Filtern
Erscheinungsjahr
Dokumenttyp
- Dissertation (62)
- Habilitation (2)
- Wissenschaftlicher Artikel (1)
Schlagworte
- Optimierung (7)
- Approximation (6)
- Approximationstheorie (6)
- Funktionentheorie (6)
- Partielle Differentialgleichung (6)
- Universalität (6)
- Funktionalanalysis (5)
- universal functions (5)
- Numerische Strömungssimulation (4)
- Optimale Kontrolle (4)
- Quadratische Optimierung (4)
- Shape Optimization (4)
- Analysis (3)
- Hadamard product (3)
- Kompositionsoperator (3)
- Numerische Mathematik (3)
- Operatortheorie (3)
- Sequentielle quadratische Optimierung (3)
- Trust-Region-Algorithmus (3)
- Universelle Funktionen (3)
- binomial (3)
- proper orthogonal decomposition (3)
- Adjungierte Differentialgleichung (2)
- Aerodynamic Design (2)
- Approximation im Komplexen (2)
- Baire's theorem (2)
- Binomial (2)
- Binomialverteilung (2)
- Dichtesatz (2)
- Faber series (2)
- Faberreihen (2)
- GPU (2)
- Gestaltoptimierung (2)
- Hadamard, Jacques (2)
- Hadamardprodukt (2)
- Homologische Algebra (2)
- Hyperzyklizität (2)
- Konvexe Optimierung (2)
- Laurentreihen (2)
- Mathematik (2)
- Monte-Carlo-Simulation (2)
- Navier-Stokes equations (2)
- Navier-Stokes-Gleichung (2)
- Nichtlineare Optimierung (2)
- One-Shot (2)
- POD-Methode (2)
- Parameteridentifikation (2)
- Parameterschätzung (2)
- Regularisierung (2)
- Robust optimization (2)
- Simulation (2)
- Statistik (2)
- Stochastischer Prozess (2)
- Strömungsmechanik (2)
- convergence (2)
- functional analysis (2)
- laurent series (2)
- optimal control (2)
- partial integro-differential equations (2)
- prescribed approximation curves (2)
- universality (2)
- universelle Funktionen (2)
- vorgegebene Approximationswege (2)
- Überkonvergenz (2)
- Adjoint (1)
- Adjoint Equation (1)
- Adjoint Method (1)
- Allokation (1)
- Alternierende Projektionen (1)
- Analytisches Funktional (1)
- Arbitrage-Pricing-Theorie (1)
- Asymptotik (1)
- Ausdehnungsoperator (1)
- Auslöschung (1)
- Banach Algebras (1)
- Banach space (1)
- Banach-Algebra (1)
- Banach-Raum (1)
- Berechnungskomplexität (1)
- Berry-Esseen (1)
- Birkhoff functions (1)
- Birkhoff-Funktionen (1)
- Borel transform (1)
- Bregman distance (1)
- Bregman-Distanz (1)
- Brownian Motion (1)
- Brownsche Bewegung (1)
- Buehler, Robert J. (1)
- Bündel-Methode (1)
- Calibration (1)
- Cancellation (1)
- Cesàro-Mittel (1)
- Chaotisches System (1)
- Codebuch (1)
- Combinatorial Optimization (1)
- Composition algebra (1)
- Composition operator (1)
- Computational Fluid Dynamics (1)
- Computational complexity (1)
- Convergence (1)
- Copositive und Vollständig positive Optimierung (1)
- Couple constraints (1)
- Cross-Border-Leasing (1)
- Césaro-Mittel (1)
- Decomposition (1)
- Dekomposition (1)
- Derivat <Wertpapier> (1)
- Dichte <Stochastik> (1)
- Direkte numerische Simulation (1)
- Discontinuous Galerkin (1)
- Diskontinuierliche Galerkin-Methode (1)
- Distribution (1)
- Distribution <Funktionalanalysis> (1)
- Doppelt nichtzentrale F-Verteilung (1)
- Doppelt nichtzentrale t-Verteilung (1)
- Doubly noncentral F-distribution (1)
- Doubly noncentral t-distribution (1)
- Downside Risk (1)
- Downside-Risiko (1)
- Dualitätstheorie (1)
- Elastizität (1)
- Electricity market equilibrium models (1)
- Entire Function (1)
- Error Estimates (1)
- Error function (1)
- Ersatzmodellierung (1)
- Expected Shortfall (1)
- Extended sign regular (1)
- Extensionsoperatoren (1)
- Faltungsoperator (1)
- Fehlerabschätzung (1)
- Fehleranalyse (1)
- Fehlerfunktion (1)
- Finanzmathematik (1)
