### Refine

#### Year of publication

- 2005 (36) (remove)

#### Document Type

- Doctoral Thesis (15)
- Preprint (14)
- Diploma Thesis (4)
- Report (3)

#### Language

- English (36) (remove)

#### Has Fulltext

- yes (36) (remove)

#### Keywords

- Mehrskalenanalyse (3)
- Wavelet (3)
- Approximation (2)
- Computeralgebra (2)
- Elastoplastizität (2)
- Galerkin-Methode (2)
- Geometric Ergodicity (2)
- Jiang's model (2)
- Jiang-Modell (2)
- Navier-Stokes-Gleichung (2)
- Poisson-Gleichung (2)
- Randwertproblem / Schiefe Ableitung (2)
- Sobolev-Raum (2)
- Ableitung höherer Ordnung (1)
- Aggregation (1)
- Algebraic dependence of commuting elements (1)
- Algebraische Abhängigkeit der kommutierende Elementen (1)
- Algebraische Geometrie (1)
- Arc distance (1)
- Ausfallrisiko (1)
- Automatische Differentiation (1)
- Barriers (1)
- Bernstejn-Polynom (1)
- Betriebsfestigkeit (1)
- Boundary Value Problem (1)
- Box Algorithms (1)
- CHAMP <Satellitenmission> (1)
- Computer Algebra System (1)
- Computeralgebra System (1)
- Container (1)
- Crane (1)
- Crash modelling (1)
- Crashmodellierung (1)
- Das Urbild von Ideal unter einen Morphismus der Algebren (1)
- Derivatives (1)
- Differentialinklusionen (1)
- Discrete Bicriteria Optimization (1)
- Dynamic Network Flow Problem (1)
- Dynamische Topographie (1)
- EM algorithm (1)
- Elastizität (1)
- Elastoplasticity (1)
- Eliminationsverfahren (1)
- Evacuation Planning (1)
- Extreme Events (1)
- FPM (1)
- Feedfoward Neural Networks (1)
- Filippov theory (1)
- Filippov-Theorie (1)
- Filtergesetz (1)
- Finite Pointset Method (1)
- Finite-Punktmengen-Methode (1)
- Firmwertmodell (1)
- GARCH (1)
- GARCH Modelle (1)
- GOCE <Satellitenmission> (1)
- GOCE <satellite mission> (1)
- GRACE (1)
- GRACE <Satellitenmission> (1)
- GRACE <satellite mission> (1)
- Geodäsie (1)
- Geodätischer Satellit (1)
- Geometrische Ergodizität (1)
- Gravitational Field (1)
- Gravitationsfeld (1)
- Gröbner-Basis (1)
- Harmonische Spline-Funktion (1)
- Hidden Markov models for Financial Time Series (1)
- Higher Order Differentials as Boundary Data (1)
- Homogenisierung <Mathematik> (1)
- Hydrological Gravity Variations (1)
- Hydrologie (1)
- Intensität (1)
- Inverses Problem (1)
- Kombinatorik (1)
- Kommutative Algebra (1)
- Konstruktive Approximation (1)
- Kontinuum <Mathematik> (1)
- Kontinuumsphysik (1)
- Kreitderivaten (1)
- Kugel (1)
- Kugelflächenfunktion (1)
- Kugelfunktion (1)
- Large-Scale Problems (1)
- Lineare Elastizitätstheorie (1)
- Lokalkompakte Kerne (1)
