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

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

Lightweight structures became more and more important over the last years. One special class of such structures are short fibre reinforced composites, produced by injection moulding. To avoid expensive experiments for testing the mechanical behaviour of these composites proper material models are needed. Thereby, the stochastic nature of the fibre orientation is the main problem. In this thesis it is looked onto the simulation of such materials in a linear thermoelastic setting. This means the material is described by its heat conduction tensor κ(p), its thermal expansion tensor T(p), and its stiffness tensor C(p). Due to the production process the internal fibre orientation p has to been understood as random variable. As a consequence the previously mentioned material quantities also become random. The classical approach is to average these quantities and solve the linear hermoelastic deformation problem with the averaged expressions. Within this thesis the incorpora- tion of this approach in a time and memory efficient manner in an existing finite element software is shown. Especially for the time and memory efficient improvement several implementation aspects of the underlying software are highlighted. For both - the classical material simulation as well as the time efficient improvement of the software - numerical results are shown. Furthermore, the aforementioned classical approach is extended within this thesis for the simulation of the thermal stresses by using the stochastic nature of the heat conduc tion. This is done by developing it into a series w.r.t. the underlying stochastic. For this series known results from uncertainty quantification are applied. With the help of these results the temperature is developed in a Taylor series. For this Taylor series a suitable expansion point is chosen. Afterwards, this series is incorporated into the computation of the thermal stresses. The advantage of this approach is shown in numerical experiments.

