Splitting methods for nonlinear evolution equations

Abweichender Titel nach Übersetzung der Verfasserin/des VerfassersDiese Arbeit motiviert und fasst die Konstruktion von Splittingschemata für Evolutionsgleichungen zusammen. Weiters werden asymptotisch korrekte lokale Fehlerschätzer vorgestellt, um die Schrittweite in der Zeitevolution anzupassen um die gesamte Rechenzeit zu reduzieren. Kommutatorfreie Magnus-Typ-Integratoren werden mit dem Splitting in der Zeitvariablen verglichen. Die Methoden werden an drei Anwendungen getestet: Modelle für Solarzellen, Reaktions-Diffusions-System von Gray-Scott und Schrödinger-Wellengleichungen.This thesis motivates and summarizes the construction of splitting schemes for evolution equations. Further asymptotic correct local error estimators are proposed, to adapt the step-size in time propagation and reduce total computation time. Commutator free Magnus-type integrators are compared to splitting off the time variable. The Methods are tested by three applications: Models for solar cells, reaction-diffusion system by Gray-Scott and Schrödinger wave equations.6

Parallel velocity extension and load-balanced Re-distancing on hierarchical grids for high performance process TCAD

Die kontinuierlichen Entwicklungen und die Miniaturisierung der Herstellungsprozesse für Halbleiterbauelemente erfordern physikalische Simulationen, um die Zahl der kostspieligen konventionellen Experimente in den Entwurfs- und Produktionsprozessen zu verringern. Am bekanntesten sind physikalische Simulationen, die einzelne physikalische Prozessschritte wie Ätzen oder Abscheiden modellieren. Diese topografieverändernden Simulationen basieren gewöhnlich auf der Level-Set-Methode, da sie komplexe dreidimensionale Bauelementstrukturen effizient darstellen kann. Die hohen Genauigkeitsanforderungen dieser Simulationen erfordern die Anwendung komplexer und daher rechenintensiver physikalischer Modelle.In dieser Arbeit werden drei parallele Algorithmen eingeführt, die zu zwei Rechenschritten der Level-Set-Methode gehören. Die Algorithmen verringern die Gesamtlaufzeit erheblich und verbessern die Genauigkeit. Die Algorithmen sind an Simulationen angepasst, die adaptive Diskretisierungen mit hierarchischen Gittern verwenden, um spitze Geometrien, z.B. Ecken und Kanten, effizient zu behandeln. Der Schwerpunkt der hier vorgestellten Forschung ist die effiziente Nutzung paralleler Rechensysteme mit gemeinsamem Speicher, um die immer anspruchsvolleren Level-Set-basierten physikalischen Simulationen zu bewältigen.Der erste Algorithmus gehört zum Rechenschritt Velocity Extension, der die Geschwindigkeit, die die Verformung einer beliebigen Struktur beschreibt, von der Oberfläche der Struktur auf das gesamte Simulationsgebiet ausdehnt. Der entwickelte Algorithmus zur Geschwindigkeitserweiterung basiert auf der Fast-Marching-Methode. Die Fast-Marching-Methode ermöglicht es, die Geschwindigkeit in einem einzigen Durchgang durch das Simulationsgebiet zu berechnen, indem die Berechnungen in einer strengen Reihenfolge durchgeführt werden. Der Hauptvorteil des entwickelten Algorithmus ist eine relaxierte Reihenfolge der Berechnungen. Diese reduziert nicht nur die Komplexität der Berechnungen, sondern ermöglicht auch Parallelität. Verschiedenen Entwicklungsstufen des Algorithmus werden durch den Vergleich der auf repräsentativen Rechensystemen gemessenen Laufzeiten bewertet. Eine Laufzeitverkürzung um den Faktor 18.5 wurde bei der Verwendung von 10 Threads erreicht.Der zweite Algorithmus gehört zum Rechenschritt Re-Distancing, der eine numerisch stabile Repräsentation der Struktur durch Berechnung des vorzeichenbehafteten Abstandsfeldes relativ zur Oberfläche der Struktur erzeugt oder wiederherstellt. Dieser Algorithmus basiert ebenfalls auf der Fast-Marching-Methode, aber wegen der selbstbezogenen Datenabhängigkeiten wurde eine andere Parallelisierungsstrategie entwickelt. Es wird eine Gebietszerlegung eingeführt, um die Granularität der parallelen Aufgaben zu erhöhen. Dies ermöglicht einen besseren impliziten Lastausgleich im Vergleich zur nativen Gebietszerlegung, die durch das gegebene hierarchische Gitter bereitgestellt wird. Eine Geschwindigkeitssteigerung von mehr als 17.4 wurde bei der Verwendung von 24 Threads erreicht. Schließlich wurde ein Bottom-up-Korrekturalgorithmus