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- Mathematik (65) (entfernen)
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.
Diese Arbeit beschäftigt sich mit (frequent) universellen Funktionen bezüglich Differentialoperatoren und gewichteten Shiftoperatoren. Hierbei wird ein Charakteristikum von Funktionen vom Exponentialtyp untersucht, das bisher im Rahmen der Universalität noch nicht betrachtet wurde: Das konjugierte Indikatordiagramm. Dabei handelt es sich um eine kompakte und konvexe Menge, die einer Funktion vom Exponentialtyp zugeordnet ist und gewisse Rückschlüsse über das Wachstum und die mögliche Nullstellenverteilung zulässt. Mittels einer speziellen Transformation werden (frequent) universelle Funktionen vom Exponentialtyp bezüglich verschiedener Differentialoperatoren ineinander überführt. Hierdurch ist eine genaue Lokalisation der konjugierten Indikatordiagramme möglicher (frequent) universeller Funktionen für diese Operatoren ableitbar. Durch Konjugation der Differentiation mit gewichteten Shiftoperatoren über das Hadamardprodukt, wird auch für diese Operatoren eine Lokalisation möglicher konjugierter Indikatordiagramme ihrer (frequent) universellen Funktionen erreicht.
This thesis introduces a calibration problem for financial market models based on a Monte Carlo approximation of the option payoff and a discretization of the underlying stochastic differential equation. It is desirable to benefit from fast deterministic optimization methods to solve this problem. To be able to achieve this goal, possible non-differentiabilities are smoothed out with an appropriately chosen twice continuously differentiable polynomial. On the basis of this so derived calibration problem, this work is essentially concerned about two issues. First, the question occurs, if a computed solution of the approximating problem, derived by applying Monte Carlo, discretizing the SDE and preserving differentiability is an approximation of a solution of the true problem. Unfortunately, this does not hold in general but is linked to certain assumptions. It will turn out, that a uniform convergence of the approximated objective function and its gradient to the true objective and gradient can be shown under typical assumptions, for instance the Lipschitz continuity of the SDE coefficients. This uniform convergence then allows to show convergence of the solutions in the sense of a first order critical point. Furthermore, an order of this convergence in relation to the number of simulations, the step size for the SDE discretization and the parameter controlling the smooth approximation of non-differentiabilites will be shown. Additionally the uniqueness of a solution of the stochastic differential equation will be analyzed in detail. Secondly, the Monte Carlo method provides only a very slow convergence. The numerical results in this thesis will show, that the Monte Carlo based calibration indeed is feasible if one is concerned about the calculated solution, but the required calculation time is too long for practical applications. Thus, techniques to speed up the calibration are strongly desired. As already mentioned above, the gradient of the objective is a starting point to improve efficiency. Due to its simplicity, finite differences is a frequently chosen method to calculate the required derivatives. However, finite differences is well known to be very slow and furthermore, it will turn out, that there may also occur severe instabilities during optimization which may lead to the break down of the algorithm before convergence has been reached. In this manner a sensitivity equation is certainly an improvement but suffers unfortunately from the same computational effort as the finite difference method. Thus, an adjoint based gradient calculation will be the method of choice as it combines the exactness of the derivative with a reduced computational effort. Furthermore, several other techniques will be introduced throughout this thesis, that enhance the efficiency of the calibration algorithm. A multi-layer method will be very effective in the case, that the chosen initial value is not already close to the solution. Variance reduction techniques are helpful to increase accuracy of the Monte Carlo estimator and thus allow for fewer simulations. Storing instead of regenerating the random numbers required for the Brownian increments in the SDE will be efficient, as deterministic optimization methods anyway require to employ the identical random sequence in each function evaluation. Finally, Monte Carlo is very well suited for a parallelization, which will be done on several central processing units (CPUs).
The subject of this thesis is a homological approach to the splitting theory of PLS-spaces, i.e. to the question for which topologically exact short sequences 0->X->Y->Z->0 of PLS-spaces X,Y,Z the right-hand map admits a right inverse. We show that the category (PLS) of PLS-spaces and continuous linear maps is an additive category in which every morphism admits a kernel and a cokernel, i.e. it is pre-abelian. However, we also show that it is neither quasi-abelian nor semi-abelian. As a foundation for our homological constructions we show the more general result that every pre-abelian category admits a largest exact structure in the sense of Quillen. In the pre-abelian category (PLS) this exact structure consists precisely of the topologically exact short sequences of PLS-spaces. Using a construction of Ext-functors due to Yoneda, we show that one can define for each PLS-space A and every natural number k the k-th abelian-group valued covariant and contravariant Ext-functors acting on the category (PLS) of PLS-spaces, which induce for every topologically exact short sequence of PLS-spaces a long exact sequence of abelian groups and group morphisms. These functors are studied in detail and we establish a connection between the Ext-functors of PLS-spaces and the Ext-functors for LS-spaces. Through this connection we arrive at an analogue of a result for Fréchet spaces which connects the first derived functor of the projective limit with the first Ext-functor and also gives sufficient conditions for the vanishing of the higher Ext-functors. Finally, we show that Ext^k(E,F) = 0 for a k greater or equal than 1, whenever E is a closed subspace and F is a Hausdorff-quotient of the space of distributions, which generalizes a result of Wengenroth that is itself a generalization of results due to Domanski and Vogt.
Large scale non-parametric applied shape optimization for computational fluid dynamics is considered. Treating a shape optimization problem as a standard optimal control problem by means of a parameterization, the Lagrangian usually requires knowledge of the partial derivative of the shape parameterization and deformation chain with respect to input parameters. For a variety of reasons, this mesh sensitivity Jacobian is usually quite problematic. For a sufficiently smooth boundary, the Hadamard theorem provides a gradient expression that exists on the surface alone, completely bypassing the mesh sensitivity Jacobian. Building upon this, the gradient computation becomes independent of the number of design parameters and all surface mesh nodes are used as design unknown in this work, effectively allowing a free morphing of shapes during optimization. Contrary to a parameterized shape optimization problem, where a smooth surface is usually created independently of the input parameters by construction, regularity is not preserved automatically in the non-parametric case. As part of this work, the shape Hessian is used in an approximative Newton method, also known as Sobolev method or gradient smoothing, to ensure a certain regularity of the updates, and thus a smooth shape is preserved while at the same time the one-shot optimization method is also accelerated considerably. For PDE constrained shape optimization, the Hessian usually is a pseudo-differential operator. Fourier analysis is used to identify the operator symbol both analytically and discretely. Preconditioning the one-shot optimization by an appropriate Hessian symbol is shown to greatly accelerate the optimization. As the correct discretization of the Hadamard form usually requires evaluating certain surface quantities such as tangential divergence and curvature, special attention is also given to discrete differential geometry on triangulated surfaces for evaluating shape gradients and Hessians. The Hadamard formula and Hessian approximations are applied to a variety of flow situations. In addition to shape optimization of internal and external flows, major focus lies on aerodynamic design such as optimizing two dimensional airfoils and three dimensional wings. Shock waves form when the local speed of sound is reached, and the gradient must be evaluated correctly at discontinuous states. To ensure proper shock resolution, an adaptive multi-level optimization of the Onera M6 wing is conducted using more than 36, 000 shape unknowns on a standard office workstation, demonstrating the applicability of the shape-one-shot method to industry size problems.