OPUS 4 | 510 Mathematik

A Penalty Branch-and-Bound Method for Piecewise Convex Objective Functions (2023)

Winkel, Lukas

Die Dissertation beschäftigt sich mit einer neuartigen Art von Branch-and-Bound Algorithmen, deren Unterschied zu klassischen Branch-and-Bound Algorithmen darin besteht, dass das Branching durch die Addition von nicht-negativen Straftermen zur Zielfunktion erfolgt anstatt durch das Hinzufügen weiterer Nebenbedingungen. Die Arbeit zeigt die theoretische Korrektheit des Algorithmusprinzips für verschiedene allgemeine Klassen von Problemen und evaluiert die Methode für verschiedene konkrete Problemklassen. Für diese Problemklassen, genauer Monotone und Nicht-Monotone Gemischtganzzahlige Lineare Komplementaritätsprobleme und Gemischtganzzahlige Lineare Probleme, präsentiert die Arbeit verschiedene problemspezifische Verbesserungsmöglichkeiten und evaluiert diese numerisch. Weiterhin vergleicht die Arbeit die neue Methode mit verschiedenen Benchmark-Methoden mit größtenteils guten Ergebnissen und gibt einen Ausblick auf weitere Anwendungsgebiete und zu beantwortende Forschungsfragen.

On Lacunary Approximation of Mergelyan Type (2022)

Born, Markus

Let K be a compact subset of the complex plane. Then the family of polynomials P is dense in A(K), the space of all continuous functions on K that are holomorphic on the interior of K, endowed with the uniform norm, if and only if the complement of K is connected. This is the statement of Mergelyan's celebrated theorem. There are, however, situations where not all polynomials are required to approximate every f ϵ A(K) but where there are strict subspaces of P that are still dense in A(K). If, for example, K is a singleton, then the subspace of all constant polynomials is dense in A(K). On the other hand, if 0 is an interior point of K, then no strict subspace of P can be dense in A(K). In between these extreme cases, the situation is much more complicated. It turns out that it is mostly determined by the geometry of K and its location in the complex plane which subspaces of P are dense in A(K). In Chapter 1, we give an overview of the known results. Our first main theorem, which we will give in Chapter 3, deals with the case where the origin is not an interior point of K. We will show that if K is a compact set with connected complement and if 0 is not an interior point of K, then any subspace Q ⊂ P which contains the constant functions and all but finitely many monomials is dense in A(K). There is a close connection between lacunary approximation and the theory of universality. At the end of Chapter 3, we will illustrate this connection by applying the above result to prove the existence of certain universal power series. To be specific, if K is a compact set with connected complement, if 0 is a boundary point of K and if A_0(K) denotes the subspace of A(K) of those functions that satisfy f(0) = 0, then there exists an A_0(K)-universal formal power series s, where A_0(K)-universal means that the family of partial sums of s forms a dense subset of A_0(K). In addition, we will show that no formal power series is simultaneously universal for all such K. The condition on the subspace Q in the main result of Chapter 3 is quite restrictive, but this should not be too surprising: The result applies to the largest possible class of compact sets. In Chapter 4, we impose a further restriction on the compact sets under consideration, and this will allow us to weaken the condition on the subspace Q. The result that we are going to give is similar to one of those presented in the first chapter, namely the one due to Anderson. In his article “Müntz-Szasz type approximation and the angular growth of lacunary integral functions”, he gives a criterion for a subspace Q of P to be dense in A(K) where K is entirely contained in some closed sector with vertex at the origin. We will consider compact sets with connected complement that are -- with the possible exception of the origin -- entirely contained in some open sector with vertex at the origin. What we are going to show is that if K\{0} is contained in an open sector of opening angle 2α and if Λ is some subset of the nonnegative integers, then the span of {z → z^λ : λ ϵ Λ} is dense in A(K) whenever 0 ϵ Λ and some Müntz-type condition is satisfied. Conversely, we will show that if a similar condition is not satisfied, then we can always find a compact set K with connected complement such that K\{0} is contained in some open sector of opening angle 2α and such that the span of {z → z^λ : λ ϵ Λ} fails to be dense in A(K).

Pre-Shape Calculus - a Unified Framework for Mesh Quality and Shape Optimization (2022)

Luft, Daniel

In common shape optimization routines, deformations of the computational mesh usually suffer from decrease of mesh quality or even destruction of the mesh. To mitigate this, we propose a theoretical framework using so-called pre-shape spaces. This gives an opportunity for a unified theory of shape optimization, and of problems related to parameterization and mesh quality. With this, we stay in the free-form approach of shape optimization, in contrast to parameterized approaches that limit possible shapes. The concept of pre-shape derivatives is defined, and according structure and calculus theorems are derived, which generalize classical shape optimization and its calculus. Tangential and normal directions are featured in pre-shape derivatives, in contrast to classical shape derivatives featuring only normal directions on shapes. Techniques from classical shape optimization and calculus are shown to carry over to this framework, and are collected in generality for future reference. A pre-shape parameterization tracking problem class for mesh quality is in- troduced, which is solvable by use of pre-shape derivatives. This class allows for non-uniform user prescribed adaptations of the shape and hold-all domain meshes. It acts as a regularizer for classical shape objectives. Existence of regularized solu- tions is guaranteed, and corresponding optimal pre-shapes are shown to correspond to optimal shapes of the original problem, which additionally achieve the user pre- scribed parameterization. We present shape gradient system modifications, which allow simultaneous nu- merical shape optimization with mesh quality improvement. Further, consistency of modified pre-shape gradient systems is established. The computational burden of our approach is limited, since additional solution of possibly larger (non-)linear systems for regularized shape gradients is not necessary. We implement and com- pare these pre-shape gradient regularization approaches for a 2D problem, which is prone to mesh degeneration. As our approach does not depend on the choice of forms to represent shape gradients, we employ and compare weak linear elasticity and weak quasilinear p-Laplacian pre-shape gradient representations. We also introduce a Quasi-Newton-ADM inspired algorithm for mesh quality, which guarantees sufficient adaption of meshes to user specification during the rou- tines. It is applicable in addition to simultaneous mesh regularization techniques. Unrelated to mesh regularization techniques, we consider shape optimization problems constrained by elliptic variational inequalities of the first kind, so-called obstacle-type problems. In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for such semi-smooth shape optimization problems. Under appropriate assumptions, we prove existence and convergence of adjoints for smooth regularizations of the VI-constraint. Moreover, we derive shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically. All previous pre-shape regularization techniques are applied to a variational inequality constrained shape optimization problem, where we also create customized targets for increased mesh adaptation of changing embedded shapes and active set boundaries of the constraining variational inequality.

Hybrid Modelling of Dynamical Systems in Mechanics (2022)

Sokolowski, Jan

Hybrid Modelling in general, describes the combination of at least two different methods to solve one specific task. As far as this work is concerned, Hybrid Models describe an approach to combine sophisticated, well-studied mathematical methods with Deep Neural Networks to solve parameter estimation tasks. To combine these two methods, the data structure of artifi- cially generated acceleration data of an approximate vehicle model, the Quarter-Car-Model, is exploited. Acceleration of individual components within a coupled dynamical system, can be described as a second order ordinary differential equation, including velocity and dis- placement of coupled states, scaled by spring - and damping-coefficient of the system. An appropriate numerical integration scheme can then be used to simulate discrete acceleration profiles of the Quarter-Car-Model with a random variation of the parameters of the system. Given explicit knowledge about the data structure, one can then investigate under which con- ditions it is possible to estimate the parameters of the dynamical system for a set of randomly generated data samples. We test, if Neural Networks are capable to solve parameter estima- tion problems in general, or if they can be used to solve several sub-tasks, which support a state-of-the-art parameter estimation method. Hybrid Models are presented for parameter estimation under uncertainties, including for instance measurement noise or incompleteness of measurements, which combine knowledge about the data structure and several Neural Networks for robust parameter estimation within a dynamical system.

Strictly and -robust counterparts of electricity market models (2021)

Kramer, Anja ; Krebs, Vanessa ; Schmidt, Martin

This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the -approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition, this result corresponds to the two classic welfare theorems, which also apply in both considered robust cases that again yield convex robustified problems. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of -robustifications of Nash–Cournot models. Thus, an analog of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.

De Rham and Cech-de Rham Cohomologies of Smooth Foliated Manifolds (2021)

Schlierkamp, Thorben

In order to classify smooth foliated manifolds, which are smooth maifolds equipped with a smooth foliation, we introduce the de Rham cohomologies of smooth foliated manifolds. These cohomologies are build in a similar way as the de Rham cohomologies of smooth manifolds. We develop some tools to compute these cohomologies. For example we proof a Mayer Vietoris theorem for foliated de Rham cohomology and show that these cohomologys are invariant under integrable homotopy. A generalization of a known Künneth formula, which relates the cohomologies of a product foliation with its factors, is discussed. In particular, this envolves a splitting theory of sequences between Frechet spaces and a theory of projective spectrums. We also prove, that the foliated de Rham cohomology is isomorphic to the Cech-de Rham cohomology and the Cech cohomology of leafwise constant functions of an underlying so called good cover.

On particles, fibers and suspension flows (2020)

Vibe, Alexander

This work studies typical mathematical challenges occurring in the modeling and simulation of manufacturing processes of paper or industrial textiles. In particular, we consider three topics: approximate models for the motion of small inertial particles in an incompressible Newtonian fluid, effective macroscopic approximations for a dilute particle suspension contained in a bounded domain accounting for a non-uniform particle distribution and particle inertia, and possibilities for a reduction of computational cost in the simulations of slender elastic fibers moving in a turbulent fluid flow. We consider the full particle-fluid interface problem given in terms of the Navier-Stokes equations coupled to momentum equations of a small rigid body. By choosing an appropriate asymptotic scaling for the particle-fluid density ratio and using an asymptotic expansion for the solution components, we derive approximations of the original interface problem. The approximate systems differ according to the chosen scaling of the density ratio in their physical behavior allowing the characterization of different inertial regimes. We extend the asymptotic approach to the case of many particles suspended in a Newtonian fluid. Under specific assumptions for the combination of particle size and particle number, we derive asymptotic approximations of this system. The approximate systems describe the particle motion which allows to use a mean field approach in order to formulate the continuity equation for the particle probability density function. The coupling of the latter with the approximation for the fluid momentum equation then reveals a macroscopic suspension description which accounts for non-uniform particle distributions in space and for small particle inertia. A slender fiber in a turbulent air flow can be modeled as a stochastic inextensible one-dimensionally parametrized Kirchhoff beam, i.e., by a stochastic partial differential algebraic equation. Its simulations involve the solution of large non-linear systems of equations by Newton's method. In order to decrease the computational time, we explore different methods for the estimation of the solution. Additionally, we apply smoothing techniques to the Wiener Process in order to regularize the stochastic force driving the fiber, exploring their respective impact on the solution and performance. We also explore the applicability of the Wiener chaos expansion as a solution technique for the simulation of the fiber dynamics.

