In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

Die Arbeit abstrahiert bekannte Konzepte der endlichdimensionalen Optimierung im Hinblick auf deren Anwendung in Banachräumen. Hierfür werden zunächst grundlegende Elemente der Funktionalanalysis wie schwache Konvergenz, Dualräume und Reflexivität vorgestellt. Anschließend erfolgt eine kurze Einführung in die Thematik der Fréchet-Differenzierbarkeit und eine Abstraktion des Begriffs der partiellen Ordnungsrelation in normierten Räumen. Nach der Formulierung eines allgemeinen Existenzsatzes für globale Optimallösungen von abstrakten Optimierungsaufgaben werden notwendige Optimalitätsbedingungen vom Karush-Kuhn-Tucker-Typ hergeleitet. Abschließend wird eine hinreichende Optimalitätsbedingung vom Karush-Kuhn-Tucker-Typ unter verallgemeinerten Konvexitätsvoraussetzungen verifiziert.

A MATTER OF NATIONAL CONCERN examines the Kennedy Administration’s contribution to the restoration of public education in Prince Edward County, Virginia, and determines if those actions support the dominant narrative of Kennedy’s overall civil rights record – a historical assessment generally generated from a few acute crises. For five consecutive years (1959-1964), in defiance of federal court orders, the county board of supervisors refused to levy taxes to operate public schools, marking Prince Edward County as the only locale in the nation without free public education. The county leadership organized a segregated private school system for the 1,400 white children, but afforded no formal education for the 1,700 African American students. The Kennedy Administration inherited the Prince Edward County school situation – a crisis that threatened to cripple a generation, and, if replicated, destroy public education. In the Prince Edward County school dilemma, the Kennedy Administration took proactive measures, proved sympathetic to the plight of African Americans, challenged Virginia’s congressional delegation, and appointed federal judges that supported President Kennedy’s civil rights agenda. The Prince Edward County story generally, and the federal government’s actions specifically, have been virtually overlooked by historians. A MATTER OF NATIONAL CONCERN challenges scholars to re-evaluate the Kennedy Administration’s civil rights record by including all of the civil rights events of the Kennedy years, thus developing a thorough, comprehensive assessment. A MATTER OF NATIONAL CONCERN is the product of the study of unpublished archival documents, oral histories, interviews, newspaper reports, and secondary sources. This work was created using Microsoft Word 2003.

