Kassel, Univ., Diss., 2008 / Dateien in unterschiedlichen Formaten

Chemnitz, Techn. Univ., Diplomarb., 2005.

Neben einer kurzen Ausführungszeit rückt bei der Optimierung von Anwendungen und Algorithmen ein geringer Energieverbrauch der genutzten Rechenressourcen in den Fokus der aktuellen Forschung. Eine hohe Energie-Effizienz von Programmen wird dabei erreicht, indem der Energieverbrauch von Programmen und Technologien reduziert wird, ohne dafür die Ausführungszeit übermäßig zu erhöhen. Im parallelen wissenschaftlichen Rechnen ist der Bedarf an energie-effizienten Programmausführungen vor allem für Algorithmen der linearen Algebra gegeben, die als Unterfunktionen in einer Vielzahl von Anwendungen eingesetzt werden. Die Vektorisierung von Programmen durch die Prozessor- und Instruktionssatzerweiterung AVX zeigt Potenzial zur energie-effizienten Ausführung von Algorithmen der linearen Algebra, wobei die erzielte Energie-Effizienz von der Umsetzung der Implementierung abhängt. Für die gezeigten Untersuchungen werden drei repräsentativ ausgewählte Algorithmen der linearen Algebra für die Ausführung auf AVX-Vektoreinheiten genutzt. Bei der AVX-Vektorisierung der Algorithmen werden verschiedene Programmvarianten erstellt, mit denen Ausführungszeit und Energieverbrauch bei der Ausführung ermittelt werden. Die Programmvarianten unterscheiden sich dabei unter anderem in der Anwendung von Programmtransformationen, wie Loop Tiling oder einer veränderten Speicherzugriffsstruktur. Zusätzlich wird gezeigt, wie die Umsetzung verschiedener Programmieransätze, wie Autovektorisierung oder unterschiedlicher Instruktionssätze, sowie Implementierungsvarianten durch die Auswahl der verwendeten Instruktionen, die Ausführungszeit und den Energieverbrauch der Programmausführung beeinflussen. Die so erstellten Programmvarianten werden auf modernen Prozessoren verschiedener Architekturfamilien mit unterschiedlichen Ausführungsparametern, wie der eingestellten Prozessorfrequenz, ausgeführt. Die Untersuchungen zeigen, dass sich Ausführungszeit und Energieverbrauch von Programmen durch die Vektorisierung reduzieren lassen. Die Auswahl der Programmtransformationen, des Programmieransatzes und der Ausführungsparameter für die energie-effiziente Ausführung von vektorisierten Programmen kann dabei anwendungsspezifisch aufgrund der Eigenschaften des ausgewählten Algorithmus getroffen werden.

Es werden effiziente Algorithmen vorgestellt, die auf algebraischen Methoden beruhen um die Unerfüllbarkeit aussagenlogischer 4-SAT Formeln zu zertifizieren. Die Algorithmen werden implementiert und auf praktische Weise hinsichtlich der Laufzeit mit Backtracking-Algorithmen verglichen.

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis.

In mathematics the parameter is used as a special kind of a variable. The classification of the terms \"variable\" and \"parameter\" is often done by intuition and changes due to different situations and needs. The history of mathematics shows that these two terms represent the same abstract object in mathematics. In today´s mathematics, compared to variables, the parameter is declared as an unknown constant measure. This interpretation of parameters can be used in set theory for describing sets with an infinite number of elements. Due to this perspective the structure of vector spaces can be developed as a special structured set theory. Further, the concept of parameters can be seen as a model for developing mathematics education in linear algebra.

Es werden effiziente Algorithmen vorgestellt, die auf algebraischen Methoden beruhen um die Unerfüllbarkeit aussagenlogischer 4-SAT Formeln zu zertifizieren. Die Algorithmen werden implementiert und auf praktische Weise hinsichtlich der Laufzeit mit Backtracking-Algorithmen verglichen.

We consider the problem of maximizing the second smallest eigenvalue of the weighted Laplacian of a (simple) graph over all nonnegative edge weightings with bounded total weight. We generalize this problem by introducing node significances and edge lengths. We give a formulation of this generalized problem as a semidefinite program. The dual program can be equivalently written as embedding problem. This is fifinding an embedding of the n nodes of the graph in n-space so that their barycenter is at the origin, the distance between adjacent nodes is bounded by the respective edge length, and the embedded nodes are spread as much as possible. (The sum of the squared norms is maximized.) We proof the following necessary condition for optimal embeddings. For any separator of the graph at least one of the components fulfills the following property: Each straight-line segment between the origin and an embedded node of the component intersects the convex hull of the embedded nodes of the separator. There exists always an optimal embedding of the graph whose dimension is bounded by the tree-width of the graph plus one. We defifine the rotational dimension of a graph. This is the minimal dimension k such that for all choices of the node significances and edge lengths an optimal embedding of the graph can be found in k-space. The rotational dimension of a graph is a minor monotone graph parameter. We characterize the graphs with rotational dimension up to two.

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis.

In mathematics the parameter is used as a special kind of a variable. The classification of the terms \"variable\" and \"parameter\" is often done by intuition and changes due to different situations and needs. The history of mathematics shows that these two terms represent the same abstract object in mathematics. In today´s mathematics, compared to variables, the parameter is declared as an unknown constant measure. This interpretation of parameters can be used in set theory for describing sets with an infinite number of elements. Due to this perspective the structure of vector spaces can be developed as a special structured set theory. Further, the concept of parameters can be seen as a model for developing mathematics education in linear algebra.