Stochastic Particle Systems and Optimization - Branching Processes, Mean Field Games and Impulse Control (2020)

Hoffmann, Daniel

This thesis addresses three different topics from the fields of mathematical finance, applied probability and stochastic optimal control. Correspondingly, it is subdivided into three independent main chapters each of which approaches a mathematical problem with a suitable notion of a stochastic particle system. In Chapter 1, we extend the branching diffusion Monte Carlo method of Henry-Labordère et. al. (2019) to the case of parabolic PDEs with mixed local-nonlocal analytic nonlinearities. We investigate branching diffusion representations of classical solutions, and we provide sufficient conditions under which the branching diffusion representation solves the PDE in the viscosity sense. Our theoretical setup directly leads to a Monte Carlo algorithm, whose applicability is showcased in two stylized high-dimensional examples. As our main application, we demonstrate how our methodology can be used to value financial positions with defaultable, systemically important counterparties. In Chapter 2, we formulate and analyze a mathematical framework for continuous-time mean field games with finitely many states and common noise, including a rigorous probabilistic construction of the state process. The key insight is that we can circumvent the master equation and reduce the mean field equilibrium to a system of forward-backward systems of (random) ordinary differential equations by conditioning on common noise events. We state and prove a corresponding existence theorem, and we illustrate our results in three stylized application examples. In the absence of common noise, our setup reduces to that of Gomes, Mohr and Souza (2013) and Cecchin and Fischer (2020). In Chapter 3, we present a heuristic approach to tackle stochastic impulse control problems in discrete time. Based on the work of Bensoussan (2008) we reformulate the classical Bellman equation of stochastic optimal control in terms of a discrete-time QVI, and we prove a corresponding verification theorem. Taking the resulting optimal impulse control as a starting point, we devise a self-learning algorithm that estimates the continuation and intervention region of such a problem. Its key features are that it explores the state space of the underlying problem by itself and successively learns the behavior of the optimally controlled state process. For illustration, we apply our algorithm to a classical example problem, and we give an outlook on open questions to be addressed in future research.

On robust small area estimation (2020)

Schoch, Tobias

Traditionell werden Zufallsstichprobenerhebungen so geplant, dass nationale Statistiken zuverlässig mit einer adäquaten Präzision geschätzt werden können. Hierbei kommen vorrangig designbasierte, Modell-unterstützte (engl. model assisted) Schätzmethoden zur Anwendung, die überwiegend auf asymptotischen Eigenschaften beruhen. Für kleinere Stichprobenumfänge, wie man sie für Small Areas (Domains bzw. Subpopulationen) antrifft, eignen sich diese Schätzmethoden eher nicht, weswegen für diese Anwendung spezielle modellbasierte Small Area-Schätzverfahren entwickelt wurden. Letztere können zwar Verzerrungen aufweisen, besitzen jedoch häufig einen kleineren mittleren quadratischen Fehler der Schätzung als dies für designbasierte Schätzer der Fall ist. Den Modell-unterstützten und modellbasierten Methoden ist gemeinsam, dass sie auf statistischen Modellen beruhen; allerdings in unterschiedlichem Ausmass. Modell-unterstützte Verfahren sind in der Regel so konstruiert, dass der Beitrag des Modells bei sehr grossen Stichprobenumfängen gering ist (bei einer Grenzwertbetrachtung sogar wegfällt). Bei modellbasierten Methoden nimmt das Modell immer eine tragende Rolle ein, unabhängig vom Stichprobenumfang. Diese Überlegungen veranschaulichen, dass das unterstellte Modell, präziser formuliert, die Güte der Modellierung für die Qualität der Small Area-Statistik von massgeblicher Bedeutung ist. Wenn es nicht gelingt, die empirischen Daten durch ein passendes Modell zu beschreiben und mit den entsprechenden Methoden zu schätzen, dann können massive Verzerrungen und / oder ineffiziente Schätzungen resultieren. Die vorliegende Arbeit beschäftigt sich mit der zentralen Frage der Robustheit von Small Area-Schätzverfahren. Als robust werden statistische Methoden dann bezeichnet, wenn sie eine beschränkte Einflussfunktion und einen möglichst hohen Bruchpunkt haben. Vereinfacht gesprochen zeichnen sich robuste Verfahren dadurch aus, dass sie nur unwesentlich durch Ausreisser und andere Anomalien in den Daten beeinflusst werden. Die Untersuchung zur Robustheit konzentriert sich auf die folgenden Modelle bzw. Schätzmethoden: i) modellbasierte Schätzer für das Fay-Herriot-Modell (Fay und Herrot, 1979, J. Amer. Statist. Assoc.) und das elementare Unit-Level-Modell (vgl. Battese et al., 1988, J. Amer. Statist. Assoc.). ii) direkte, Modell-unterstützte Schätzer unter der Annahme eines linearen Regressionsmodells. Das Unit-Level-Modell zur Mittelwertschätzung beruht auf einem linearen gemischten Gauss'schen Modell (engl. mixed linear model, MLM) mit blockdiagonaler Kovarianzmatrix. Im Gegensatz zu bspw. einem multiplen linearen Regressionsmodell, besitzen MLM-Modelle keine nennenswerten Invarianzeigenschaften, so dass eine Kontamination der abhängigen Variablen unvermeidbar zu verzerrten Parameterschätzungen führt. Für die Maximum-Likelihood-Methode kann die resultierende Verzerrung nahezu beliebig groß werden. Aus diesem Grund haben Richardson und Welsh (1995, Biometrics) die robusten Schätzmethoden RML 1 und RML 2 entwickelt, die bei kontaminierten Daten nur eine geringe Verzerrung aufweisen und wesentlich effizienter sind als die Maximum-Likelihood-Methode. Eine Abwandlung von Methode RML 2 wurde Sinha und Rao (2009, Canad. J. Statist.) für die robuste Schätzung von Unit-Level-Modellen vorgeschlagen. Allerdings erweisen sich die gebräuchlichen numerischen Verfahren zur Berechnung der RML-2-Methode (dies gilt auch für den Vorschlag von Sinha und Rao) als notorisch unzuverlässig. In dieser Arbeit werden zuerst die Konvergenzprobleme der bestehenden Verfahren erörtert und anschließend ein numerisches Verfahren vorgeschlagen, das sich durch wesentlich bessere numerische Eigenschaften auszeichnet. Schließlich wird das vorgeschlagene Schätzverfahren im Rahmen einer Simulationsstudie untersucht und anhand eines empirischen Beispiels zur Schätzung von oberirdischer Biomasse in norwegischen Kommunen illustriert. Das Modell von Fay-Herriot kann als Spezialfall eines MLM mit blockdiagonaler Kovarianzmatrix aufgefasst werden, obwohl die Varianzen des Zufallseffekts für die Small Areas nicht geschätzt werden müssen, sondern als bereits bekannte Größen betrachtet werden. Diese Eigenschaft kann man sich nun zunutze machen, um die von Sinha und Rao (2009) vorgeschlagene Robustifizierung des Unit-Level-Modells direkt auf das Fay-Herriot Model zu übertragen. In der vorliegenden Arbeit wird jedoch ein alternativer Vorschlag erarbeitet, der von der folgenden Beobachtung ausgeht: Fay und Herriot (1979) haben ihr Modell als Verallgemeinerung des James-Stein-Schätzers motiviert, wobei sie sich einen empirischen Bayes-Ansatz zunutze machen. Wir greifen diese Motivation des Problems auf und formulieren ein analoges robustes Bayes'sches Verfahren. Wählt man nun in der robusten Bayes'schen Problemformulierung die ungünstigste Verteilung (engl. least favorable distribution) von Huber (1964, Ann. Math. Statist.) als A-priori-Verteilung für die Lokationswerte der Small Areas, dann resultiert als Bayes-Schätzer [=Schätzer mit dem kleinsten Bayes-Risk] die Limited-Translation-Rule (LTR) von Efron und Morris (1971, J. Amer. Statist. Assoc.). Im Kontext der frequentistischen Statistik kann die Limited-Translation-Rule nicht verwendet werden, weil sie (als Bayes-Schätzer) auf unbekannten Parametern beruht. Die unbekannten Parameter können jedoch nach dem empirischen Bayes-Ansatz an der Randverteilung der abhängigen Variablen geschätzt werden. Hierbei gilt es zu beachten (und dies wurde in der Literatur vernachlässigt), dass die Randverteilung unter der ungünstigsten A-priori-Verteilung nicht einer Normalverteilung entspricht, sondern durch die ungünstigste Verteilung nach Huber (1964) beschrieben wird. Es ist nun nicht weiter erstaunlich, dass es sich bei den Maximum-Likelihood-Schätzern von Regressionskoeffizienten und Modellvarianz unter der Randverteilung um M-Schätzer mit der Huber'schen psi-Funktion handelt. Unsere theoriegeleitete Herleitung von robusten Schätzern zum Fay-Herriot-Modell zeigt auf, dass bei kontaminierten Daten die geschätzte LTR (mit Parameterschätzungen nach der M-Schätzmethodik) optimal ist und, dass die LTR ein integraler Bestandteil der Schätzmethodik ist (und nicht als ``Zusatz'' o.Ä. zu betrachten ist, wie dies andernorts getan wird). Die vorgeschlagenen M-Schätzer sind robust bei Vorliegen von atypischen Small Areas (Ausreissern), wie dies auch die Simulations- und Fallstudien zeigen. Um auch Robustheit bei Vorkommen von einflussreichen Beobachtungen in den unabhängigen Variablen zu erzielen, wurden verallgemeinerte M-Schätzer (engl. generalized M-estimator) für das Fay-Herriot-Modell entwickelt.