- Fledermäuse (1)
- Formenräume (1)
- Formoptimierung (1)
- Fréchet-Algebra (1)
- Functor (1)
- Funktor (1)
- Gaussian measures (1)
- Gauß-Maß (1)
- Gebietszerlegung (1)
- Gittererzeugung (1)
- Globale Konvergenz (1)
- Globale Optimierung (1)
- Graphentheorie (1)
- Graphikprozessor (1)
- Grenzüberschreitendes Leasing (1)
- Grundwasserstrom (1)
- Gärung (1)
- HPC (1)
- Hadamard cycle (1)
- Hadamardzyklus (1)
- Hassler Whitney (1)
- Hauptkomponentenanalyse (1)
- Hypercyclicity (1)
- Hypergeometric 3-F-1 Polynomials (1)
- Hypergeometrische 3-F-1 Polynome (1)
- Hypergeometrische Funktionen (1)
- Hypoelliptischer Operator (1)
- Individuenbasiertes Modell (1)
- Induktiver Limes (1)
- Inkorrekt gestelltes Problem (1)
- Innere-Punkte-Methode (1)
- Integer Optimization (1)
- Integrodifferentialgleichung (1)
- Intervallalgebra (1)
- Kegel (1)
- Kleinman (1)
- Kombinatorische Optimierung (1)
- Komplexe Approximation (1)
- Kompositionsalgebra (1)
- Konfidenzbereich (1)
- Konfidenzintervall (1)
- Konfidenzintervalle (1)
- Konfluente hypergeometrische Funktion (1)
- Kontrolltheorie (1)
- Konvektions-Diffusionsgleichung (1)
- Konvergenz (1)
- Konvergenztheorie (1)
- Korovkin-Satz (1)
- Kriging (1)
- Krylov subspace methods (1)
- Krylov-Verfahren (1)
- LB-Algebra (1)
- Laplace Method (1)
- Laplace Methode (1)
- Level Set Methode (1)
- Level constraints (1)
- Linear complementarity problems (1)
- Lineare Dynamik (1)
- Lineare Funktionalanalysis (1)
- Linearer partieller Differentialoperator (1)
- Lückenapproximation (1)
- Lückenreihe (1)
- Markov Inkrement (1)
- Markov-Kette (1)
- Matching (1)
- Matching polytope (1)
- Matrixcone (1)
- Matrixzerlegung (1)
- Mehrgitterverfahren (1)
- Mellin transformation (1)
- Mellin-Transformierte (1)
- Menage (1)
- Mesh Generation (1)
- Methode der kleinsten Quadrate (1)
- Methode der logarithmischen Barriere (1)
- Mischung (1)
- Mittag-Leffler Funktion (1)
- Mittag-Leffler function (1)
- Modellprädiktive Regelung (1)
- Modified Bessel function (1)
- Modifizierte Besselfunktion (1)
- Monte Carlo Simulation (1)
- Monte-Carlo Methods (1)
- Multinomial (1)
- Multiplikationssatz (1)
- Ménage Polynome (1)
- Ménage Polynomials (1)
- Nash–Cournot competition (1)
- Nebenbedingung (1)
- Newton (1)
- Newton-Verfahren (1)
- Nichtfortsetzbare Potenzreihe (1)
- Nichtglatte Optimierung (1)
- Nichtkonvexe Optimierung (1)
- Nonlinear Optimization (1)
- Normalverteilung (1)
- Nullstellen (1)
- Numerical Optimization (1)
- Numerisches Verfahren (1)
- Optimierung bei nichtlinearen partiellen Differentialgleichungen (1)
- Optimierung unter Unsicherheiten (1)
- Optimization under Uncertainty (1)
- Optionspreis (1)
- Orthogonale Zerlegung (1)
- Overconvergence (1)
- Overconvergent power series and matrix-transforms (1)
- P-Konvexität für Träger (1)
- P-Konvexität für singuläre Träger (1)
- P-convexity for singular supports (1)
- P-convexity for supports (1)
- PDE Beschränkungen (1)
- PDE Constraints (1)
- PDE-constrained optimization (1)
- Parameter dependence of solutions of linear partial differential equations (1)
- Parameterabhängige Lösungen linearer partieller Differentialgeichungen (1)
- Parameterabhängigkeit (1)
- Parametrische Optimierung (1)
- Perfect competition (1)
- Poisson (1)
- Polyeder (1)
- Polynom (1)
- Polynom-Interpolationsverfahren (1)
- Populationsmodellierung (1)
- Potenzialtheorie (1)
- Projective Limit (1)
- Projektiver Limes (1)
- Proper Orthogonal Decomposition (1)
- Proximal-Punkt-Verfahren (1)
- Public Sector Financing (1)
- Quantisierung (1)
