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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.
Optimal Control of Partial Integro-Differential Equations and Analysis of the Gaussian Kernel
(2018)
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.
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.
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.
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.
In this thesis, we investigate the quantization problem of Gaussian measures on Banach spaces by means of constructive methods. That is, for a random variable X and a natural number N, we are searching for those N elements in the underlying Banach space which give the best approximation to X in the average sense. We particularly focus on centered Gaussians on the space of continuous functions on [0,1] equipped with the supremum-norm, since in that case all known methods failed to achieve the optimal quantization rate for important Gauss-processes. In fact, by means of Spline-approximations and a scheme based on the Best-Approximations in the sense of the Kolmogorov n-width we were able to attain the optimal rate of convergence to zero for these quantization problems. Moreover, we established a new upper bound for the quantization error, which is based on a very simple criterion, the modulus of smoothness of the covariance function. Finally, we explicitly constructed those quantizers numerically.
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.
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.
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.
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.