New Concise Extended Formulations for Circular Structures in Optimization Problems (2020)

Perscheid, Bernd

Many NP-hard optimization problems that originate from classical graph theory, such as the maximum stable set problem and the maximum clique problem, have been extensively studied over the past decades and involve the choice of a subset of edges or vertices. There usually exist combinatorial methods that can be applied to solve them directly in the graph. The most simple method is to enumerate feasible solutions and select the best. It is not surprising that this method is very slow oftentimes, so the task is to cleverly discard fruitless search space during the search. An alternative method to solve graph problems is to formulate integer linear programs, such that their solution yields an optimal solution to the original optimization problem in the graph. In order to solve integer linear programs, one can start with relaxing the integer constraints and then try to find inequalities for cutting off fractional extreme points. In the best case, it would be possible to describe the convex hull of the feasible region of the integer linear program with a set of inequalities. In general, giving a complete description of this convex hull is out of reach, even if it has a polynomial number of facets. Thus, one tries to strengthen the (weak) relaxation of the integer linear program best possible via strong inequalities that are valid for the convex hull of feasible integer points. Many classes of valid inequalities are of exponential size. For instance, a graph can have exponentially many odd cycles in general and therefore the number of odd cycle inequalities for the maximum stable set problem is exponential. It is sometimes possible to check in polynomial time if some given point violates any of the exponentially many inequalities. This is indeed the case for the odd cycle inequalities for the maximum stable set problem. If a polynomial time separation algorithm is known, there exists a formulation of polynomial size that contains a given point if and only if it does not violate one of the (potentially exponentially many) inequalities. This thesis can be divided into two parts. The first part is the main part and it contains various new results. We present new extended formulations for several optimization problems, i.e. the maximum stable set problem, the nonconvex quadratic program with box constraints and the p-median problem. In the second part we modify a very fast algorithm for finding a maximum clique in very large sparse graphs. We suggest and compare three alternative versions of this algorithm to the original version and compare their strengths and weaknesses.

Robust Training of Artificial Neural Networks via p-Quasinorms (2020)

Geisen, Stefan

Data used for the purpose of machine learning are often erroneous. In this thesis, p-quasinorms (p<1) are employed as loss functions in order to increase the robustness of training algorithms for artificial neural networks. Numerical issues arising from these loss functions are addressed via enhanced optimization algorithms (proximal point methods; Frank-Wolfe methods) based on the (non-monotonic) Armijo-rule. Numerical experiments comprising 1100 test problems confirm the effectiveness of the approach. Depending on the parametrization, an average reduction of the absolute residuals of up to 64.6% is achieved (aggregated over 100 test problems).

On port-Hamiltonian modeling and structure-preserving model reduction (2020)

Liljegren-Sailer, Björn

In this thesis we study structure-preserving model reduction methods for the efficient and reliable approximation of dynamical systems. A major focus is the approximation of a nonlinear flow problem on networks, which can, e.g., be used to describe gas network systems. Our proposed approximation framework guarantees so-called port-Hamiltonian structure and is general enough to be realizable by projection-based model order reduction combined with complexity reduction. We divide the discussion of the flow problem into two parts, one concerned with the linear damped wave equation and the other one with the general nonlinear flow problem on networks. The study around the linear damped wave equation relies on a Galerkin framework, which allows for convenient network generalizations. Notable contributions of this part are the profound analysis of the algebraic setting after space-discretization in relation to the infinite dimensional setting and its implications for model reduction. In particular, this includes the discussion of differential-algebraic structures associated to the network-character of our problem and the derivation of compatibility conditions related to fundamental physical properties. Amongst the different model reduction techniques, we consider the moment matching method to be a particularly well-suited choice in our framework. The Galerkin framework is then appropriately extended to our general nonlinear flow problem. Crucial supplementary concepts are required for the analysis, such as the partial Legendre transform and a more careful discussion of the underlying energy-based modeling. The preservation of the port-Hamiltonian structure after the model-order- and complexity-reduction-step represents a major focus of this work. Similar as in the analysis of the model order reduction, compatibility conditions play a crucial role in the analysis of our complexity reduction, which relies on a quadrature-type ansatz. Furthermore, energy-stable time-discretization schemes are derived for our port-Hamiltonian approximations, as structure-preserving methods from literature are not applicable due to our rather unconventional parametrization of the solution. Apart from the port-Hamiltonian approximation of the flow problem, another topic of this thesis is the derivation of a new extension of moment matching methods from linear systems to quadratic-bilinear systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem rather than to a multi-dimensional interpolation as for the multivariate approaches, i.e., it involves fewer interpolation frequencies to be chosen. The notion of signal-generator-driven systems, variational expansions of the resulting autonomous systems as well as the derivation of convenient tensor-structured approximation conditions are the main ingredients of this part. Notably, our approach allows for the incorporation of general input relations in the state equations, not only affine-linear ones as in existing system-theoretic methods.

Nonlocal models with truncated interaction kernels - analysis, finite element methods and shape optimization (2019)

Vollmann, Christian

Nonlocal operators are used in a wide variety of models and applications due to many natural phenomena being driven by nonlocal dynamics. Nonlocal operators are integral operators allowing for interactions between two distinct points in space. The nonlocal models investigated in this thesis involve kernels that are assumed to have a finite range of nonlocal interactions. Kernels of this type are used in nonlocal elasticity and convection-diffusion models as well as finance and image analysis. Also within the mathematical theory they arouse great interest, as they are asymptotically related to fractional and classical differential equations. The results in this thesis can be grouped according to the following three aspects: modeling and analysis, discretization and optimization. Mathematical models demonstrate their true usefulness when put into numerical practice. For computational purposes, it is important that the support of the kernel is clearly determined. Therefore nonlocal interactions are typically assumed to occur within an Euclidean ball of finite radius. In this thesis we consider more general interaction sets including norm induced balls as special cases and extend established results about well-posedness and asymptotic limits. The discretization of integral equations is a challenging endeavor. Especially kernels which are truncated by Euclidean balls require carefully designed quadrature rules for the implementation of efficient finite element codes. In this thesis we investigate the computational benefits of polyhedral interaction sets as well as geometrically approximated interaction sets. In addition to that we outline the computational advantages of sufficiently structured problem settings. Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface-dependent kernels. We derive the shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques.

Cardinality-Constrained Discrete Optimization for Regression (2019)

Kreber, Dennis

We consider a linear regression model for which we assume that some of the observed variables are irrelevant for the prediction. Including the wrong variables in the statistical model can either lead to the problem of having too little information to properly estimate the statistic of interest, or having too much information and consequently describing fictitious connections. This thesis considers discrete optimization to conduct a variable selection. In light of this, the subset selection regression method is analyzed. The approach gained a lot of interest in recent years due to its promising predictive performance. A major challenge associated with the subset selection regression is the computational difficulty. In this thesis, we propose several improvements for the efficiency of the method. Novel bounds on the coefficients of the subset selection regression are developed, which help to tighten the relaxation of the associated mixed-integer program, which relies on a Big-M formulation. Moreover, a novel mixed-integer linear formulation for the subset selection regression based on a bilevel optimization reformulation is proposed. Finally, it is shown that the perspective formulation of the subset selection regression is equivalent to a state-of-the-art binary formulation. We use this insight to develop novel bounds for the subset selection regression problem, which show to be highly effective in combination with the proposed linear formulation. In the second part of this thesis, we examine the statistical conception of the subset selection regression and conclude that it is misaligned with its intention. The subset selection regression uses the training error to decide on which variables to select. The approach conducts the validation on the training data, which oftentimes is not a good estimate of the prediction error. Hence, it requires a predetermined cardinality bound. Instead, we propose to select variables with respect to the cross-validation value. The process is formulated as a mixed-integer program with the sparsity becoming subject of the optimization. Usually, a cross-validation is used to select the best model out of a few options. With the proposed program the best model out of all possible models is selected. Since the cross-validation is a much better estimate of the prediction error, the model can select the best sparsity itself. The thesis is concluded with an extensive simulation study which provides evidence that discrete optimization can be used to produce highly valuable predictive models with the cross-validation subset selection regression almost always producing the best results.

Optimization in Tensor Spaces for Data Science and Scientific Computing (2019)

Heidel, Gennadij

In this thesis, we consider the solution of high-dimensional optimization problems with an underlying low-rank tensor structure. Due to the exponentially increasing computational complexity in the number of dimensions—the so-called curse of dimensionality—they present a considerable computational challenge and become infeasible even for moderate problem sizes. Multilinear algebra and tensor numerical methods have a wide range of applications in the fields of data science and scientific computing. Due to the typically large problem sizes in practical settings, efficient methods, which exploit low-rank structures, are essential. In this thesis, we consider an application each in both of these fields. Tensor completion, or imputation of unknown values in partially known multiway data is an important problem, which appears in statistics, mathematical imaging science and data science. Under the assumption of redundancy in the underlying data, this is a well-defined problem and methods of mathematical optimization can be applied to it. Due to the fact that tensors of fixed rank form a Riemannian submanifold of the ambient high-dimensional tensor space, Riemannian optimization is a natural framework for these problems, which is both mathematically rigorous and computationally efficient. We present a novel Riemannian trust-region scheme, which compares favourably with the state of the art on selected application cases and outperforms known methods on some test problems. Optimization problems governed by partial differential equations form an area of scientific computing which has applications in a variety of areas, ranging from physics to financial mathematics. Due to the inherent high dimensionality of optimization problems arising from discretized differential equations, these problems present computational challenges, especially in the case of three or more dimensions. An even more challenging class of optimization problems has operators of integral instead of differential type in the constraint. These operators are nonlocal, and therefore lead to large, dense discrete systems of equations. We present a novel solution method, based on separation of spatial dimensions and provably low-rank approximation of the nonlocal operator. Our approach allows the solution of multidimensional problems with a complexity which is only slightly larger than linear in the univariate grid size; this improves the state of the art for a particular test problem problem by at least two orders of magnitude.