- Quantisierungkugel (1)
- Quantisierungsradius (1)
- Quantization (1)
- Randverhalten (1)
- Rechteckwahrscheinlichkeit (1)
- Regularisierungsverfahren (1)
- Robustheit (1)
- Rundungsfehler (1)
- Scan Statistik (1)
- Schalenkonstruktionen (1)
- Schnittebenen (1)
- Selbst-Concordanz (1)
- Semiinfinite Optimierung (1)
- Shape Kalkül (1)
- Shape SQP Methods (1)
- Shape Spaces (1)
- Spektrum <Mathematik> (1)
- Spezielle Funktionen (1)
- Splitting (1)
- Stark stetige Halbgruppe (1)
- Stichprobe (1)
- Stochastic Differential Equation (1)
- Stochastische Approximation (1)
- Stochastische Differentialgleichungen (1)
- Stochastische Konvergenz (1)
- Stochastische Quantisierung (1)
- Stochastische optimale Kontrolle (1)
- Stratified sampling (1)
- Strukturoptimierung (1)
- Survey Statistics (1)
- Survey statistics (1)
- Survey-Statistik (1)
- Taylor Shift Operator (1)
- Taylor shift operator (1)
- Theorie (1)
- Topological Algebra (1)
- Topologieoptimierung (1)
- Topologische Algebra (1)
- Topologische Algebra mit Gewebe (1)
- Topologische Sensitivität (1)
- Transaktionskosten (1)
- Transitivität (1)
- Trust Region (1)
- US-Lease (1)
- Ueberkonvergenz (1)
- Ultradistribut (1)
- Unimodality (1)
- Unimodalität (1)
- Universal approximation (1)
- Universal functions (1)
- Universal overconvergence (1)
- Universal power series (1)
- Universalitäten (1)
- Universelle Approximation (1)
- Universelle Funktion (1)
- Universelle Potenzreihen (1)
- Universelle trigonometrische Reihe (1)
- Universelle ueberkonvergente Potenzreihen und Matrix-Transformierte (1)
- Universelle Überkonvergenz (1)
- Value-at-Risk (1)
- Variationsungleichung (1)
- Versuchsplanung (1)
- Verteilungsapproximation (1)
- Volkszählung (1)
- Vorkonditionierung (1)
- Vorzeichenreguläre Funktionen (1)
- Wahrscheinlichkeitsverteilung (1)
- Webbed Spaces (1)
- Weingärung (1)
- Wertpapie (1)
- Whitney jets (1)
- Whitney's extension problem (1)
- Whitneys Extensionsproblem (1)
- Windkraftwerk (1)
- Zwillingsformel (1)
- alternating projections (1)
- amarts (1)
- analytic functional (1)
- approximation (1)
- approximation in the complex plane (1)
- asymptotically optimal codebooks (1)
- asymptotisch optimale Codebücher (1)
- auxiliary problem principle (1)
- boundary behavior (1)
- bundle-method (1)
- combinatorial optimization (1)
- completely positive (1)
- completely positive cone (1)
- completely positive modelling and optimization (1)
- complex analysis (1)
- complex approximation (1)
- complex dynamics (1)
- complexity reduction (1)
- composition operator (1)
- computational fluid dynamics (1)
- confidence intervals (1)
- confidence region (1)
- confluent hypergeometric function (1)
- convergence theory (1)
- convolution operator (1)
- copositive cone (1)
- copositive optimization (1)
- cutting planes (1)
- design of experiments (1)
- domain decomposition (1)
- eigenfunction expansion (1)
- exponential type (1)
- extension operator (1)
- final set (1)
- financial derivatives (1)
- flow control (1)
- frequently hypercyclic operator (1)
- ganze Funktion (1)
- gap power series (1)
- gewöhnliche Differentialgleichungen (1)
- growth (1)
- homological algebra (1)
- homological methods (1)
- homologische Methoden (1)
- hypercyclic operator (1)
- hypercyclicity (1)
- hypergeometric functions (1)
- individual based model (1)
- inexact (1)
- inexact Gauss-Newton methods (1)
- kombinatorische Optimierung (1)
- kommunales Sonderfinanzierungsinstrument (1)
- komplexe Dynamik (1)
- konvexe Reforumlierungen (1)