- Maximal Cohen-Macaulay modules (1)
- Maximale Cohen-Macaulay Moduln (1)
- Maximum Likelihood Estimation (1)
- Maximum-Likelihood-Schätzung (1)
- Mikroelektronik (1)
- Minimum Cost Network Flow Problem (1)
- Modellierung (1)
- Morphismus (1)
- Multileaf collimator (1)
- Multiobjective programming (1)
- Multiple objective optimization (1)
- Netzwerksynthese (1)
- Nichtkommutative Algebra (1)
- Nichtlineare Zeitreihenanalyse (1)
- Numerische Mathematik (1)
- Papiermaschine (1)
- Parameter identification (1)
- Parameteridentifikation (1)
- Portfoliomanagement (1)
- Poröser Stoff (1)
- Preimage of an ideal under a morphism of algebras (1)
- Regularisierung (1)
- Restricted Regions (1)
- Richtungsableitung (1)
- Scheduling (1)
- Sensitivitäten (1)
- Spannungs-Dehn (1)
- Spherical Harmonics (1)
- Spherical Location Problem (1)
- Spherical Wavelets (1)
- Sphäre (1)
- Sphärische Wavelets (1)
- Spline-Wavelets (1)
- Split Operator (1)
- Split-Operator (1)
- Stabile Vektorbundle (1)
- Stable vector bundles (1)
- Stochastisches Feld (1)
- Test for Changepoint (1)
- Time Series (1)
- Transportation Problem (1)
- Unschärferelation (1)
- Upwind-Verfahren (1)
- Wavelet-Analyse (1)
- Wavelets auf der Kugel und der Sphäre (1)
- Zeitliche Veränderungen (1)
- analoge Mikroelektronik (1)
- automatic differentiation (1)
- ball (1)
- combinatorics (1)
- computeralgebra (1)
- constructive approximation (1)
- default time (1)
- derivative-free iterative method (1)
- differential inclusions (1)
- durability (1)
- dynamical topography (1)
- efficient solution (1)
- elastoplasticity (1)
- epsilon-constraint method (1)
- face value (1)
- facets (1)
- float glass (1)
- heat radiation (1)
- hub covering (1)
- hub location (1)
- initial temperature (1)
- initial temperature reconstruction (1)
- integer programming (1)
- intensity (1)
- inverse problem (1)
- large scale integer programming (1)
- locally compact kernels (1)
- lokalisierende Kerne (1)
- mixed convection (1)
- network flows (1)
- network synthesis (1)
- nichtlineare Netzwerke (1)
- nonlinear circuits (1)
- nonlinear heat equation (1)
- nonlinear inverse problem (1)
- numerics (1)
- optimization (1)
- portfolio (1)
- properly efficient solution (1)
- radiative heat transfer (1)
- regular surface (1)
- regularization (1)
- reguläre Fläche (1)
- representative systems (1)
- scalarization (1)
- sensitivities (1)
- spline-wavelets (1)
- superposed fluids (1)
- translinear circuits (1)
- translineare Schaltungen (1)