Copositivity in Infinite Dimension (2019)

Adams, Claudia

Many combinatorial optimization problems on finite graphs can be formulated as conic convex programs, e.g. the stable set problem, the maximum clique problem or the maximum cut problem. Especially NP-hard problems can be written as copositive programs. In this case the complexity is moved entirely into the copositivity constraint. Copositive programming is a quite new topic in optimization. It deals with optimization over the so-called copositive cone, a superset of the positive semidefinite cone, where the quadratic form x^T Ax has to be nonnegative for only the nonnegative vectors x. Its dual cone is the cone of completely positive matrices, which includes all matrices that can be decomposed as a sum of nonnegative symmetric vector-vector-products. The related optimization problems are linear programs with matrix variables and cone constraints. However, some optimization problems can be formulated as combinatorial problems on infinite graphs. For example, the kissing number problem can be formulated as a stable set problem on a circle. In this thesis we will discuss how the theory of copositive optimization can be lifted up to infinite dimension. For some special cases we will give applications in combinatorial optimization.

Optimal Control of Partial Integro-Differential Equations and Analysis of the Gaussian Kernel (2018)

Zimmer, Lukas Aaron

An important field of applied mathematics is the simulation of complex financial, mechanical, chemical, physical or medical processes with mathematical models. In addition to the pure modeling of the processes, the simultaneous optimization of an objective function by changing the model parameters is often the actual goal. Models in fields such as finance, biology or medicine benefit from this optimization step. While many processes can be modeled using an ordinary differential equation (ODE), partial differential equations (PDEs) are needed to optimize heat conduction and flow characteristics, spreading of tumor cells in tissue as well as option prices. A partial integro-differential equation (PIDE) is a parital differential equation involving an integral operator, e.g., the convolution of the unknown function with a given kernel function. PIDEs occur for example in models that simulate adhesive forces between cells or option prices with jumps. In each of the two parts of this thesis, a certain PIDE is the main object of interest. In the first part, we study a semilinear PIDE-constrained optimal control problem with the aim to derive necessary optimality conditions. In the second, we analyze a linear PIDE that includes the convolution of the unknown function with the Gaussian kernel.

The Nonlocal Spatial Ramsey Model with Endogenous Productivity Growth (2018)

Somorowsky, Laura Viktoria

The economic growth theory analyses which factors affect economic growth and tries to analyze how it can last. A popular neoclassical growth model is the Ramsey-Cass-Koopmans model, which aims to determine how much of its income a nation or an economy should save in order to maximize its welfare. In this thesis, we present and analyze an extended capital accumulation equation of a spatial version of the Ramsey model, balancing diffusive and agglomerative effects. We model the capital mobility in space via a nonlocal diffusion operator which allows for jumps of the capital stock from one location to an other. Moreover, this operator smooths out heterogeneities in the factor distributions slower, which generated a more realistic behavior of capital flows. In addition to that, we introduce an endogenous productivity-production operator which depends on time and on the capital distribution in space. This operator models the technological progress of the economy. The resulting mathematical model is an optimal control problem under a semilinear parabolic integro-differential equation with initial and volume constraints, which are a nonlocal analog to local boundary conditions, and box-constraints on the state and the control variables. In this thesis, we consider this problem on a bounded and unbounded spatial domain, in both cases with a finite time horizon. We derive existence results of weak solutions for the capital accumulation equations in both settings and we proof the existence of a Ramsey equilibrium in the unbounded case. Moreover, we solve the optimal control problem numerically and discuss the results in the economic context.

A Method for Completely Positive and Nonnegative Matrix Factorization (2018)

Groetzner, Patrick Hermann

A matrix A is called completely positive if there exists an entrywise nonnegative matrix B such that A = BB^T. These matrices can be used to obtain convex reformulations of for example nonconvex quadratic or combinatorial problems. One of the main problems with completely positive matrices is checking whether a given matrix is completely positive. This is known to be NP-hard in general. rnrnFor a given matrix completely positive matrix A, it is nontrivial to find a cp-factorization A=BB^T with nonnegative B since this factorization would provide a certificate for the matrix to be completely positive. But this factorization is not only important for the membership to the completely positive cone, it can also be used to recover the solution of the underlying quadratic or combinatorial problem. In addition, it is not a priori known how many columns are necessary to generate a cp-factorization for the given matrix. The minimal possible number of columns is called the cp-rank of A and so far it is still an open question how to derive the cp-rank for a given matrix. Some facts on completely positive matrices and the cp-rank will be given in Chapter 2. Moreover, in Chapter 6, we will see a factorization algorithm, which, for a given completely positive matrix A and a suitable starting point, computes the nonnegative factorization A=BB^T. The algorithm therefore returns a certificate for the matrix to be completely positive. As introduced in Chapter 3, the fundamental idea of the factorization algorithm is to start from an initial square factorization which is not necessarily entrywise nonnegative, and extend this factorization to a matrix for which the number of columns is greater than or equal to the cp-rank of A. Then it is the goal to transform this generated factorization into a cp-factorization. This problem can be formulated as a nonconvex feasibility problem, as shown in Section 4.1, and solved by a method which is based on alternating projections, as proven in Chapter 6. On the topic of alternating projections, a survey will be given in Chapter 5. Here we will see how to apply this technique to several types of sets like subspaces, convex sets, manifolds and semialgebraic sets. Furthermore, we will see some known facts on the convergence rate for alternating projections between these types of sets. Considering more than two sets yields the so called cyclic projections approach. Here some known facts for subspaces and convex sets will be shown. Moreover, we will see a new convergence result on cyclic projections among a sequence of manifolds in Section 5.4. In the context of cp-factorizations, a local convergence result for the introduced algorithm will be given. This result is based on the known convergence for alternating projections between semialgebraic sets. To obtain cp-facrorizations with this first method, it is necessary to solve a second order cone problem in every projection step, which is very costly. Therefore, in Section 6.2, we will see an additional heuristic extension, which improves the numerical performance of the algorithm. Extensive numerical tests in Chapter 7 will show that the factorization method is very fast in most instances. In addition, we will see how to derive a certificate for the matrix to be an element of the interior of the completely positive cone. As a further application, this method can be extended to find a symmetric nonnegative matrix factorization, where we consider an additional low-rank constraint. Here again, the method to derive factorizations for completely positive matrices can be used, albeit with some further adjustments, introduced in Section 8.1. Moreover, we will see that even for the general case of deriving a nonnegative matrix factorization for a given rectangular matrix A, the key aspects of the completely positive factorization approach can be used. To this end, it becomes necessary to extend the idea of finding a completely positive factorization such that it can be used for rectangular matrices. This yields an applicable algorithm for nonnegative matrix factorization in Section 8.2. Numerical results for this approach will suggest that the presented algorithms and techniques to obtain completely positive matrix factorizations can be extended to general nonnegative factorization problems.

Dynamics of the Taylor Shift on Spaces of Holomorphic Functions (2018)

Thelen, Maike

We will consider discrete dynamical systems (X,T) which consist of a state space X and a linear operator T acting on X. Given a state x in X at time zero, its state at time n is determined by the n-th iteration T^n(x). We are interested in the long-term behaviour of this system, that means we want to know how the sequence (T^n (x))_(n in N) behaves for increasing n and x in X. In the first chapter, we will sum up the relevant definitions and results of linear dynamics. In particular, in topological dynamics the notions of hypercyclic, frequently hypercyclic and mixing operators will be presented. In the setting of measurable dynamics, the most important definitions will be those of weakly and strongly mixing operators. If U is an open set in the (extended) complex plane containing 0, we can define the Taylor shift operator on the space H(U) of functions f holomorphic in U as Tf(z) = (f(z)- f(0))/z if z is not equal to 0 and otherwise Tf(0) = f'(0). In the second chapter, we will start examining the Taylor shift on H(U) endowed with the topology of locally uniform convergence. Depending on the choice of U, we will study whether or not the Taylor shift is weakly or strongly mixing in the Gaussian sense. Next, we will consider Banach spaces of functions holomorphic on the unit disc D. The first section of this chapter will sum up the basic properties of Bergman and Hardy spaces in order to analyse the dynamical behaviour of the Taylor shift on these Banach spaces in the next part. In the third section, we study the space of Cauchy transforms of complex Borel measures on the unit circle first endowed with the quotient norm of the total variation and then with a weak-* topology. While the Taylor shift is not even hypercyclic in the first case, we show that it is mixing for the latter case. In Chapter 4, we will first introduce Bergman spaces A^p(U) for general open sets and provide approximation results which will be needed in the next chapter where we examine the Taylor shift on these spaces on its dynamical properties. In particular, for 1<=p<2 we will find sufficient conditions for the Taylor shift to be weakly mixing or strongly mixing in the Gaussian sense. For p>=2, we consider specific Cauchy transforms in order to determine open sets U such that the Taylor shift is mixing on A^p(U). In both sections, we will illustrate the results with appropriate examples. Finally, we apply our results to universal Taylor series. The results of Chapter 5 about the Taylor shift allow us to consider the behaviour of the partial sums of the Taylor expansion of functions in general Bergman spaces outside its disc of convergence.