- kopositiver Kegel (1)
- lacunary approximation (1)
- large scale problems (1)
- linear dynamics (1)
- linear elasticity (1)
- lineare Elastizität (1)
- local quantization error (1)
- logarithmic-quadratic distance function (1)
- logarithmisch-quadratische Distanzfunktion (1)
- lokaler Quantisierungsfehler (1)
- markov increment (1)
- meromorphic functions (1)
- minimal compliance (1)
- minimale Nachgiebigkeit (1)
- mixing (1)
- model order reduction (1)
- model predictive control (1)
- monotone (1)
- multigrid (1)
- multinomial (1)
- n.a. (1)
- nichtnegativ (1)
- nonnegative (1)
- normal approximation (1)
- optimal continuity estimates (1)
- optimal quantization (1)
- optimale Quantisierung (1)
- optimale Stetigkeitsabschätzungen (1)
- optimization (1)
- ordinary differential equations (1)
- orthotrope Materialien (1)
- orthotropic material (1)
- parameter dependence (1)
- parameter estimation (1)
- parameter identification (1)
- partial differential equations (1)
- partial differential operators of first order as generators of C0-semigroups (1)
- partial integro-differential equation (1)
- partielle Differentialgleichungen (1)
- partielle Differentialoperatoren erster Ordnung als Erzeuger von C0-Halbgruppen (1)
- partielle Integro Differentialgleichung (1)
- partielle Integro-Differentialgleichungen (1)
- partielle Integrodifferentialgleichungen (1)
- population modelling (1)
- port-Hamiltonian (1)
- preconditioning (1)
- pricing (1)
- principal component analysis (1)
- quantization ball (1)
- quantization radius (1)
- rationale und meromorphe Approximation (1)
- rectangular probabilities (1)
- reduced order modelling (1)
- reduced-order modelling (1)
- robustness (1)
- scan statistics (1)
- second order cone (1)
- self-concodrance (1)
- series expansion (1)
- shape calculus (1)
- shape optimization (1)
- shell construction (1)
- special functions (1)
- splitting (1)
- starke und schwache Asymptotiken (1)
- statistics (1)
- stochastic Predictor-Corrector-Scheme (1)
- stochastic processes (1)
- strong and weak asymptotics (1)
- structural optimization (1)
- structure-preserving (1)
- sukzessive Ableitungen (1)
- surrogate modeling (1)
- topological derivative (1)
- topology optimization (1)
- transaction costs (1)
- transitivity (1)
- trust-region method (1)
- trust-region methods (1)
- underdetermined nonlinear least squares problem (1)
- universal (1)
- universal trigonometric series (1)
- universalities (1)
- vollständig positiv (1)
- vollständig positiver Kegel (1)
- wine fermentation (1)
- zeros (1)
Institut
- Mathematik (65) (entfernen)
In dieser Arbeit untersuchen wir das Optimierungsproblem der optimalen Materialausrichtung orthotroper Materialien in der Hülle von dreidimensionalen Schalenkonstruktionen. Ziel der Optimierung ist dabei die Minimierung der Gesamtnachgiebigkeit der Konstruktion, was der Suche nach einem möglichst steifen Design entspricht. Sowohl die mathematischen als auch die mechanischen Grundlagen werden in kompakter Form zusammengetragen und basierend darauf werden sowohl gradientenbasierte als auch auf mechanischen Prinzipien beruhende, neue Erweiterungen punktweise formulierter Optimierungsverfahren entwickelt und implementiert. Die vorgestellten Verfahren werden anhand des Beispiels des Modells einer Flugzeugtragfläche mit praxisrelevanter Problemgröße getestet und verglichen. Schließlich werden die untersuchten Methoden in ihrer Koppelung mit einem Verfahren zur Topologieoptimierung, basierend auf dem topologischen Gradienten untersucht.