#### Faculty / Organisational entity

- Fachbereich Mathematik (36) (remove)

A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.

By means of the limit and jump relations of classical potential theory with respect to the vectorial Helmholtz equation a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging scalar, vectorial and tensorial potential kernels act as scaling functions. Corresponding wavelets are defined via a canonical refinement equation. A tree algorithm for fast decomposition of a complex-valued vector field given on a regular surface is developed based on numerical integration rules. By virtue of this tree algorithm, an effcient numerical method for the solution of vectorial Fredholm integral equations on regular surfaces is discussed in more detail. The resulting multiscale formulation is used to solve boundary-value problems for the time harmonic Maxwell's equations corresponding to regular surfaces.

Aggregation of Large-Scale Network Flow Problems with Application to Evacuation Planning at SAP
(2005)

Our initial situation is as follows: The blueprint of the ground floor of SAP’s main building the EVZ is given and the open question on how mathematic can support the evacuation’s planning process ? To model evacuation processes in advance as well as for existing buildings two models can be considered: macro- and microscopic models. Microscopic models emphasize the individual movement of evacuees. These models consider individual parameters such as walking speed, reaction time or physical abilities as well as the interaction of evacuees during the evacuation process. Because of the fact that the microscopic model requires lots of data, simulations are taken for implementation. Most of the current approaches concerning simulation are based on cellular automats. In contrast to microscopic models, macroscopic models do not consider individual parameters such as the physical abilities of the evacuees. This means that the evacuees are treated as a homogenous group for which only common characteristics are considered; an average human being is assumed. We do not have that much data as in the case of the microscopic models. Therefore, the macroscopic models are mainly based on optimization approaches. In most cases, a building or any other evacuation object is represented through a static network. A time horizon T is added, in order to be able to describe the evolution of the evacuation process over time. Connecting these two components we finally get a dynamic network. Based on this network, dynamic network flow problems are formulated, which can map evacuation processes. We focused on the macroscopic model in our thesis. Our main focus concerning the transfer from the real world problem (e.g. supporting the evacuation planning) will be the modeling of the blueprint as a dynamic network. After modeling the blueprint as a dynamic network, it will be no problem to give a formulation of a dynamic network flow problem, the so-called evacuation problem, which seeks for an optimal evacuation time. However, we have to solve a static large-scale network flow problem to derive a solution for this formulation. In order to reduce the network size, we will examine the possibility of applying aggregation to the evacuation problem. Aggregation (lat. aggregare = piling, affiliate; lat. aggregatio = accumulation, union; the act of gathering something together) was basically used to reduce the size of general large-scale linear or integer programs. The results gained for the general problem definitions were then applied to the transportation problem and the minimum cost network flow problem. We review this theory in detail and look on how results derived there can be used for the evacuation problem, too.

This thesis contains the mathematical treatment of a special class of analog microelectronic circuits called translinear circuits. The goal is to provide foundations of a new coherent synthesis approach for this class of circuits. The mathematical methods of the suggested synthesis approach come from graph theory, combinatorics, and from algebraic geometry, in particular symbolic methods from computer algebra. Translinear circuits form a very special class of analog circuits, because they rely on nonlinear device models, but still allow a very structured approach to network analysis and synthesis. Thus, translinear circuits play the role of a bridge between the "unknown space" of nonlinear circuit theory and the very well exploited domain of linear circuit theory. The nonlinear equations describing the behavior of translinear circuits possess a strong algebraic structure that is nonetheless flexible enough for a wide range of nonlinear functionality. Furthermore, translinear circuits offer several technical advantages like high functional density, low supply voltage and insensitivity to temperature. This unique profile is the reason that several authors consider translinear networks as the key to systematic synthesis methods for nonlinear circuits. The thesis proposes the usage of a computer-generated catalog of translinear network topologies as a synthesis tool. The idea to compile such a catalog has grown from the observation that on the one hand, the topology of a translinear network must satisfy strong constraints which severely limit the number of "admissible" topologies, in particular for networks with few transistors, and on the other hand, the topology of a translinear network already fixes its essential behavior, at least for static networks, because the so-called translinear principle requires the continuous parameters of all transistors to be the same. Even though the admissible topologies are heavily restricted, it is a highly nontrivial task to compile such a catalog. Combinatorial techniques have been adapted to undertake this task. In a catalog of translinear network topologies, prototype network equations can be stored along with each topology. When a circuit with a specified behavior is to be designed, one can search the catalog for a network whose equations can be matched with the desired behavior. In this context, two algebraic problems arise: To set up a meaningful equation for a network in the catalog, an elimination of variables must be performed, and to test whether a prototype equation from the catalog and a specified equation of desired behavior can be "matched", a complex system of polynomial equations must be solved, where the solutions are restricted to a finite set of integers. Sophisticated algorithms from computer algebra are applied in both cases to perform the symbolic computations. All mentioned algorithms have been implemented using C++, Singular, and Mathematica, and are successfully applied to actual design problems of humidity sensor circuitry at Analog Microelectronics GmbH, Mainz. As result of the research conducted, an exhaustive catalog of all static formal translinear networks with at most eight transistors is available. The application for the humidity sensor system proves the applicability of the developed synthesis approach. The details and implementations of the algorithms are worked out only for static networks, but can easily be adopted for dynamic networks as well. While the implementation of the combinatorial algorithms is stand-alone software written "from scratch" in C++, the implementation of the algebraic algorithms, namely the symbolic treatment of the network equations and the match finding, heavily rely on the sophisticated Gröbner basis engine of Singular and thus on more than a decade of experience contained in a special-purpose computer algebra system. It should be pointed out that the thesis contains the new observation that the translinear loop equations of a translinear network are precisely represented by the toric ideal of the network's translinear digraph. Altogether, this thesis confirms and strengthenes the key role of translinear circuits as systematically designable nonlinear circuits.