Extension Operators with Optimal Continuity Estimates (2018)

Jakobs, Arne

Given a compact set K in R^d, the theory of extension operators examines the question, under which conditions on K, the linear and continuous restriction operators r_n:E^n(R^d)→E^n(K),f↦(∂^α f|_K)_{|α|≤n}, n in N_0 and r:E(R^d)→E(K),f↦(∂^α f|_K)_{α in N_0^d}, have a linear and continuous right inverse. This inverse is called extension operator and this problem is known as Whitney's extension problem, named after Hassler Whitney. In this context, E^n(K) respectively E(K) denote spaces of Whitney jets of order n respectively of infinite order. With E^n(R^d) and E(R^d), we denote the spaces of n-times respectively infinitely often continuously partially differentiable functions on R^d. Whitney already solved the question for finite order completely. He showed that it is always possible to construct a linear and continuous right inverse E_n for r_n. This work is concerned with the question of how the existence of a linear and continuous right inverse of r, fulfilling certain continuity estimates, can be characterized by properties of K. On E(K), we introduce a full real scale of generalized Whitney seminorms (|·|_{s,K})_{s≥0}, where |·|_{s,K} coincides with the classical Whitney seminorms for s in N_0. We equip also E(R^d) with a family (|·|_{s,L})_{s≥0} of those seminorms, where L shall be a a compact set with K in L-°. This family of seminorms on E(R^d) suffices to characterize the continuity properties of an extension operator E, since we can without loss of generality assume that E(E(K)) in D^s(L). In Chapter 2, we introduce basic concepts and summarize the classical results of Whitney and Stein. In Chapter 3, we modify the classical construction of Whitney's operators E_n and show that |E_n(·)|_{s,L}≤C|·|_{s,K} for s in[n,n+1). In Chapter 4, we generalize a result of Frerick, Jordá and Wengenroth and show that LMI(1) for K implies the existence of an extension operator E without loss of derivatives, i.e. we have it fulfils |E(·)|_{s,L}≤C|·|_{s,K} for all s≥0. We show that a large class of self similar sets, which includes the Cantor set and the Sierpinski triangle, admits an extensions operator without loss of derivatives. In Chapter 5 we generalize a result of Frerick, Jordá and Wengenroth and show that WLMI(r) for r≥1 implies the existence of a tame linear extension operator E having a homogeneous loss of derivatives, such that |E(·)|_{s,L}≤C|·|_{(r+ε)s,K} for all s≥0 and all ε>0. In the last chapter we characterize the existence of an extension operator having an arbitrary loss of derivatives by the existence of measures on K.

Modeling, Simulation and Optimization of Wine Fermentation (2018)

Schenk, Christina

Industrial companies mainly aim for increasing their profit. That is why they intend to reduce production costs without sacrificing the quality. Furthermore, in the context of the 2020 energy targets, energy efficiency plays a crucial role. Mathematical modeling, simulation and optimization tools can contribute to the achievement of these industrial and environmental goals. For the process of white wine fermentation, there exists a huge potential for saving energy. In this thesis mathematical modeling, simulation and optimization tools are customized to the needs of this biochemical process and applied to it. Two different models are derived that represent the process as it can be observed in real experiments. One model takes the growth, division and death behavior of the single yeast cell into account. This is modeled by a partial integro-differential equation and additional multiple ordinary integro-differential equations showing the development of the other substrates involved. The other model, described by ordinary differential equations, represents the growth and death behavior of the yeast concentration and development of the other substrates involved. The more detailed model is investigated analytically and numerically. Thereby existence and uniqueness of solutions are studied and the process is simulated. These investigations initiate a discussion regarding the value of the additional benefit of this model compared to the simpler one. For optimization, the process is described by the less detailed model. The process is identified by a parameter and state estimation problem. The energy and quality targets are formulated in the objective function of an optimal control or model predictive control problem controlling the fermentation temperature. This means that cooling during the process of wine fermentation is controlled. Parameter and state estimation with nonlinear economic model predictive control is applied in two experiments. For the first experiment, the optimization problems are solved by multiple shooting with a backward differentiation formula method for the discretization of the problem and a sequential quadratic programming method with a line search strategy and a Broyden-Fletcher-Goldfarb-Shanno update for the solution of the constrained nonlinear optimization problems. Different rounding strategies are applied to the resulting post-fermentation control profile. Furthermore, a quality assurance test is performed. The outcomes of this experiment are remarkable energy savings and tasty wine. For the next experiment, some modifications are made, and the optimization problems are solved by using direct transcription via orthogonal collocation on finite elements for the discretization and an interior-point filter line-search method for the solution of the constrained nonlinear optimization problems. The second experiment verifies the results of the first experiment. This means that by the use of this novel control strategy energy conservation is ensured and production costs are reduced. From now on tasty white wine can be produced at a lower price and with a clearer conscience at the same time.

Quadratic Optimization: Copositive Modelling, Algorithms and Aspects of Duality (2018)

Nguyen, Duy Van

Quadratische Optimierungsprobleme (QP) haben ein breites Anwendungsgebiet, wie beispielsweise kombinatorische Probleme einschließlich des maximalen Cliquenroblems. Motzkin und Straus [25] zeigten die Äquivalenz zwischen dem maximalen Cliquenproblem und dem standard quadratischen Problem. Auch mathematische Statistik ist ein weiteres Anwendungsgebiet von (QP), sowie eine Vielzahl von ökonomischen Modellen basieren auf (QP), z.B. das quadratische Rucksackproblem. In [5] Bomze et al. haben das standard quadratische Optimierungsproblem (StQP) in ein Copositive-Problem umformuliert. Im Folgenden wurden Algorithmen zur Lösung dieses copositiviten Problems von Bomze und de Klerk in [6] und Dür und Bundfuss in [9] entwickelt. Während die Implementierung dieser Algorithmen einige vielversprechende numerische Ergebnisse hervorbrachten, konnten die Autoren nur die copositive Neuformulierung des (StQP)s lösen. In [11] präsentierte Burer eine vollständig positive Umformulierung für allgemeine (QP)s, sogar mit binären Nebenbedingungen. Leider konnte er keine Methode zur Lösung für ein solches vollständig positives Problem präsentieren, noch wurde eine copositive Formulierung vorgeschlagen, auf die man die oben erwähnten Algorithmen modifizieren und anwenden könnte, um diese zu lösen. Diese Arbeit wird einen neuen endlichen Algorithmus zur Lösung eines standard quadratischen Optimierungsproblems aufstellen. Desweiteren werden in dieser Thesis copositve Darstellungen für ungleichungsbeschränkte sowie gleichungsbeschränkte quadratische Optimierungsprobleme vorgestellt. Für den ersten Ansatz wurde eine vollständig positive Umformulierung des (QP) entwickelt. Die copositive Umformulierung konnte durch Betrachtung des dualen Problems des vollständig positiven Problems erhalten werden. Ein direkterer Ansatz wurde gemacht, indem das Lagrange-Duale eines äquivalenten quadratischen Optimierungsproblems betrachtet wurde, das durch eine semidefinite quadratische Nebenbedingung beschränkt wurde. In diesem Zusammenhang werden Bedingungen für starke Dualität vorgeschlagen.

Applications of the Adjoint Method in Stochastic Financial Modelling (2015)

Groß, Bastian Peter

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.

Numerical modelling of the flight behaviour of bats to estimate the collision risk with wind turbines (2018)

Müller, Christina

In this thesis, we present a new approach for estimating the effects of wind turbines for a local bat population. We build an individual based model (IBM) which simulates the movement behaviour of every single bat of the population with its own preferences, foraging behaviour and other species characteristics. This behaviour is normalized by a Monte-Carlo simulation which gives us the average behaviour of the population. The result is an occurrence map of the considered habitat which tells us how often the bat and therefore the considered bat population frequent every region of this habitat. Hence, it is possible to estimate the crossing rate of the position of an existing or potential wind turbine. We compare this individual based approach with a partial differential equation based method. This second approach produces a lower computational effort but, unfortunately, we lose information about the movement trajectories at the same time. Additionally, the PDE based model only gives us a density profile. Hence, we lose the information how often each bat crosses special points in the habitat in one night. In a next step we predict the average number of fatalities for each wind turbine in the habitat, depending on the type of the wind turbine and the behaviour of the considered bat species. This gives us the extra mortality caused by the wind turbines for the local population. This value is used for a population model and finally we can calculate whether the population still grows or if there already is a decline in population size which leads to the extinction of the population. Using the combination of all these models, we are able to evaluate the conflict of wind turbines and bats and to predict the result of this conflict. Furthermore, it is possible to find better positions for wind turbines such that the local bat population has a better chance to survive. Since bats tend to move in swarm formations under certain circumstances, we introduce swarm simulation using partial integro-differential equations. Thereby, we have a closer look at existence and uniqueness properties of solutions.

Optimierung der Materialausrichtung von orthotropen Materialien in Schalenkonstruktionen (2017)

Zorn, Heinz

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.

On the Accuracy of Loader's Algorithm for the Binomial Density and Algorithms for Rectangle Probabilities for Markov Increments (2016)

Dimitriadis, Jannis

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.

Efficient PDE Constrained Shape Optimization in Shape Spaces (2016)

Welker, Kathrin

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 optimal Berry-Esseen constant in the binomial case (2016)

Schulz, Jona

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.

Matching Problems with Additional Resource Constraints (2015)

Thomas, Dirk Johannes

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.

Buehler Confidence Regions and the Comparison of Diagnostic Tests (2015)

Dinev, Todor

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.

Universality of composition operators with applications to complex dynamics (2015)

Jung, Andreas

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 âˆˆ X at time zero, its state at time n âˆˆ 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 âˆˆ X as the time n increases. On the one hand, it is possible that there exist certain states x âˆˆ X such that the system behaves stably, which means that f^n(x) approaches a state of equilibrium for nâ†’âˆž. On the other hand, it might be the case that the system runs unstably for some initial states x âˆˆ 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 (gâˆ˜f^n) on D, where D is an open subset of F_f with f(D)âŠ‚ 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 (gâˆ˜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 âˆˆ I} of continuous functions T_i:Xâ†’Y, an element x âˆˆ X is called universal for the family {T_i: i âˆˆ I} if the set {T_i(x): i âˆˆ 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 âˆˆ X is called hypercyclic for T if it is universal for the family {T^n: n âˆˆ 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)âŠ‚ 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)â†’H(D), C_f(g):=gâˆ˜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 âŠ‚ 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 âˆˆ H(D) which are hypercyclic for C_f. Indeed, for each hypercyclic function g for C_f, the set of compositions {gâˆ˜f^n|_D: n âˆˆ N} is dense in H(D) so that the sequence of compositions (gâˆ˜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 (gâˆ˜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.