Shape optimization is of interest in many fields of application. In particular, shape optimization problems arise frequently in technological processes which are modelled by partial differential equations (PDEs). In a lot of practical circumstances, the shape under investigation is parametrized by a finite number of parameters, which, on the one hand, allows the application of standard optimization approaches, but, on the other hand, unnecessarily limits the space of reachable shapes. Shape calculus presents a way to circumvent this dilemma. However, so far shape optimization based on shape calculus is mainly performed using gradient descent methods. One reason for this is the lack of symmetry of second order shape derivatives or shape Hessians. A major difference between shape optimization and the standard PDE constrained optimization framework is the lack of a linear space structure on shape spaces. If one cannot use a linear space structure, then the next best structure is a Riemannian manifold structure, in which one works with Riemannian shape Hessians. They possess the often sought property of symmetry, characterize well-posedness of optimization problems and define sufficient optimality conditions. In general, shape Hessians are used to accelerate gradient-based shape optimization methods. This thesis deals with shape optimization problems constrained by PDEs and embeds these problems in the framework of optimization on Riemannian manifolds to provide efficient techniques for PDE constrained shape optimization problems on shape spaces. A Lagrange-Newton and a quasi-Newton technique in shape spaces for PDE constrained shape optimization problems are formulated. These techniques are based on the Hadamard-form of shape derivatives, i.e., on the form of integrals over the surface of the shape under investigation. It is often a very tedious, not to say painful, process to derive such surface expressions. Along the way, volume formulations in the form of integrals over the entire domain appear as an intermediate step. This thesis couples volume integral formulations of shape derivatives with optimization strategies on shape spaces in order to establish efficient shape algorithms reducing analytical effort and programming work. In this context, a novel shape space is proposed.
The main achievement of this thesis is an analysis of the accuracy of computations with Loader's algorithm for the binomial density. This analysis in later progress of work could be used for a theorem about the numerical accuracy of algorithms that compute rectangle probabilities for scan statistics of a multinomially distributed random variable. An example that shall illustrate the practical use of probabilities for scan statistics is the following, which arises in epidemiology: Let n patients arrive at a clinic in d = 365 days, each of the patients with probability 1/d at each of these d days and all patients independently from each other. The knowledge of the probability, that there exist 3 adjacent days, in which together more than k patients arrive, helps deciding, after observing data, if there is a cluster which we would not suspect to have occurred randomly but for which we suspect there must be a reason. Formally, this epidemiological example can be described by a multinomial model. As multinomially distributed random variables are examples of Markov increments, which is a fact already used implicitly by Corrado (2011) to compute the distribution function of the multinomial maximum, we can use a generalized version of Corrado's Algorithm to compute the probability described in our example. To compute its result, the algorithm for rectangle probabilities for Markov increments always uses transition probabilities of the corresponding Markov Chain. In the multinomial case, the transition probabilities of the corresponding Markov Chain are binomial probabilities. Therefore, we start an analysis of accuracy of Loader's algorithm for the binomial density, which for example the statistical software R uses. With the help of accuracy bounds for the binomial density we would be able to derive accuracy bounds for the computation of rectangle probabilities for scan statistics of multinomially distributed random variables. To figure out how sharp derived accuracy bounds are, in examples these can be compared to rigorous upper bounds and rigorous lower bounds which we obtain by interval-arithmetical computations.
