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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.

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.

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.

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.