Proximität von Binomial- und Poisson-Verteilungen und Konfidenzintervalle für durchschnittliche Erfolgswahrscheinlichkeiten (2015)

Tasto, Christoph

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.

Analytische Eigenschaften doppelt nichtzentraler Verteilungen (2014)

Gessner, Yves

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.

Surjectivity of augmented linear partial differential operators with constant coefficients and a conjecture of Trèves (2012)

Kalmes, Thomas

The main topic of this treatise is the solution of two problems from the general theory of linear partial differential equations with constant coefficients. While surjectivity criteria for linear partial differential operators in spaces of smooth functions over an open subset of euclidean space and distributions were proved by B. Malgrange and L. Hörmander in 1955, respectively 1962, concrete evaluation of these criteria is still a highly non-trivial task. In particular, it is well-known that surjectivity in the space of smooth functions over an open subset of euclidean space does not automatically imply surjectivity in the space of distributions. Though, examples for this fact all live in three or higher dimensions. In 1966, F. Trèves conjectured that in the two dimensional setting surjectivity of a linear partial differential operator on the smooth functions indeed implies surjectivity on the space of distributions. An affirmative solution to this problem is presented in this treatise. The second main result solves the so-called problem of (distributional) parameter dependence for solutions of linear partial differential equations with constant coefficients posed by J. Bonet and P. Domanski in 2006. It is shown that, in dimensions three or higher, this problem in general has a negative solution even for hypoelliptic operators. Moreover, it is proved that the two dimensional case is again an exception, because in this setting the problem of parameter dependence always has a positive solution.

Stable elements in topological algebras (2013)

Riefer, Markus

In a paper of 1996 the british mathematician Graham R. Allan posed the question, whether the product of two stable elements is again stable. Here stability describes the solvability of a certain infinite system of equations. Using a method from the theory of homological algebra, it is proved that in the case of topological algebras with multiplicative webs, and thus in all common locally convex topological algebras that occur in standard analysis, the answer of Allan's question is affirmative.

Splitting theory for PLH spaces (2014)

Dierolf, Bernhard

In splitting theory of locally convex spaces we investigate evaluable characterizations of the pairs (E, X) of locally convex spaces such that each exact sequence 0 -> X -> G -> E -> 0 of locally convex spaces splits, i.e. either X -> G has a continuous linear left inverse or G -> E has a continuous linear right inverse. In the thesis at hand we deal with splitting of short exact sequences of so-called PLH spaces, which are defined as projective limits of strongly reduced spectra of strong duals of Fréchet-Hilbert spaces. This class of locally convex spaces contains most of the spaces of interest for application in the theory of partial differential operators as the space of Schwartz distributions , the space of real analytic functions and various spaces of ultradifferentiable functions and ultradistributions. It also contains non-Schwartz spaces as B(2,k,loc)(Ω) and spaces of smooth and square integrable functions that are not covered by the current theory for PLS spaces. We prove a complete characterizations of the above problem in the case of X being a PLH space and E either being a Fréchet-Hilbert space or a strong dual of one by conditions of type (T ). To this end, we establish the full homological toolbox of Yoneda Ext functors in exact categories for the category of PLH spaces including the long exact sequence, which in particular involves a thorough discussion of the proper concept of exactness. Furthermore, we exhibit the connection to the parameter dependence problem via the Hilbert tensor product for hilbertizable locally convex spaces. We show that the Hilbert tensor product of two PLH spaces is again a PLH space which in particular proves the positive answer to Grothendieck- problème des topologies. In addition to that we give a complete characterization of the vanishing of the first derivative of the functor proj for tensorized PLH spectra if one of the PLH spaces E and X meets some nuclearity assumptions. To apply our results to concrete cases we establish sufficient conditions of (DN)-(Ω) type and apply them to the parameter dependence problem for partial differential operators with constant coefficients on B(2,k,loc)(Ω) spaces as well as to the smooth and square integrable parameter dependence problem. Concluding we give a complete solution of all the problems under consideration for PLH spaces of Köthe type.

Konstruktive und generische Gewinnung universeller Funktionen, 2. geänderte Fassung (2004)

Mayenberger, Daniel

In dieser Dissertation beschäftigen wir uns mit der konstruktiven und generischen Gewinnung universeller Funktionen. Unter einer universellen Funktion verstehen wie dabei eine solche holomorphe Funktion, die in gewissem Sinne ganze Klassen von Funktionen enthält. Die konstruktive Methode beinhaltet die explizite Konstruktion einer universellen Funktion über einen Grenzprozess, etwa als Polynomreihe. Die generische Methode definiert zunächst rein abstrakt die jeweils gewünschte Klasse von universellen Funktionen. Mithilfe des Baireschen Dichtesatzes wird dann gezeigt, dass die Klasse dieser Funktionen nicht nur nichtleer, sondern sogar G_delta und dicht in dem betrachteten Funktionenraum ist. Beide Methoden bedienen sich der Approximationssätze von Runge und von Mergelyan. Die Hauptergebnisse sind die folgenden: (1) Wir haben konstruktiv die Existenz von universellen Laurentreihen auf mehrfach zusammenhängenden Gebieten bewiesen. Zusätzlich haben wir gezeigt, dass die Menge solcher universeller Laurentreihen dicht im Raum der auf dem betrachteten Gebiet holomorphen Funktionen ist. (2) Die Existenz von universellen Faberreihen auf gewissen Gebieten wurde sowohl konstruktiv als auch generisch bewiesen. (3) Zum einen haben wir konstruktiv gezeigt, dass es so genannte ganze T-universelle Funktionen mit vorgegebenen Approximationswegen gibt. Die Approximationswege sind durch eine hinreichend variable funktionale Form vorgegeben. Die Menge solcher Funktionen ist im Raum der ganzen Funktionen eine dichte G_delta-Menge. Zum anderen haben wir generisch die Existenz von auf einem beschränkten Gebiet T-universellen Funktionen bezüglich gewisser vorgegebener Approximationswege bewiesen. Die Approximationswege sind auch hier genügend allgemein.

Discontinuous Galerkin Approaches for HPC Flow Simulations on Stream Processors (2013)

Siebenborn, Martin

This work investigates the industrial applicability of graphics and stream processors in the field of fluid simulations. For this purpose, an explicit Runge-Kutta discontinuous Galerkin method in arbitrarily high order is implemented completely for the hardware architecture of GPUs. The same functionality is simultaneously realized for CPUs and compared to GPUs. Explicit time steppings as well as established implicit methods are under consideration for the CPU. This work aims at the simulation of inviscid, transsonic flows over the ONERA M6 wing. The discontinuities which typically arise in hyperbolic equations are treated with an artificial viscosity approach. It is further investigated how this approach fits into the explicit time stepping and works together with the special architecture of the GPU. Since the treatment of artificial viscosity is close to the simulation of the Navier-Stokes equations, it is reviewed how GPU-accelerated methods could be applied for computing viscous flows. This work is based on a nodal discontinuous Galerkin approach for linear hyperbolic problems. Here, it is extended to non-linear problems, which makes the application of numerical quadrature obligatory. Moreover, the representation of complex geometries is realized using isoparametric mappings. Higher order methods are typically very sensitive with respect to boundaries which are not properly resolved. For this purpose, an approach is presented which fits straight-sided DG meshes to curved geometries which are described by NURBS surfaces. The mesh is modeled as an elastic body and deformed according to the solution of closest point problems in order to minimize the gap to the original spline surface. The sensitivity with respect to geometry representations is reviewed in the end of this work in the context of shape optimization. Here, the aerodynamic drag of the ONERA M6 wing is minimized according to the shape gradient which is implicitly smoothed within the mesh deformation approach. In this context a comparison to the classical Laplace-Beltrami operator is made in a Stokes flow situation.

Hadamard convolution operators on spaces of holomorphic functions (2013)

Lorson, Tobias

The Hadamard product of two holomorphic functions which is defined via a convolution integral constitutes a generalization of the Hadamard product of two power series which is obtained by pointwise multiplying their coefficients. Based on the integral representation mentioned above, an associative law for this convolution is shown. The main purpose of this thesis is the examination of the linear and continuous Hadamard convolution operators. These operators map between spaces of holomorphic functions and send - with a fixed function phi - a function f to the convolution of phi and f. The transposed operator is computed and turns out to be a Hadamard convolution operator, too, mapping between spaces of germs of holomorphic functions. The kernel of Hadamard convolution operators is investigated and necessary and sufficient conditions for those operators to be injective or to have dense range are given. In case that the domain of holomorphy of the function phi allows a Mellin transform of phi, certain (generalized) monomials are identified as eigenfunctions of the corresponding operator. By means of this result and some extract of the theory of growth of entire functions, further propositions concerning the injectivity, the denseness of the range or the surjectivity of Hadamard convolution operators are shown. The relationship between Hadamard convolution operators, operators which are defined via the convolution with an analytic functional and differential operators of infinite order is investigated and the results which are obtained in the thesis are put into the research context. The thesis ends with an application of the results to the approximation of holomorphic functions by lacunary polynomials. On the one hand, the question under which conditions lacunary polynomials are dense in the space of all holomorphic functions is investigated and on the other hand, the rate of approximation is considered. In this context, a result corresponding to the Bernstein-Walsh theorem is formulated.

Some aspects of the optimization over the copositive and the completely positive cone (2013)

Witzel, Julia

Copositive programming is concerned with the problem of optimizing a linear function over the copositive cone, or its dual, the completely positive cone. It is an active field of research and has received a growing amount of attention in recent years. This is because many combinatorial as well as quadratic problems can be formulated as copositive optimization problems. The complexity of these problems is then moved entirely to the cone constraint, showing that general copositive programs are hard to solve. A better understanding of the copositive and the completely positive cone can therefore help in solving (certain classes of) quadratic problems. In this thesis, several aspects of copositive programming are considered. We start by studying the problem of computing the projection of a given matrix onto the copositive and the completely positive cone. These projections can be used to compute factorizations of completely positive matrices. As a second application, we use them to construct cutting planes to separate a matrix from the completely positive cone. Besides the cuts based on copositive projections, we will study another approach to separate a triangle-free doubly nonnegative matrix from the completely positive cone. A special focus is on copositive and completely positive programs that arise as reformulations of quadratic optimization problems. Among those we start by studying the standard quadratic optimization problem. We will show that for several classes of objective functions, the relaxation resulting from replacing the copositive or the completely positive cone in the conic reformulation by a tractable cone is exact. Based on these results, we develop two algorithms for solving standard quadratic optimization problems and discuss numerical results. The methods presented cannot immediately be adapted to general quadratic optimization problems. This is illustrated with examples.