The present work considers the normal approximation of the binomial distribution and yields estimations of the supremum distance of the distribution functions of the binomial- and the corresponding standardized normal distribution. The type of the estimations correspond to the classical Berry-Esseen theorem, in the special case that all random variables are identically Bernoulli distributed. In this case we state the optimal constant for the Berry-Esseen theorem. In the proof of these estimations several inequalities regarding the density as well as the distribution function of the binomial distribution are presented. Furthermore in the estimations mentioned above the distribution function is replaced by the probability of arbitrary, not only unlimited intervals and in this new situation we also present an upper bound.
In the first part of this work we generalize a method of building optimal confidence bounds provided in Buehler (1957) by specializing an exhaustive class of confidence regions inspired by Sterne (1954). The resulting confidence regions, also called Buehlerizations, are valid in general models and depend on a designated statistic'' that can be chosen according to some desired monotonicity behaviour of the confidence region. For a fixed designated statistic, the thus obtained family of confidence regions indexed by their confidence level is nested. Buehlerizations have furthermore the optimality property of being the smallest (w.r.t. set inclusion) confidence regions that are increasing in their designated statistic. The theory is eventually applied to normal, binomial, and exponential samples. The second part deals with the statistical comparison of pairs of diagnostic tests and establishes relations 1. between the sets of lower confidence bounds, 2. between the sets of pairs of comparable lower confidence bounds, and 3. between the sets of admissible lower confidence bounds in various models for diverse parameters of interest.
Matching problems with additional resource constraints are generalizations of the classical matching problem. The focus of this work is on matching problems with two types of additional resource constraints: The couple constrained matching problem and the level constrained matching problem. The first one is a matching problem which has imposed a set of additional equality constraints. Each constraint demands that for a given pair of edges either both edges are in the matching or none of them is in the matching. The second one is a matching problem which has imposed a single equality constraint. This constraint demands that an exact number of edges in the matching are so-called on-level edges. In a bipartite graph with fixed indices of the nodes, these are the edges with end-nodes that have the same index. As a central result concerning the couple constrained matching problem we prove that this problem is NP-hard, even on bipartite cycle graphs. Concerning the complexity of the level constrained perfect matching problem we show that it is polynomially equivalent to three other combinatorial optimization problems from the literature. For different combinations of fixed and variable parameters of one of these problems, the restricted perfect matching problem, we investigate their effect on the complexity of the problem. Further, the complexity of the assignment problem with an additional equality constraint is investigated. In a central part of this work we bring couple constraints into connection with a level constraint. We introduce the couple and level constrained matching problem with on-level couples, which is a matching problem with a special case of couple constraints together with a level constraint imposed on it. We prove that the decision version of this problem is NP-complete. This shows that the level constraint can be sufficient for making a polynomially solvable problem NP-hard when being imposed on that problem. This work also deals with the polyhedral structure of resource constrained matching problems. For the polytope corresponding to the relaxation of the level constrained perfect matching problem we develop a characterization of its non-integral vertices. We prove that for any given non-integral vertex of the polytope a corresponding inequality which separates this vertex from the convex hull of integral points can be found in polynomial time. Regarding the calculation of solutions of resource constrained matching problems, two new algorithms are presented. We develop a polynomial approximation algorithm for the level constrained matching problem on level graphs, which returns solutions whose size is at most one less than the size of an optimal solution. We then describe the Objective Branching Algorithm, a new algorithm for exactly solving the perfect matching problem with an additional equality constraint. The algorithm makes use of the fact that the weighted perfect matching problem without an additional side constraint is polynomially solvable. In the Appendix, experimental results of an implementation of the Objective Branching Algorithm are listed.