Competing Neural Networks as Models for Non Stationary Financial Time Series -Changepoint Analysis-
(2005)

The problem of structural changes (variations) play a central role in many scientific fields. One of the most current debates is about climatic changes. Further, politicians, environmentalists, scientists, etc. are involved in this debate and almost everyone is concerned with the consequences of climatic changes. However, in this thesis we will not move into the latter direction, i.e. the study of climatic changes. Instead, we consider models for analyzing changes in the dynamics of observed time series assuming these changes are driven by a non-observable stochastic process. To this end, we consider a first order stationary Markov Chain as hidden process and define the Generalized Mixture of AR-ARCH model(GMAR-ARCH) which is an extension of the classical ARCH model to suit to model with dynamical changes. For this model we provide sufficient conditions that ensure its geometric ergodic property. Further, we define a conditional likelihood given the hidden process and a pseudo conditional likelihood in turn. For the pseudo conditional likelihood we assume that at each time instant the autoregressive and volatility functions can be suitably approximated by given Feedfoward Networks. Under this setting the consistency of the parameter estimates is derived and versions of the well-known Expectation Maximization algorithm and Viterbi Algorithm are designed to solve the problem numerically. Moreover, considering the volatility functions to be constants, we establish the consistency of the autoregressive functions estimates given some parametric classes of functions in general and some classes of single layer Feedfoward Networks in particular. Beside this hidden Markov Driven model, we define as alternative a Weighted Least Squares for estimating the time of change and the autoregressive functions. For the latter formulation, we consider a mixture of independent nonlinear autoregressive processes and assume once more that the autoregressive functions can be approximated by given single layer Feedfoward Networks. We derive the consistency and asymptotic normality of the parameter estimates. Further, we prove the convergence of Backpropagation for this setting under some regularity assumptions. Last but not least, we consider a Mixture of Nonlinear autoregressive processes with only one abrupt unknown changepoint and design a statistical test that can validate such changes.

The following three papers present recent developments in nonlinear Galerkin schemes for solving the spherical Navier-Stokes equation, in wavelet theory based on the 3-dimensional ball, and in multiscale solutions of the Poisson equation inside the ball, that have been presented at the 76th GAMM Annual Meeting in Luxemburg. Part A: A Nonlinear Galerkin Scheme Involving Vectorial and Tensorial Spherical Wavelets for Solving the Incompressible Navier-Stokes Equation on the Sphere The spherical Navier-Stokes equation plays a fundamental role in meteorology by modelling meso-scale (stratified) atmospherical flows. This article introduces a wavelet based nonlinear Galerkin method applied to the Navier-Stokes equation on the rotating sphere. In detail, this scheme is implemented by using divergence free vectorial spherical wavelets, and its convergence is proven. To improve numerical efficiency an extension of the spherical panel clustering algorithm to vectorial and tensorial kernels is constructed. This method enables the rapid computation of the wavelet coefficients of the nonlinear advection term. Thereby, we also indicate error estimates. Finally, extensive numerical simulations for the nonlinear interaction of three vortices are presented. Part B: Methods of Resolution for the Poisson Equation on the 3D Ball Within the article at hand, we investigate the Poisson equation solved by an integral operator, originating from an ansatz by Greens functions. This connection between mass distributions and the gravitational force is essential to investigate, especially inside the Earth, where structures and phenomena are not sufficiently known and plumbable. Since the operator stated above does not solve the equation for all square-integrable functions, the solution space will be decomposed by a multiscale analysis in terms of scaling functions. Classical Euclidean wavelet theory appears not to be the appropriate choice. Ansatz functions are chosen to be reflecting the rotational invariance of the ball. In these terms, the operator itself is finally decomposed and replaced by versions more manageable, revealing structural information about itself. Part C: Wavelets on the 3–dimensional Ball In this article wavelets on a ball in R^3 are introduced. Corresponding properties like an approximate identity and decomposition/reconstruction (scale step property) are proved. The advantage of this approach compared to a classical Fourier analysis in orthogonal polynomials is a better localization of the used ansatz functions.