This thesis is on the numerical computation of eigenvalues of symmetric hierarchical matrices. The numerical algorithms used for this computation are derivations of the LR Cholesky algorithm, the preconditioned inverse iteration, and a bisection method based on LDLT factorizations. The investigation of QR decompositions for H-matrices leads to a new QR decomposition. It has some properties that are superior to the existing ones, which is shown by experiments using the HQR decompositions to build a QR (eigenvalue) algorithm for H-matrices does not progress to a more efficient algorithm than the LR Cholesky algorithm. The implementation of the LR Cholesky algorithm for hierarchical matrices together with deflation and shift strategies yields an algorithm that require O(n) iterations to find all eigenvalues. Unfortunately, the local ranks of the iterates show a strong growth in the first steps. These H-fill-ins makes the computation expensive, so that O(n³) flops and O(n²) storage are required. Theorem 4.3.1 explains this behavior and shows that the LR Cholesky algorithm is efficient for the simple structured Hl-matrices. There is an exact LDLT factorization for Hl-matrices and an approximate LDLT factorization for H-matrices in linear-polylogarithmic complexity. This factorizations can be used to compute the inertia of an H-matrix. With the knowledge of the inertia for arbitrary shifts, one can compute an eigenvalue by bisectioning. The slicing the spectrum algorithm can compute all eigenvalues of an Hl-matrix in linear-polylogarithmic complexity. A single eigenvalue can be computed in O(k²n log^4 n). Since the LDLT factorization for general H-matrices is only approximative, the accuracy of the LDLT slicing algorithm is limited. The local ranks of the LDLT factorization for indefinite matrices are generally unknown, so that there is no statement on the complexity of the algorithm besides the numerical results in Table 5.7. The preconditioned inverse iteration computes the smallest eigenvalue and the corresponding eigenvector. This method is efficient, since the number of iterations is independent of the matrix dimension. If other eigenvalues than the smallest are searched, then preconditioned inverse iteration can not be simply applied to the shifted matrix, since positive definiteness is necessary. The squared and shifted matrix (M-mu I)² is positive definite. Inner eigenvalues can be computed by the combination of folded spectrum method and PINVIT. Numerical experiments show that the approximate inversion of (M-mu I)² is more expensive than the approximate inversion of M, so that the computation of the inner eigenvalues is more expensive. We compare the different eigenvalue algorithms. The preconditioned inverse iteration for hierarchical matrices is better than the LDLT slicing algorithm for the computation of the smallest eigenvalues, especially if the inverse is already available. The computation of inner eigenvalues with the folded spectrum method and preconditioned inverse iteration is more expensive. The LDLT slicing algorithm is competitive to H-PINVIT for the computation of inner eigenvalues. In the case of large, sparse matrices, specially tailored algorithms for sparse matrices, like the MATLAB function eigs, are more efficient. If one wants to compute all eigenvalues, then the LDLT slicing algorithm seems to be better than the LR Cholesky algorithm. If the matrix is small enough to be handled in dense arithmetic (and is not an Hl(1)-matrix), then dense eigensolvers, like the LAPACK function dsyev, are superior. The H-PINVIT and the LDLT slicing algorithm require only an almost linear amount of storage. They can handle larger matrices than eigenvalue algorithms for dense matrices. For Hl-matrices of local rank 1, the LDLT slicing algorithm and the LR Cholesky algorithm need almost the same time for the computation of all eigenvalues. For large matrices, both algorithms are faster than the dense LAPACK function dsyev.:List of Figures xi List of Tables xiii List of Algorithms xv List of Acronyms xvii List of Symbols xix Publications xxi 1 Introduction 1 1.1 Notation 2 1.2 Structure of this Thesis 3 2 Basics 5 2.1 Linear Algebra and Eigenvalues 6 2.1.1 The Eigenvalue Problem 7 2.1.2 Dense Matrix Algorithms 9 2.2 Integral Operators and Integral Equations 14 2.2.1 Definitions 14 2.2.2 Example - BEM 16 2.3 Introduction to Hierarchical Arithmetic 17 2.3.1 Main Idea 17 2.3.2 Definitions 19 2.3.3 Hierarchical Arithmetic 24 2.3.4 Simple Hierarchical Matrices (Hl-Matrices) 30 2.4 Examples 33 2.4.1 FEM Example 33 2.4.2 BEM Example 36 2.4.3 Randomly Generated Examples 37 2.4.4 Application Based Examples 38 2.4.5 One-Dimensional Integral Equation 38 2.5 Related Matrix Formats 39 2.5.1 H2-Matrices 40 2.5.2 Diagonal plus Semiseparable Matrices 40 2.5.3 Hierarchically Semiseparable Matrices 42 2.6 Review of Existing Eigenvalue Algorithms 44 2.6.1 Projection Method 44 2.6.2 Divide-and-Conquer for Hl(1)-Matrices 45 2.6.3 Transforming Hierarchical into Semiseparable Matrices 46 2.7 Compute Cluster Otto 47 3 QR Decomposition of Hierarchical Matrices 49 3.1 Introduction 49 3.2 Review of Known QR Decompositions for H-Matrices 50 3.2.1 Lintner’s H-QR Decomposition 50 3.2.2 Bebendorf’s H-QR Decomposition 52 3.3 A new Method for Computing the H-QR Decomposition 54 3.3.1 Leaf Block-Column 54 3.3.2 Non-Leaf Block Column 56 3.3.3 Complexity 57 3.3.4 Orthogonality 60 3.3.5 Comparison to QR Decompositions for Sparse Matrices 61 3.4 Numerical Results 62 3.4.1 Lintner’s H-QR decomposition 62 3.4.2 Bebendorf’s H-QR decomposition 66 3.4.3 The new H-QR decomposition 66 3.5 Conclusions 67 4 QR-like Algorithms for Hierarchical Matrices 69 4.1 Introduction 70 4.1.1 LR Cholesky Algorithm 70 4.1.2 QR Algorithm 70 4.1.3 Complexity 71 4.2 LR Cholesky Algorithm for Hierarchical Matrices 72 4.2.1 Algorithm 72 4.2.2 Shift Strategy 72 4.2.3 Deflation 73 4.2.4 Numerical Results 73 4.3 LR Cholesky Algorithm for Diagonal plus Semiseparable Matrices 75 4.3.1 Theorem 75 4.3.2 Application to Tridiagonal and Band Matrices 79 4.3.3 Application to Matrices with Rank Structure 79 4.3.4 Application to H-Matrices 80 4.3.5 Application to Hl-Matrices 82 4.3.6 Application to H2-Matrices 83 4.4 Numerical Examples 84 4.5 The Unsymmetric Case 84 4.6 Conclusions 88 5 Slicing the Spectrum of Hierarchical Matrices 89 5.1 Introduction 89 5.2 Slicing the Spectrum by LDLT Factorization 91 5.2.1 The Function nu(M − µI) 91 5.2.2 LDLT Factorization of Hl-Matrices 92 5.2.3 Start-Interval [a, b] 96 5.2.4 Complexity 96 5.3 Numerical Results 97 5.4 Possible Extensions 100 5.4.1 LDLT Slicing Algorithm for HSS Matrices 103 5.4.2 LDLT Slicing Algorithm for H-Matrices 103 5.4.3 Parallelization 105 5.4.4 Eigenvectors 107 5.5 Conclusions 107 6 Computing Eigenvalues by Vector Iterations 109 6.1 Power Iteration 109 6.1.1 Power Iteration for Hierarchical Matrices 110 6.1.2 Inverse Iteration 111 6.2 Preconditioned Inverse Iteration for Hierarchical Matrices 111 6.2.1 Preconditioned Inverse Iteration 113 6.2.2 The Approximate Inverse of an H-Matrix 115 6.2.3 The Approximate Cholesky Decomposition of an H-Matrix 116 6.2.4 PINVIT for H-Matrices 117 6.2.5 The Interior of the Spectrum 120 6.2.6 Numerical Results 123 6.2.7 Conclusions 130 7 Comparison of the Algorithms and Numerical Results 133 7.1 Theoretical Comparison 133 7.2 Numerical Comparison 135 8 Conclusions 141 Theses 143 Bibliography 145 Index 153