Die vorliegende Arbeit teilt sich in die zwei titelgebenden Themengebiete. Inhalt des ersten Teils dieser Arbeit ist die Untersuchung der Proximität, also einer gewissen Messung der Nähe, von Binomial- und Poisson-Verteilungen. Speziell wird die uniforme Struktur des Totalvariationsabstandes auf der abgeschlossenen Menge aller Binomial- und Poisson-Verteilungen charakterisiert, und zwar mit Hilfe der die Verteilungen eindeutig bestimmenden zugehörigen Erwartungswerte und Varianzen. Insbesondere wird eine obere Abschätzung des Totalvariationsabstandes auf der Menge der Binomial- und Poisson-Verteilungen durch eine entsprechende Funktion der zugehörigen Erwartungswerte und Varianzen angegeben. Der zweite Teil der Arbeit widmet sich Konfidenzintervallen für Durchschnitte von Erfolgswahrscheinlichkeiten. Eine der ersten und bekanntesten Arbeiten zu Konfidenzintervallen von Erfolgswahrscheinlichkeiten ist die von Clopper und Pearson (1934). Im Binomialmodell werden hier bei bekanntem Stichprobenumfang und Konfidenzniveau Konfidenzintervalle für die unbekannte Erfolgswahrscheinlichkeit entwickelt. Betrachtet man bei festem Stichprobenumfang statt einer Binomialverteilung, also dem Bildmaß einer homogenen Bernoulli-Kette unter der Summationsabbildung, das entsprechende Bildmaß einer inhomogenen Bernoulli-Kette, so erhält man eine Bernoulli-Faltung mit den entsprechenden Erfolgswahrscheinlichkeiten. Für das Schätzen der durchschnittlichen Erfolgswahrscheinlichkeit im größeren Bernoulli-Faltungs-Modell sind z. B. die einseitigen Clopper-Pearson-Intervalle im Allgemeinen nicht gültig. Es werden hier optimale einseitige und gültige zweiseitige Konfidenzintervalle für die durchschnittliche Erfolgswahrscheinlichkeit im Bernoulli-Faltungs-Modell entwickelt. Die einseitigen Clopper-Pearson-Intervalle sind im Allgemeinen auch nicht gültig für das Schätzen der Erfolgswahrscheinlichkeit im hypergeometrischen Modell, das ein Teilmodell des Bernoulli-Faltungs-Modells ist. Für das hypergeometrische Modell mit festem Stichprobenumfang und bekannter Urnengröße sind die optimalen einseitigen Konfidenzintervalle bekannt. Bei festem Stichprobenumfang und unbekannter Urnengröße werden aus den im Bernoulli-Faltungs-Modell optimalen Konfidenzintervallen optimale Konfidenzintervalle für das hypergeometrische Modell entwickelt. Außerdem wird der Fall betrachtet, dass eine obere Schranke für die unbekannte Urnengröße gegeben ist.
This thesis is divided into three main parts: The description of the calibration problem, the numerical solution of this problem and the connection to optimal stochastic control problems. Fitting model prices to given market prices leads to an abstract least squares formulation as calibration problem. The corresponding option price can be computed by solving a stochastic differential equation via the Monte-Carlo method which seems to be preferred by most practitioners. Due to the fact that the Monte-Carlo method is expensive in terms of computational effort and requires memory, more sophisticated stochastic predictor-corrector schemes are established in this thesis. The numerical advantage of these predictor-corrector schemes ispresented and discussed. The adjoint method is applied to the calibration. The theoretical advantage of the adjoint method is discussed in detail. It is shown that the computational effort of gradient calculation via the adjoint method is independent of the number of calibration parameters. Numerical results confirm the theoretical results and summarize the computational advantage of the adjoint method. Furthermore, provides the connection to optimal stochastic control problems is proven in this thesis.