This diploma thesis examines logistic problems occurring in a container terminal. The thesis focuses on the scheduling of cranes handling containers in a port. Two problems are discussed in detail: the yard crane scheduling of rubber-tired gantry cranes (RMGC) which move freely among the container blocks, and the scheduling of rail-mounted gantry cranes (RMGC) which can only move within a yard zone. The problems are formulated as integer programs. For each of the two problems discussed, two models are presented: In one model, the crane tasks are interpreted as jobs with release times and processing times while in the other model, it is assumed that the tasks can be modeled as generic workload measured in crane minutes. It is shown that the problems are NP-hard in the strong sense. Heuristic solution procedures are developed and evaluated by numerical results. Further ideas which could lead to other solution procedures are presented and some interesting special cases are discussed.

We will give explicit differentiation and integration rules for homogeneous harmonic polynomial polynomials and spherical harmonics in IR^3 with respect to the following differential operators: partial_1, partial_2, partial_3, x_3 partial_2 - x_2 partial_3, x_3 partial_1 - x_1 partial_3, x_2 partial_1 - x_1 partial_2 and x_1 partial_1 + x_2 partial_2 + x_3 partial_3. A numerical application to the problem of determining the geopotential field will be shown.

In this dissertation a model of melt spinning (by Doufas, McHugh and Miller) has been investigated. The model (DMM model) which takes into account effects of inertia, air drag, gravity and surface tension in the momentum equation and heat exchange between air and fibre surface, viscous dissipation and crystallization in the energy equation also has a complicated coupling with the microstructure. The model has two parts, before onset of crystallization (BOC) and after onset of crystallization (AOC) with the point of onset of crystallization as the unknown interface. Mathematically the model has been formulated as a Free boundary value problem. Changes have been introduced in the model with respect to the air drag and an interface condition at the free boundary. The mathematical analysis of the nonlinear, coupled free boundary value problem shows that the solution of this problem depends heavily on initial conditions and parameters which renders the global analysis impossible. But by defining a physically acceptable solution, it is shown that for a more restricted set of initial conditions if a unique solution exists for IVP BOC then it is physically acceptable. For this the important property of the positivity of the conformation tensor variables has been proved. Further it is shown that if a physically acceptable solution exists for IVP BOC then under certain conditions it also exists for IVP AOC. This gives an important relation between the initial conditions of IVP BOC and the existence of a physically acceptable solution of IVP AOC. A new investigation has been done for the melt spinning process in the framework of classical mechanics. A Hamiltonian formulation has been done for the melt spinning process for which appropriate Poisson brackets have been derived for the 1-d, elongational flow of a viscoelastic fluid. From the Hamiltonian, cross sectionally averaged balance mass and momentum equations of melt spinning can be derived along with the microstructural equations. These studies show that the complicated problem of melt spinning can also be studied under the framework of classical mechanics. This work provides the basic groundwork on which further investigations on the dynamics of a fibre could be carried out. The Free boundary value problem has been solved numerically using shooting method. Matlab routines have been used to solve the IVPs arising in the problem. Some numerical case studies have been done to study the sensitivity of the ODE systems with respect to the initial guess and parameters. These experiments support the analysis done and throw more light on the stiff nature and ill posedness of the ODE systems. To validate the model, simulations have been performed on sets of data provided by the company. Comparison of numerical results (axial velocity profiles) has been done with the experimental profiles provided by the company. Numerical results have been found to be in excellent agreement with the experimental profiles.