We describe an implementation of a multi-threaded NFFT (nonequispaced fast Fourier transform) software library and present the used parallelization approaches. Besides the NFFT kernel, the NFFT on the two-sphere and the fast summation based on NFFT are also parallelized. Thereby, the parallelization is based on OpenMP and the multi-threaded FFTW library. Furthermore, benchmarks for various cases are performed. The results show that an efficiency higher than 0.50 and up to 0.79 can still be achieved at 12 threads.

In this paper we describe a parallel algorithm for calculating nonequispaced fast Fourier transforms on massively parallel distributed memory architectures. These algorithms are implemented in an open source software library called PNFFT. Furthermore, we derive a parallel fast algorithm for the computation of the Coulomb potentials and forces in a charged particle system, which is based on the parallel nonequispaced fast Fourier transform. To prove the high scalability of our algorithms we provide performance results on a BlueGene/P system using up to 65536 cores.

The present thesis considers the problem of reconstructing a function f that is defined on the d-dimensional unit sphere from its mean values along hyperplane sections. In case of the two-dimensional sphere, these plane sections are circles. In many tomographic applications, however, only limited data is available. Therefore, one is interested in the reconstruction of the function f from its mean values with respect to only some subfamily of all hyperplane sections of the sphere. Compared with the full data case, the limited data problem is more challenging and raises several questions. The first one is the injectivity, i.e., can any function be uniquely reconstructed from the available data? Further issues are the stability of the reconstruction, which is closely connected with a description of the range, as well as the demand for actual inversion methods or algorithms. We provide a detailed coverage and answers of these questions for different families of hyperplane sections of the sphere such as vertical slices, sections with hyperplanes through a common point and also incomplete great circles. Such reconstruction problems arise in various practical applications like Compton camera imaging, magnetic resonance imaging, photoacoustic tomography, Radar imaging or seismic imaging. Furthermore, we apply our findings about spherical means to the cone-beam transform and prove its singular value decomposition. / Die vorliegende Arbeit beschäftigt sich mit dem Problem der Rekonstruktion einer Funktion f, die auf der d-dimensionalen Einheitssphäre definiert ist, anhand ihrer Mittelwerte entlang von Schnitten mit Hyperebenen. Im Fall d=2 sind diese Schnitte genau die Kreise auf der Sphäre. In vielen tomografischen Anwendungen sind aber nur eingeschränkte Daten verfügbar. Deshalb besteht das Interesse an der Rekonstruktion der Funktion f nur anhand der Mittelwerte bestimmter Familien von Hyperebenen-Schnitten der Sphäre. Verglichen mit dem Fall vollständiger Daten birgt dieses Problem mehrere Herausforderungen und Fragen. Die erste ist die Injektivität, also können alle Funktionen anhand der gegebenen Daten eindeutig rekonstruiert werden? Weitere Punkte sind die die Frage nach der Stabilität der Rekonstruktion, welche eng mit einer Beschreibung der Bildmenge verbunden ist, sowie der praktische Bedarf an Rekonstruktionsmethoden und -algorithmen. Diese Arbeit gibt einen detaillierten Überblick und Antworten auf diese Fragen für verschiedene Familien von Hyperebenen-Schnitten, angefangen von vertikalen Schnitten über Schnitte mit Hyperebenen durch einen festen Punkt sowie Kreisbögen. Solche Rekonstruktionsprobleme treten in diversen Anwendungen auf wie der Bildgebung mittels Compton-Kamera, Magnetresonanztomografie, fotoakustischen Tomografie, Radar-Bildgebung sowie der Tomografie seismischer Wellen. Weiterhin nutzen wir unsere Ergebnisse über sphärische Mittelwerte, um eine Singulärwertzerlegung für die Kegelstrahltomografie zu zeigen.

Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient. For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods. We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference. The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice. / Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt. Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden. Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz. Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert.

The Mindlin-Reissner plate model is widely used for the elastic deformation simulation of moderately thick plates. Shear locking occurs in the case of thin plates, which means slow convergence with respect to the mesh size. The Kirchhoff plate model does not show locking effects, but is valid only for thin plates. One would like to have a method suitable for both thick and thin plates. Several approaches are known to deal with the shear locking in the Mindlin-Reissner plate model. In addition to the well-known MITC elements and other approaches based on a mixed formulation, hierarchical methods have been developed in the recent years. These are based on the Kirchhoff model and add terms to account for shear deformations. We present some of these methods and develop a new hierarchic plate formulation. This new model can be discretised by a combination of C0 and C1 finite elements. Numerical tests show that the new formulation is locking-free and numerically efficient. We also give an extension of the model to a hierarchical Naghdi shell based on a Koiter shell formulation with unknowns in Cartesian coordinates.:1 Introduction 2 Plate theory 3 Shell theory 4 Conclusion

The present thesis is dedicated to the efficient computation of electrostatic interactions in particle systems, which is of great importance in the field of molecular dynamics simulations. In order to compute the therefor required physical quantities with only O(N log N) arithmetic operations, so called particle-mesh methods make use of the well-known Ewald summation approach and the fast Fourier transform (FFT). Typically, such methods are able to handle systems of point charges subject to periodic boundary conditions in all spatial directions. However, periodicity is not always desired in all three dimensions and, moreover, also interactions to dipoles play an important role in many applications. Within the scope of the present work, we consider the particle-particle NFFT method (P²NFFT), a particle-mesh approach based on the fast Fourier transform for nonequispaced data (NFFT). An extension of this method for mixed periodic as well as open boundary conditions is presented. Furthermore, the method is appropriately modified in order to treat particle systems containing both charges and dipoles. Consequently, an efficient algorithm for mixed charge-dipole systems, that additionally allows a unified handling of various types of periodic boundary conditions, is presented for the first time. Appropriate error estimates as well as parameter tuning strategies are developed and verified by numerical examples. / Die vorliegende Arbeit widmet sich der Berechnung elektrostatischer Wechselwirkungen in Partikelsystemen, was beispielsweise im Bereich der molekulardynamischen Simulationen eine zentrale Rolle spielt. Um die dafür benötigten physikalischen Größen mit lediglich O(N log N) arithmetischen Operationen zu berechnen, nutzen sogenannte Teilchen-Gitter-Methoden die Ewald-Summation sowie die schnelle Fourier-Transformation (FFT). Typischerweise können derartige Verfahren Systeme von Punktladungen unter periodischen Randbedingungen in allen Raumrichtungen handhaben. Periodizität ist jedoch nicht immer bezüglich aller drei Dimensionen erwünscht. Des Weiteren spielen auch Wechselwirkungen zu Dipolen in vielen Anwendungen eine wichtige Rolle. Zentraler Gegenstand dieser Arbeit ist die Partikel-Partikel-NFFT Methode (P²NFFT), ein Teilchen-Gitter-Verfahren, welches auf der schnellen Fouriertransformation für nichtäquidistante Daten (NFFT) basiert. Eine Erweiterung dieses Verfahrens auf gemischt periodische sowie offene Randbedingungen wird vorgestellt. Außerdem wird die Methode für die Behandlung von Partikelsystemen, in denen sowohl Ladungen als auch Dipole vorliegen, angepasst. Somit wird erstmalig ein effizienter Algorithmus für gemischte Ladungs-Dipol-Systeme präsentiert, der zusätzlich die Behandlung sämtlicher Arten von Randbedingungen mit einem einheitlichen Zugang erlaubt. Entsprechende Fehlerabschätzungen sowie Strategien für die Parameterwahl werden entwickelt und anhand numerischer Beispiele verifiziert.