Structural Optimization of Coupled Problems (2013)

Stoffel, Roland

Design and structural optimization has become a very important field in industrial applications over the last years. Due to economical and ecological reasons, the efficient use of material is of highly industrial interest. Therefore, computational tools based on optimization theory have been developed and studied in the last decades. In this work, different structural optimization methods are considered. Special attention lies on the applicability to three-dimensional, large-scale, multiphysic problems, which arise from different areas of the industry. Based on the theory of PDE-constraint optimization, descent methods in structural optimization require knowledge of the (partial) derivatives with respect to shape or topology variations. Therefore, shape and topology sensitivity analysis is introduced and the connection between both sensitivities is given by the Topological-Shape Sensitivity Method. This method leads to a systematic procedure to compute the topological derivative by terms of the shape sensitivity. Due to the framework of moving boundaries in structural optimization, different interface tracking techniques are presented. If the topology of the domain is preserved during the optimization process, explicit interface tracking techniques, combined with mesh-deformation, are used to capture the interface. This techniques fit very well the requirements in classical shape optimization. Otherwise, an implicit representation of the interface is of advantage if the optimal topology is unknown. In this case, the level set method is combined with the concept of the topological derivative to deal with topological perturbation. The resulting methods are applied to different industrial problems. On the one hand, interface shape optimization for solid bodies subject to a transient heat-up phase governed by both linear elasticity and thermal stresses is considered. Therefore, the shape calculus is applied to coupled heat and elasticity problems and a generalized compliance objective function is studied. The resulting thermo-elastic shape optimization scheme is used for compliance reduction of realistic hotplates. On the other hand, structural optimization based on the topological derivative for three-dimensional elasticity problems is observed. In order to comply typical volume constraints, a one-shot augmented Lagrangian method is proposed. Additionally, a multiphase optimization approach based on mesh-refinement is used to reduce the computational costs and the method is illustrated by classical minimum compliance problems. Finally, the topology optimization algorithm is applied to aero-elastic problems and numerical results are presented.

Numerical Optimization in Survey Statistics (2013)

Wagner, Matthias

In der modernen Survey-Statistik treten immer häufifiger Optimierungsprobleme auf, die es zu lösen gilt. Diese sind oft von hoher Dimension und Simulationsstudien erfordern das mehrmalige Lösen dieser Optimierungsprobleme. Um dies in angemessener Zeit durchführen zu können, sind spezielle Algorithmen und Lösungsansätze erforderlich, welche in dieser Arbeit entwickelt und untersucht werden. Bei den Optimierungsproblemen handelt es sich zum einen um Allokationsprobleme zur Bestimmung optimaler Teilstichprobenumfänge. Hierbei werden neben auf einem Nullstellenproblem basierende, stetige Lösungsmethoden auch ganzzahlige, auf der Greedy-Idee basierende Lösungsmethoden untersucht und die sich ergebenden Optimallösungen miteinander verglichen.Zum anderen beschäftigt sich diese Arbeit mit verschiedenen Kalibrierungsproblemen. Hierzu wird ein alternativer Lösungsansatz zu den bisher praktizierten Methoden vorgestellt. Dieser macht das Lösen eines nichtglatten Nullstellenproblemes erforderlich, was mittels desrnnichtglatten Newton Verfahrens erfolgt. Im Zusammenhang mit nichtglatten Optimierungsalgorithmen spielt die Schrittweitensteuerung eine große Rolle. Hierzu wird ein allgemeiner Ansatz zur nichtmonotonen Schrittweitensteuerung bei Bouligand-differenzierbaren Funktionen betrachtet. Neben der klassischen Kalibrierung wird ferner ein Kalibrierungsproblem zur kohärenten Small Area Schätzung unter relaxierten Nebenbedingungen und zusätzlicher Beschränkung der Variation der Designgewichte betrachtet. Dieses Problem lässt sich in ein hochdimensionales quadratisches Optimierungsproblem umwandeln, welches die Verwendung von Lösern für dünn besetzte Optimierungsprobleme erfordert.Die in dieser Arbeit betrachteten numerischen Probleme können beispielsweise bei Zensen auftreten. In diesem Zusammenhang werden die vorgestellten Ansätze abschließend in Simulationsstudien auf eine mögliche Anwendung auf den Zensus 2011 untersucht, die im Rahmen des Zensus-Stichprobenforschungsprojektes untersucht wurden.

Efficient Global Surrogate Models for Responses of Expensive Simulations (2013)

Rosenbaum, Benjamin

In this thesis, global surrogate models for responses of expensive simulations are investigated. Computational fluid dynamics (CFD) have become an indispensable tool in the aircraft industry. But simulations of realistic aircraft configurations remain challenging and computationally expensive despite the sustained advances in computing power. With the demand for numerous simulations to describe the behavior of an output quantity over a design space, the need for surrogate models arises. They are easy to evaluate and approximate quantities of interest of a computer code. Only a few number of evaluations of the simulation are stored for determining the behavior of the response over a whole range of the input parameter domain. The Kriging method is capable of interpolating highly nonlinear, deterministic functions based on scattered datasets. Using correlation functions, distinct sensitivities of the response with respect to the input parameters can be considered automatically. Kriging can be extended to incorporate not only evaluations of the simulation, but also gradient information, which is called gradient-enhanced Kriging. Adaptive sampling strategies can generate more efficient surrogate models. Contrary to traditional one-stage approaches, the surrogate model is built step-by-step. In every stage of an adaptive process, the current surrogate is assessed in order to determine new sample locations, where the response is evaluated and the new samples are added to the existing set of samples. In this way, the sampling strategy learns about the behavior of the response and a problem-specific design is generated. Critical regions of the input parameter space are identified automatically and sampled more densely for reproducing the response's behavior correctly. The number of required expensive simulations is decreased considerably. All these approaches treat the response itself more or less as an unknown output of a black-box. A new approach is motivated by the assumption that for a predefined problem class, the behavior of the response is not arbitrary, but rather related to other instances of the mutual problem class. In CFD, for example, responses of aerodynamic coefficients share structural similarities for different airfoil geometries. The goal is to identify the similarities in a database of responses via principal component analysis and to use them for a generic surrogate model. Characteristic structures of the problem class can be used for increasing the approximation quality in new test cases. Traditional approaches still require a large number of response evaluations, in order to achieve a globally high approximation quality. Validating the generic surrogate model for industrial relevant test cases shows that they generate efficient surrogates, which are more accurate than common interpolations. Thus practical, i.e. affordable surrogates are possible already for moderate sample sizes. So far, interpolation problems were regarded as separate problems. The new approach uses the structural similarities of a mutual problem class innovatively for surrogate modeling. Concepts from response surface methods, variable-fidelity modeling, design of experiments, image registration and statistical shape analysis are connected in an interdisciplinary way. Generic surrogate modeling is not restricted to aerodynamic simulation. It can be applied, whenever expensive simulations can be assigned to a larger problem class, in which structural similarities are expected.

Composition operators witch closed range between spaces of smooth functions (2012)

Kenessey, Nicolas François

One of the main tasks in mathematics is to answer the question whether an equation possesses a solution or not. In the 1940- Thom and Glaeser studied a new type of equations that are given by the composition of functions. They raised the following question: For which functions Î¨ does the equation F(Î¨)=f always have a solution. Of course this question only makes sense if the right hand side f satisfies some a priori conditions like being contained in the closure of the space of all compositions with Î¨ and is easy to answer if F and f are continuous functions. Considering further restrictions to these functions, especially to F, extremely complicates the search for an adequate solution. For smooth functions one can already find deep results by Bierstone and Milman which answer the question in the case of a real-analytic function Î¨. This work contains further results for a different class of functions, namely those Î¨ that are smooth and injective. In the case of a function Î¨ of a single real variable, the question can be fully answered and we give three conditions that are both sufficient and necessary in order for the composition equation to always have a solution. Furthermore one can unify these three conditions to show that they are equivalent to the fact that Î¨ has a locally Hölder-continuous inverse. For injective functions Î¨ of several real variables we give necessary conditions for the composition equation to be solvable. For instance Î¨ should satisfy some form of local distance estimate for the partial derivatives. Under the additional assumption of the Whitney-regularity of the image of Î¨, we can give sufficient conditions for flat functions f on the critical set of Î¨ to possess a solution F(Î¨)=f.

