp Heßlerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3527Wed, 05 Jun 2013 08:32:35 +0200Coupling traffic flow networks to pedestrian motion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3529
In the present paper scalar macroscopic models for traffic and pedestrian flows are coupled and the resulting system is investigated numerically. For the traffic flow the classical
Lighthill-Whitham model on a network of roads and for the pedestrian flow the Hughes
model are used. These models are coupled via terms in the fundamental diagrams mod-
eling an influence of the traffic and pedestrian flow on the maximal velocities of the
corresponding models. Several physical situations, where pedestrians and cars interact,
are investigated.
Raul Borsche; Anne Meurerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3529Wed, 05 Jun 2013 08:17:08 +0200