This thesis addresses optimal control problems with elasticity for large deformations. A hyperelastic model with a polyconvex energy density is employed to describe the elastic behavior of a body. The two approaches to derive the nonlinear partial differential equation, a balance of forces and an energy minimization, are compared. Besides the conventional volume and boundary loads, two novel internal loads are presented. Furthermore, curvilinear coordinates and a hierarchical plate model can be incorporated into the formulation of the elastic forward problem. The forward problem can be solved with Newton\\\'s method, though a globalization technique should be used to avoid divergence of Newton\\\'s method. The repeated solution of the Newton system is done by a CG or MinRes method with a multigrid V-cycle as a preconditioner. The optimal control problem consists of the displacement (as the state) and a load (as the control). Besides the standard tracking-type objective, alternative objective functionals are presented for problems where a reasonable desired state cannot be provided. Two methods are proposed to solve the optimal control problem: an all-at-once approach by a Lagrange-Newton method and a reduced formulation by a quasi-Newton method with an inverse limited-memory BFGS update. The algorithms for the solution of the forward problem and the optimal control problem are implemented in the finite-element software FEniCS, with the geometrical multigrid extension FMG. Numerical experiments are performed to demonstrate the mesh independence of the algorithms and both optimization methods.

We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1 Chebyshev lattices based on a component-by-component approach. Moreover, we give a method for the fast, exact and stable reconstruction.

Der Schwerpunkt der vorliegenden Arbeit liegt auf Methoden zur Lösung nichtlinearer Optimierungsprobleme mit der Anforderung, jedes globale Optimum garantiert zu finden und mit einer im Voraus festgesetzten Genauigkeit zu approximieren. Eng verbunden mit dieser deterministischen Optimierung ist die Berechnung von Schranken für den Wertebereich einer Funktion über einem gegebenen Hyperquader. Verschiedene Ansätze, z. B. auf Basis der Intervallarithmetik, werden vorgestellt und analysiert. Im Besonderen werden Methoden zur Schrankengenerierung für multivariate ganz- und gebrochenrationale Polynome mit Hilfe der Darstellung in der Basis der Bernsteinpolynome weiterentwickelt. Weiterhin werden in der Arbeit schrittweise die Bausteine eines deterministischen Optimierungsverfahrens unter Verwendung der berechneten Wertebereichsschranken beschrieben und Besonderheiten für die Optimierung polynomialer Aufgaben näher untersucht. Die Analyse und Bearbeitung einer Aufgabenstellung aus dem Entwicklungsprozess für Fahrzeuggetriebe zeigt, wie die erarbeiteten Ansätze zur Lösung nichtlinearer Optimierungsprobleme die Suche nach energieeffizienten Getrieben mit einer optimalen Struktur unterstützen kann. Kontakt zum Autor: [Nachname] [.] [Vorname] [@] gmx [.] de

In this thesis a theory for large deformation of plates is presented. Herein aspects of the common 3D-theory for large deformation with the Kirchhoff hypothesis for reducing the dimension from 3D to 2D is combined. Even though the Kirchhoff assumption was developed for small strain and linear material laws, the deformation of thin plates made of isotropic non-linear material was investigated in a numerical experiment. Finally a heavily deformed shell without any change in thickness arises. This way of modeling leads to a two-dimensional strain tensor essentially depending on the first two fundamental forms of the deformed mid surface. Minimizing the resulting deformation energy one ends up with a nonlinear equation system defining the unknown displacement vector U. The aim of this thesis was to apply the incremental Newton technique with a conformal, C^1-continuous finite element discretization. For this the computation of the second derivative of the energy functional is the key difficulty and the most time consuming part of the algorithm. The practicability and fast convergence are demonstrated by different numerical experiments.

We present a MPI based software library for computing the fast Fourier transforms on massively parallel, distributed memory architectures. Similar to established transpose FFT algorithms, we propose a parallel FFT framework that is based on a combination of local FFTs, local data permutations and global data transpositions. This framework can be generalized to arbitrary multi-dimensional data and process meshes. All performance relevant building blocks can be implemented with the help of the FFTW software library. Therefore, our library offers great flexibility and portable performance. Likewise FFTW, we are able to compute FFTs of complex data, real data and even- or odd-symmetric real data. All the transforms can be performed completely in place. Furthermore, we propose an algorithm to calculate pruned FFTs more efficiently on distributed memory architectures. For example, we provide performance measurements of FFTs of size 512^3 and 1024^3 up to 262144 cores on a BlueGene/P architecture.

In der vorliegenden Arbeit wird ein neues Tensorformat vorgestellt und eingehend analysiert. Das hierarchische Format verwendet einen binären Baum, um den Tensorraum der Ordnung d mit einer geschachtelten Unterraumstruktur zu versehen. Der Speicheraufwand für diese Darstellung ist von der Größenordnung O(dnr + dr^3), wobei n den Speicheraufwand in den Ansatzräumen kennzeichnet und r ein Rangparameter ist, der durch die Dimensionen der geschachtelten Unterräume bestimmt wird. Das hierarchische Format umfasst verschiedene Standardformate zur Tensordarstellung wie das kanonische oder r-Term-Format und die Unterraum-/Tucker-Darstellung. Die in dieser Arbeit entwickelte zugehörige Arithmetik inklusive mehrerer Approximationsmethoden basiert auf stabilen Methoden der Linearen Algebra, insbesondere die Singulärwertzerlegung und die QR-Zerlegung sind von zentraler Bedeutung. Die rechnerische Komplexität ist hierbei O(dnr^2+dr^4). Die lineare Abhängigkeit von der Ordnung d des Tensorraumes ist hervorzuheben. Für die verschiedenen Approximationsmethoden, deren Effizienz und Effektivität für die Anwendbarkeit des neuen Formates entscheidend sind, werden qualitative und quantitative Fehlerabschätzungen gezeigt. Umfassende numerische Experimente mit einem Fokus auf den Approximationsmethoden bestätigen zum einen die theoretischen Resultate und belegen die Stärken der neuen Tensordarstellung, zeigen aber zum anderen auch weitere, eher überraschende positive Eigenschaften der mit FastHOSVD bezeichneten schnellsten Kürzungsmethode. / In this dissertation we present and a new format for the representation of tensors and analyse its properties. The hierarchical format uses a binary tree in order to define a hierarchical structure of nested subspaces in the tensor space of order d. The strorage requirements are O(dnr+dr^3) where n is determined by the storage requirements in the ansatz spaces and r is a rank parameter determined by the dimensions of the nested subspaces. The hierarchichal representation contains the standard representation like canonical or r-term representation and subspace or Tucker representation. The arithmetical operations that have been developed in this work, including several approximation methods, are based on stable Linear Alebra methods, especially the singular value decomposition (SVD) and the QR decomposition are of importance. The computational complexity is O(dnr^2+dr^4). The linear dependence from the order d of the tensor space is important. The approximation methods are one of the key ingredients for the applicability of the new format and we present qualitative and quantitative error estimates. Numerical experiments approve the theoretical results and show some additional, but unexpected positive aspects of the fastest method called FastHOSVD.

Diese Arbeit behandelt die effiziente Vorkonditionierung von Finite-Elemente-Matrizen unter Verwendung hierarchischer Matrizen.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.