Multigrid Optimization Methods for High Performance Computing (2012)

Wagner, Christian

Optimal control problems are optimization problems governed by ordinary or partial differential equations (PDEs). A general formulation is given byrn \min_{(y,u)} J(y,u) with subject to e(y,u)=0, assuming that e_y^{-1} exists and consists of the three main elements: 1. The cost functional J that models the purpose of the control on the system. 2. The definition of a control function u that represents the influence of the environment of the systems. 3. The set of differential equations e(y,u) modeling the controlled system, represented by the state function y:=y(u) which depends on u. These kind of problems are well investigated and arise in many fields of application, for example robot control, control of biological processes, test drive simulation and shape and topology optimization. In this thesis, an academic model problem of the form \min_{(y,u)} J(y,u):=\min_{(y,u)}\frac{1}{2}\|y-y_d\|^2_{L^2(\Omega)}+\frac{\alpha}{2}\|u\|^2_{L^2(\Omega)} subject to -\div(A\grad y)+cy=f+u in \Omega, y=0 on \partial\Omega and u\in U_{ad} is considered. The objective is tracking type with a given target function y_d and a regularization term with parameter \alpha. The control function u takes effect on the whole domain \Omega. The underlying partial differential equation is assumed to be uniformly elliptic. This problem belongs to the class of linear-quadratic elliptic control problems with distributed control. The existence and uniqueness of an optimal solution for problems of this type is well-known and in a first step, following the paradigm 'first optimize, then discretize', the necessary and sufficient optimality conditions are derived by means of the adjoint equation which ends in a characterization of the optimal solution in form of an optimality system. In a second step, the occurring differential operators are approximated by finite differences and the hence resulting discretized optimality system is solved with a collective smoothing multigrid method (CSMG). In general, there are several optimization methods for solving the optimal control problem: an application of the implicit function theorem leads to so-called black-box approaches where the PDE-constrained optimization problem is transformed into an unconstrained optimization problem and the reduced gradient for these reduced functional is computed via the adjoint approach. Another possibilities are Quasi-Newton methods, which approximate the Hessian by a low-rank update based on gradient evaluations, Krylov-Newton methods or (reduced) SQP methods. The use of multigrid methods for optimization purposes is motivated by its optimal computational complexity, i.e. the number of required computer iterations scales linearly with the number of unknowns and the rate of convergence, which is independent of the grid size. Originally multigrid methods are a class of algorithms for solving linear systems arising from the discretization of partial differential equations. The main part of this thesis is devoted to the investigation of the implementability and the efficiency of the CSMG on commodity graphics cards. GPUs (graphic processing units) are designed for highly parallelizable graphics computations and possess many cores of SIMD-architecture, which are able to outperform the CPU regarding to computational power and memory bandwidth. Here they are considered as prototype for prospective multi-core computers with several hundred of cores. When using GPUs as streamprocessors, two major problems arise: data have to be transferred from the CPU main memory to the GPU main memory, which can be quite slow and the limited size of the GPU main memory. Furthermore, only when the streamprocessors are fully used to capacity, a remarkable speed-up comparing to a CPU is achieved. Therefore, new algorithms for the solution of optimal control problems are designed in this thesis. To this end, a nonoverlapping domain decomposition method is introduced which allows the exploitation of the computational power of many GPUs resp. CPUs in parallel. This algorithm is based on preliminary work for elliptic problems and enhanced for the application to optimal control problems. For the domain decomposition into two subdomains the linear system for the unknowns on the interface is solved with a Schur complement method by using a discrete approximation of the Steklov-Poincare operator. For the academic optimal control problem, the arising capacitance matrix can be inverted analytically. On this basis, two different algorithms for the nonoverlapping domain decomposition for the case of many subdomains are proposed in this thesis: on the one hand, a recursive approach and on the other hand a simultaneous approach. Numerical test compare the performance of the CSMG for the one domain case and the two approaches for the multi-domain case on a GPU and CPU for different variants.

Adaptive Trust-Region POD Methods and their Application in Finance (2012)

Schu, Matthias

Bei der Preisberechnung von Finanzderivaten bieten sogenannte Jump-diffusion-Modelle mit lokaler Volatilität viele Vorteile. Aus mathematischer Sicht jedoch sind sie sehr aufwendig, da die zugehörigen Modellpreise mittels einer partiellen Integro-Differentialgleichung (PIDG) berechnet werden. Wir beschäftigen uns mit der Kalibrierung der Parameter eines solchen Modells. In einem kleinste-Quadrate-Ansatz werden hierzu Marktpreise von europäischen Standardoptionen mit den Modellpreisen verglichen, was zu einem Problem optimaler Steuerung führt. Ein wesentlicher Teil dieser Arbeit beschäftigt sich mit der Lösung der PIDG aus theoretischer und vor allem aus numerischer Sicht. Die durch ein implizites Zeitdiskretisierungsverfahren entstandenen, dicht besetzten Gleichungssysteme werden mit einem präkonditionierten GMRES-Verfahren gelöst, was zu beinahe linearem Aufwand bezüglich Orts- und Zeitdiskretisierung führt. Trotz dieser effizienten Lösungsmethode sind Funktionsauswertungen der kleinste-Quadrate-Zielfunktion immer noch teuer, so dass im Hauptteil der Arbeit Modelle reduzierter Ordnung basierend auf Proper Orthogonal Decomposition Anwendung finden. Lokale a priori Fehlerabschätzungen für die reduzierte Differentialgleichung sowie für die reduzierte Zielfunktion, kombiniert mit einem Trust-Region-Ansatz zur Globalisierung liefern einen effizienten Algorithmus, der die Rechenzeit deutlich verkürzt. Das Hauptresultat der Arbeit ist ein Konvergenzbeweis für diesen Algorithmus für eine weite Klasse von Optimierungsproblemen, in die auch das betrachtete Kalibrierungsproblem fällt.

Some Preconditioners in Optimization with Partial Differential Equations (2013)

Ye, Xuancan

Krylov subspace methods are often used to solve large-scale linear equations arising from optimization problems involving partial differential equations (PDEs). Appropriate preconditioning is vital for designing efficient iterative solvers of this type. This research consists of two parts. In the first part, we compare two different kinds of preconditioners for a conjugate gradient (CG) solver attacking one partial integro-differential equation (PIDE) in finance, both theoretically and numerically. An analysis on mesh independence and rate of convergence of the CG solver is included. The knowledge of preconditioning the PIDE is applied to a relevant optimization problem. The second part aims at developing a new preconditioning technique by embedding reduced order models of nonlinear PDEs, which are generated by proper orthogonal decomposition (POD), into deflated Krylov subspace algorithms in solving corresponding optimization problems. Numerical results are reported for a series of test problems.

Geometry of optimal codebooks and constructive quantization (2012)

Junglen, Stefan

In this thesis, we mainly investigate geometric properties of optimal codebooks for random elements $X$ in a seperable Banach space $E$. Here, for a natural number $ N $ and a random element $X$ , an $N$-optimal codebook is an $ N $-subset in the underlying Banach space $E$ which gives a best approximation to $ X $ in an average sense. We focus on two types of geometric properties: The global growth behaviour (growing in $N$) for a sequence of $N$-optimal codebooks is described by the maximal (quantization) radius and a so-called quantization ball. For many distributions, such as central-symmetric distributions on $R^d$ as well as Gaussian distributions on general Banach spaces, we are able to estimate the asymptotics of the quantization radius as well as the quantization ball. Furthermore, we investigate local properties of optimal codebooks, in particular the local quantization error and the weights of the Voronoi cells induced by an optimal codebook. In the finite-dimensional setting, we are able to proof for many interesting distributions classical conjectures on the asymptotic behaviour of those properties. Finally, we propose a method to construct sequences of asymptotically optimal codebooks for random elements in infinite dimensional Banach spaces and apply this method to construct codebooks for stochastic processes, such as fractional Brownian Motions.

Extension of the Proximal Auxiliary Problem Method using Logarithmic-quadratic Distances - Convergence Theory and Numerical Investigations (2011)

Jager, Christina

Variational inequality problems constitute a common basis to investigate the theory and algorithms for many problems in mathematical physics, in economy as well as in natural and technical sciences. They appear in a variety of mathematical applications like convex programming, game theory and economic equilibrium problems, but also in fluid mechanics, physics of solid bodies and others. Many variational inequalities arising from applications are ill-posed. This means, for example, that the solution is not unique, or that small deviations in the data can cause large deviations in the solution. In such a situation, standard solution methods converge very slowly or even fail. In this case, so-called regularization methods are the methods of choice. They have the advantage that an ill-posed original problem is replaced by a sequence of well-posed auxiliary problems, which have better properties (like, e.g., a unique solution and a better conditionality). Moreover, a suitable choice of the regularization term can lead to unconstrained auxiliary problems that are even equivalent to optimization problems. The development and improvement of such methods are a focus of current research, in which we take part with this thesis. We suggest and investigate a logarithmic-quadratic proximal auxiliary problem (LQPAP) method that combines the advantages of the well-known proximal-point algorithm and the so-called auxiliary problem principle. Its exploration and convergence analysis is one of the main results in this work. The LQPAP method continues the recent developments of regularization methods. It includes different techniques presented in literature to improve the numerical stability: The logarithmic-quadratic distance function constitutes an interior point effect which allows to treat the auxiliary problems as unconstrained ones. Furthermore, outer operator approximations are considered. This simplifies the numerical solution of variational inequalities with multi-valued operators since, for example, bundle-techniques can be applied. With respect to the numerical practicability, inexact solutions of the auxiliary problems are allowed using a summable-error criterion that is easy to implement. As a further advantage of the logarithmic-quadratic distance we verify that it is self-concordant (in the sense of Nesterov/Nemirovskii). This motivates to apply the Newton method for the solution of the auxiliary problems. In the numerical part of the thesis the LQPAP method is applied to linearly constrained, differentiable and nondifferentiable convex optimization problems, as well as to nonsymmetric variational inequalities with co-coercive operators. It can often be observed that the sequence of iterates reaches the boundary of the feasible set before being close to an optimal solution. Against this background, we present the strategy of under-relaxation, which robustifies the LQPAP method. Furthermore, we compare the results with an appropriate method based on Bregman distances (BrPAP method). For differentiable, convex optimization problems we describe the implementation of the Newton method to solve the auxiliary problems and carry out different numerical experiments. For example, an adaptive choice of the initial regularization parameter and a combination of an Armijo and a self-concordance step size are evaluated. Test examples for nonsymmetric variational inequalities are hardly available in literature. Therefore, we present a geometric and an analytic approach to generate test examples with known solution(s). To solve the auxiliary problems in the case of nondifferentiable, convex optimization problems we apply the well-known bundle technique. The implementation is described in detail and the involved functions and sequences of parameters are discussed. As far as possible, our analysis is substantiated by new theoretical results. Furthermore, it is explained in detail how the bundle auxiliary problems are solved with a primal-dual interior point method. Such investigations have by now only been published for Bregman distances. The LQPAP bundle method is again applied to several test examples from literature. Thus, this thesis builds a bridge between theoretical and numerical investigations of solution methods for variational inequalities.

510 Mathematik

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