This thesis addresses issues related to the simulation and optimization of the injection molding of short fiber-reinforced plastics (SFRPs). The injection molding process is modeled by a two phase flow problem. The simulation of the two phase flow is accompanied by the solution of the Folgar-Tucker equation (FTE) for the simulation of the moments of fiber orientation densities. The FTE requires the solution of the so called 'closure problem'', i.e. the representation of the 4th order moments in terms of the 2nd order moments. In the absence of fiber-fiber interactions and isotropic initial fiber density, the FTE admits an analytical solution in terms of elliptic integrals. From these elliptic integrals, the closure problem can be solved by a simple numerical inversion. Part of this work derives approximate inverses and analytical inverses for special cases of fiber orientation densities. Furthermore a method is presented to generate rational functions for the computation of arbitrary moments in terms of the 2nd order closure parameters. Another part of this work treats the determination of effective material properties for SFRPs by the use of FFT-based homogenization methods. For these methods a novel discretization scheme, the 'staggered grid'' method, was developed and successfully tested. Furthermore the so called 'composite voxel'' approach was extended to nonlinear elasticity, which improves the approximation of material properties at the interfaces and allows the reduction of the model order by several magnitudes compared to classical approaches. Related the homogenization we investigate optimal experimental designs to robustly determine effective elastic properties of SFRPs with the least number of computer simulations. Finally we deal with the topology optimization of injection molded parts, by extending classical SIMP-based topology optimization with an approximate model for the fiber orientations. Along with the compliance minimization by topology optimization we also present a simple shape optimization method for compensation of part warpage for an black-box production process.:Acknowledgments v Abstract vii Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Nomenclature 3 Chapter 2. Numerical simulation of SFRP injection molding 5 2.1 Introduction 5 2.2 Injection molding technology 5 2.3 Process simulation 6 2.4 Governing equations 8 2.5 Numerical implementation 18 2.6 Numerical examples 25 2.7 Conclusions and outlook 27 Chapter 3. Numerical and analytical methods for the exact closure of the Folgar-Tucker equation 35 3.1 Introduction 35 3.2 The ACG as solution of Jeffery's equation 35 3.3 The exact closure 36 3.4 Carlson-type elliptic integrals 37 3.5 Inversion of R_D-system 40 3.6 Moment tensors of the angular central Gaussian distribution on the n-sphere 49 3.7 Experimental evidence for ACG distribution hypothesis 54 3.8 Conclusions and outlook 60 Chapter 4. Homogenization of SFRP materials 63 4.1 Introduction 63 4.2 Microscopic and macroscopic model of SFRP materials 63 4.3 Effective linear elastic properties 65 4.4 The staggered grid method 68 4.5 Model order reduction by composite voxels 80 4.6 Optimal experimental design for parameter identification 93 Chapter 5. Optimization of parts produced by SFRP injection molding 103 5.1 Topology optimization 103 5.2 Warpage compensation 110 Chapter 6. Conclusions and perspectives 115 Appendix A. Appendix 117 A.1 Evaluation of R_D in Python 117 A.2 Approximate inverse for R_D in Python 117 A.3 Inversion of R_D using Newton's/Halley's method in Python 117 A.4 Inversion of R_D using fixed point method in Python 119 A.5 Moment computation using SymPy 120 A.6 Fiber collision test 122 A.7 OED calculation of the weighting matrix 123 A.8 OED Jacobian of objective and constraints 123 Appendix B. Theses 125 Bibliography 127 / Diese Arbeit befasst sich mit Fragen der Simulation und Optimierung des Spritzgießens von kurzfaserverstärkten Kunststoffen (SFRPs). Der Spritzgussprozess wird durch ein Zweiphasen-Fließproblem modelliert. Die Simulation des Zweiphasenflusses wird von der Lösung der Folgar-Tucker-Gleichung (FTE) zur Simulation der Momente der Faserorientierungsdichten begleitet. Die FTE erfordert die Lösung des sogenannten 'Abschlussproblems'', d. h. die Darstellung der Momente 4. Ordnung in Form der Momente 2. Ordnung. In Abwesenheit von Faser-Faser-Wechselwirkungen und anfänglich isotroper Faserdichte lässt die FTE eine analytische Lösung durch elliptische Integrale zu. Aus diesen elliptischen Integralen kann das Abschlussproblem durch eine einfache numerische Inversion gelöst werden. Ein Teil dieser Arbeit leitet approximative Inverse und analytische Inverse für spezielle Fälle von Faserorientierungsdichten her. Weiterhin wird eine Methode vorgestellt, um rationale Funktionen für die Berechnung beliebiger Momente in Bezug auf die Abschlussparameter 2. Ordnung zu generieren. Ein weiterer Teil dieser Arbeit befasst sich mit der Bestimmung effektiver Materialeigenschaften für SFRPs durch FFT-basierte Homogenisierungsmethoden. Für diese Methoden wurde ein neuartiges Diskretisierungsschema 'staggerd grid'' entwickelt und erfolgreich getestet. Darüber hinaus wurde der sogenannte 'composite voxel''-Ansatz auf die nichtlineare Elastizität ausgedehnt, was die Approximation der Materialeigenschaften an den Grenzflächen verbessert und die Reduzierung der Modellordnung um mehrere Größenordnungen im Vergleich zu klassischen Ansätzen ermöglicht. Im Zusammenhang mit der Homogenisierung untersuchen wir optimale experimentelle Designs, um die effektiven elastischen Eigenschaften von SFRPs mit der geringsten Anzahl von Computersimulationen zuverlässig zu bestimmen. Schließlich beschäftigen wir uns mit der Topologieoptimierung von Spritzgussteilen, indem wir die klassische SIMP-basierte Topologieoptimierung um ein Näherungsmodell für die Faserorientierungen erweitern. Neben der Compliance-Minimierung durch Topologieoptimierung stellen wir eine einfache Formoptimierungsmethode zur Kompensation von Teileverzug für einen Black-Box-Produktionsprozess vor.:Acknowledgments v Abstract vii Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Nomenclature 3 Chapter 2. Numerical simulation of SFRP injection molding 5 2.1 Introduction 5 2.2 Injection molding technology 5 2.3 Process simulation 6 2.4 Governing equations 8 2.5 Numerical implementation 18 2.6 Numerical examples 25 2.7 Conclusions and outlook 27 Chapter 3. Numerical and analytical methods for the exact closure of the Folgar-Tucker equation 35 3.1 Introduction 35 3.2 The ACG as solution of Jeffery's equation 35 3.3 The exact closure 36 3.4 Carlson-type elliptic integrals 37 3.5 Inversion of R_D-system 40 3.6 Moment tensors of the angular central Gaussian distribution on the n-sphere 49 3.7 Experimental evidence for ACG distribution hypothesis 54 3.8 Conclusions and outlook 60 Chapter 4. Homogenization of SFRP materials 63 4.1 Introduction 63 4.2 Microscopic and macroscopic model of SFRP materials 63 4.3 Effective linear elastic properties 65 4.4 The staggered grid method 68 4.5 Model order reduction by composite voxels 80 4.6 Optimal experimental design for parameter identification 93 Chapter 5. Optimization of parts produced by SFRP injection molding 103 5.1 Topology optimization 103 5.2 Warpage compensation 110 Chapter 6. Conclusions and perspectives 115 Appendix A. Appendix 117 A.1 Evaluation of R_D in Python 117 A.2 Approximate inverse for R_D in Python 117 A.3 Inversion of R_D using Newton's/Halley's method in Python 117 A.4 Inversion of R_D using fixed point method in Python 119 A.5 Moment computation using SymPy 120 A.6 Fiber collision test 122 A.7 OED calculation of the weighting matrix 123 A.8 OED Jacobian of objective and constraints 123 Appendix B. Theses 125 Bibliography 127