In recent years, the study of dynamical systems has developed into a central research area in mathematics. Actually, in combination with keywords such as "chaos" or "butterfly effect", parts of this theory have been incorporated in other scientific fields, e.g. in physics, biology, meteorology and economics. In general, a discrete dynamical system is given by a set X and a self-map f of X. The set X can be interpreted as the state space of the system and the function f describes the temporal development of the system. If the system is in state x ∈ X at time zero, its state at time n ∈ N is denoted by f^n(x), where f^n stands for the n-th iterate of the map f. Typically, one is interested in the long-time behaviour of the dynamical system, i.e. in the behaviour of the sequence (f^n(x)) for an arbitrary initial state x ∈ X as the time n increases. On the one hand, it is possible that there exist certain states x ∈ X such that the system behaves stably, which means that f^n(x) approaches a state of equilibrium for n→∞. On the other hand, it might be the case that the system runs unstably for some initial states x ∈ X so that the sequence (f^n(x)) somehow shows chaotic behaviour. In case of a non-linear entire function f, the complex plane always decomposes into two disjoint parts, the Fatou set F_f of f and the Julia set J_f of f. These two sets are defined in such a way that the sequence of iterates (f^n) behaves quite "wildly" or "chaotically" on J_f whereas, on the other hand, the behaviour of (f^n) on F_f is rather "nice" and well-understood. However, this nice behaviour of the iterates on the Fatou set can "change dramatically" if we compose the iterates from the left with just one other suitable holomorphic function, i.e. if we consider sequences of the form (g∘f^n) on D, where D is an open subset of F_f with f(D)⊂ D and g is holomorphic on D. The general aim of this work is to study the long-time behaviour of such modified sequences. In particular, we will prove the existence of holomorphic functions g on D having the property that the behaviour of the sequence of compositions (g∘f^n) on the set D becomes quite similarly chaotic as the behaviour of the sequence (f^n) on the Julia set of f. With this approach, we immerse ourselves into the theory of universal families and hypercyclic operators, which itself has developed into an own branch of research. In general, for topological spaces X, Y and a family {T_i: i ∈ I} of continuous functions T_i:X→Y, an element x ∈ X is called universal for the family {T_i: i ∈ I} if the set {T_i(x): i ∈ I} is dense in Y. In case that X is a topological vector space and T is a continuous linear operator on X, a vector x ∈ X is called hypercyclic for T if it is universal for the family {T^n: n ∈ N}. Thus, roughly speaking, universality and hypercyclicity can be described via the following two aspects: There exists a single object which allows us, via simple analytical operations, to approximate every element of a whole class of objects. In the above situation, i.e. for a non-linear entire function f and an open subset D of F_f with f(D)⊂ D, we endow the space H(D) of holomorphic functions on D with the topology of locally uniform convergence and we consider the map C_f:H(D)→H(D), C_f(g):=g∘f|_D, which is called the composition operator with symbol f. The transform C_f is a continuous linear operator on the Fréchet space H(D). In order to show that the above-mentioned "nice" behaviour of the sequence of iterates (f^n) on the set D ⊂ F_f can "change dramatically" if we compose the iterates from the left with another suitable holomorphic function, our aim consists in finding functions g ∈ H(D) which are hypercyclic for C_f. Indeed, for each hypercyclic function g for C_f, the set of compositions {g∘f^n|_D: n ∈ N} is dense in H(D) so that the sequence of compositions (g∘f^n|_D) is kind of "maximally divergent" " meaning that each function in H(D) can be approximated locally uniformly on D via subsequences of (g∘f^n|_D). This kind of behaviour stands in sharp contrast to the fact that the sequence of iterates (f^n) itself converges, behaves like a rotation or shows some "wandering behaviour" on each component of F_f. To put it in a nutshell, this work combines the theory of non-linear complex dynamics in the complex plane with the theory of dynamics of continuous linear operators on spaces of holomorphic functions. As far as the author knows, this approach has not been investigated before.
Zu den klassischen Verteilungen der mathematischen Statistik zählen die zentralen F- und t-Verteilungen. Die vorliegende Arbeit untersucht Verallgemeinerungen dieser Verteilungen, die sogenannten doppelt nichtzentralen F- und t-Verteilungen, welche in der statistischen Testtheorie von Bedeutung sind. Die Tatsache, dass die zugehörigen Wahrscheinlichkeitsdichten nur in Form von Parameterintegral- bzw. Doppelreihendarstellungen gegeben sind, stellt eine große Herausforderung bei der Untersuchung analytischer Eigenschaften dar. Unter Verwendung von Techniken aus der Theorie der vorzeichenregulären Funktionen gelingt es, die bisher vermutete, jedoch lediglich aus Approximationen abgeleitete, strikt unimodale Gestalt der Dichtefunktion für eine große Klasse doppelt nichtzentraler Verteilungen zu zeigen. Dieses Resultat gestattet die Untersuchung des eindeutig bestimmten Modus als Funktion gewisser Nichtzentralitätsparameter. Hier erweist sich die Theorie der vorzeichenregulären Funktionen als wichtiges Hilfsmittel, um monotone Abhängigkeiten nachzuweisen.