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- Mathematik (65) (entfernen)
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 Ψ does the equation F(Ψ)=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 Ψ 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 Ψ. This work contains further results for a different class of functions, namely those Ψ that are smooth and injective. In the case of a function Ψ 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 Ψ has a locally Hölder-continuous inverse. For injective functions Ψ of several real variables we give necessary conditions for the composition equation to be solvable. For instance Ψ should satisfy some form of local distance estimate for the partial derivatives. Under the additional assumption of the Whitney-regularity of the image of Ψ, we can give sufficient conditions for flat functions f on the critical set of Ψ to possess a solution F(Ψ)=f.
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.
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.
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.
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.
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.
Das erste Beispiel einer so genannten universellen holomorphen Funktion stammt von Birkhoff, welcher im Jahre 1929 die Existenz einer ganzen Funktion beweisen konnte, die gewissermaßen jede ganze Funktion durch geeignete Translationen approximieren kann. In der Folgezeit hat sich der Bereich der "universellen Approximation" zu einem eigenständigen Gebiet innerhalb der komplexen Approximationstheorie entwickelt, und es gibt eine Vielzahl an Ergebnissen über universelle Funktionen. Hierbei wurde sich allerdings fast ausschließlich auf das Studium holomorpher und ganzer Funktionen beschränkt, insbesondere die Klasse der meromorphen Funktionen wurde bisher kaum auf das Phänomen der Universalität hin untersucht. Die vorliegende Arbeit beschäftigt sich mit universeller meromorpher Approximation, und geht der Fragestellung nach, ob meromorphe Funktionen mit gewissen Universalitätseigenschaften existieren, und ob die klassischen Ergebnisse aus der universellen holomorphen Approximation auf den meromorphen Fall erweiterbar sind. Hierbei wird zunächst zwischen Translations- und Streckungsuniversalität unterschieden und bewiesen, dass in beiden Fällen jeweils eine im Raum der meromorphen Funktionen residuale Menge an universellen Funktionen existiert. Weiterhin werden die Eigenschaften dieser Funktionen ausführlich studiert. Anschließend werden meromorphe Funktionen auf Ableitungsuniversalität hin untersucht. Hierbei wird einerseits gezeigt, dass im Allgemeinen keine positiven Ergebnisse möglich sind, während andererseits eine spezielle Klasse meromorpher Funktionen betrachtet wird, für welche universelles Verhalten der sukzessiven Ableitungen nachgewiesen werden kann.
Recently, optimization has become an integral part of the aerodynamic design process chain. However, because of uncertainties with respect to the flight conditions and geometrical uncertainties, a design optimized by a traditional design optimization method seeking only optimality may not achieve its expected performance. Robust optimization deals with optimal designs, which are robust with respect to small (or even large) perturbations of the optimization setpoint conditions. The resulting optimization tasks become much more complex than the usual single setpoint case, so that efficient and fast algorithms need to be developed in order to identify, quantize and include the uncertainties in the overall optimization procedure. In this thesis, a novel approach towards stochastic distributed aleatory uncertainties for the specific application of optimal aerodynamic design under uncertainties is presented. In order to include the uncertainties in the optimization, robust formulations of the general aerodynamic design optimization problem based on probabilistic models of the uncertainties are discussed. Three classes of formulations, the worst-case, the chance-constrained and the semi-infinite formulation, of the aerodynamic shape optimization problem are identified. Since the worst-case formulation may lead to overly conservative designs, the focus of this thesis is on the chance-constrained and semi-infinite formulation. A key issue is then to propagate the input uncertainties through the systems to obtain statistics of quantities of interest, which are used as a measure of robustness in both robust counterparts of the deterministic optimization problem. Due to the highly nonlinear underlying design problem, uncertainty quantification methods are used in order to approximate and consequently simplify the problem to a solvable optimization task. Computationally demanding evaluations of high dimensional integrals resulting from the direct approximation of statistics as well as from uncertainty quantification approximations arise. To overcome the curse of dimensionality, sparse grid methods in combination with adaptive refinement strategies are applied. The reduction of the number of discretization points is an important issue in the context of robust design, since the computational effort of the numerical quadrature comes up in every iteration of the optimization algorithm. In order to efficiently solve the resulting optimization problems, algorithmic approaches based on multiple-setpoint ideas in combination with one-shot methods are presented. A parallelization approach is provided to overcome the amount of additional computational effort involved by multiple-setpoint optimization problems. Finally, the developed methods are applied to 2D and 3D Euler and Navier-Stokes test cases verifying their industrial usability and reliability. Numerical results of robust aerodynamic shape optimization under uncertain flight conditions as well as geometrical uncertainties are presented. Further, uncertainty quantification methods are used to investigate the influence of geometrical uncertainties on quantities of interest in a 3D test case. The results demonstrate the significant effect of uncertainties in the context of aerodynamic design and thus the need for robust design to ensure a good performance in real life conditions. The thesis proposes a general framework for robust aerodynamic design attacking the additional computational complexity of the treatment of uncertainties, thus making robust design in this sense possible.
Extension of inexact Kleinman-Newton methods to a general monotonicity preserving convergence theory
(2011)
The thesis at hand considers inexact Newton methods in combination with algebraic Riccati equation. A monotone convergence behaviour is proven, which enables a non-local convergence. Above relation is transferred to a general convergence theory for inexact Newton methods securing the monotonicity of the iterates for convex or concave mappings. Several application prove the pratical benefits of the new developed theory.
Die Ménage-Polynome (engl.: ménage hit polynomials) ergeben sich in natürlicher Weise aus den in der Kombinatorik auftretenden Ménage-Zahlen. Eine Verbindung zu einer gewissen Klasse hypergeometrischer Polynome führt auf die Untersuchung spezieller Folgen von Polynomen vom Typ 3-F-1. Unter Verwendung einer Modifikation der komplexen Laplace-Methode zur gleichmäßigen asymptotischen Auswertung von Parameterintegralen sowie einiger Hilfsmittel aus der Potentialtheorie der komplexen Ebene werden starke und schwache Asymptotiken für die in Rede stehenden Polynomfolgen hergeleitet.