entwickelt, der ebenfalls zum Rechenschritt Re-Distancing gehört und die Genauigkeit des vom zweiten Algorithmus berechneten vorzeichenbehafteten Abstandsfeldes erhöht. Dieser Korrekturalgorithmus benutzt das vorzeichenbehaftete Abstandsfeld in höher aufgelösten Gebieten des hierarchischen Gitters, um auch den Fehler in niedriger aufgelösten Gebieten zu reduzieren. Der entwickelte Algorithmus fügt dem zweiten Algorithmus einen vernachlässigbaren Rechenaufwand hinzu, reduziert aber den Fehler bei Ecken um einen Faktor von bis zu 2.7.Durch die Kombination aller entwickelten Algorithmen wird gezeigt, dass sich die Gesamtlaufzeit einer repräsentativen physikalischen Simulation mehr als halbiert, während die Genauigkeit weiter verbessert wird.The continuous developments and miniaturization of manufacturing processes for semiconductor devices require physical simulations to reduce the number of costly conventional experiments involved in the design and production processes. Most prominent are physical simulations which model individual physical processing steps like etching or deposition. These topography-changing simulations are commonly based on the level-set method, because of its capability to efficiently represent complex three-dimensional device structures. High accuracy demands of those simulations require the application of complex and, therefore, computationally expensive physical models. In this work, three parallel algorithms belonging to two computational steps of the level-set method are introduced. The algorithms significantly reduce overall run-time and improve accuracy. The algorithms are tailored to simulations using adaptive discretizations with hierarchical grids to efficiently handle sharp features, e.g., corners and edges. The focus of the presented research is to efficiently utilize shared-memory parallel computing systems to stem the increasingly demanding level-set based physical simulations. The first algorithm belongs to the computational step Velocity Extension which extends the velocity describing the deformation of an arbitrary structure from the structure's surface to the entire computational domain. The developed velocity extension algorithm is based on the fast marching method. The fast marching method allows to extend the velocity in a single pass through the computational domain by means of a strict ordering of the computations. The key advantage of the developed velocity extension algorithm is a relaxed ordering of the computations. This not only reduces the computational complexity but also enables parallelism. Different stages of the developed algorithm are evaluated by comparing run-times measured on representative computing systems. A run-time reduction by a factor of 18.5 using 10 threads has been achieved. The second algorithm belongs to the computational step Re-Distancing which creates or restores a numerically stable representation of the structure by computing the signed-distance field relative to the surface of the structure. This algorithm is also based on the fast marching method, but because of self-referred data dependencies a different parallelization strategy was developed. A domain decomposition is introduced to increase the granularity of the parallel tasks. This enables a better implicit load-balancing compared to the native decomposition provided by the given hierarchical grid. A speedup of more than 17.4 has been achieved when using 24 threads. Finally, a bottom-up correction algorithm was developed, also belonging to the computational step Re-Distancing, which increases the accuracy of the signed-distance field computed by the second algorithm. This correction algorithm utilizes the signed-distance field on higher resolved regions of hierarchical grids to also reduce the error in lower resolved regions. The developed algorithm adds negligible computational overhead to the second algorithm, yet reduces the error around corners by a factor of up to 2.7. Combining all developed algorithms, it is shown that the run-time of a representative physical simulation is more than halved whilst the accuracy is further improved.13

Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray–Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators

error estimation for Magnus-type integrators

We study high-order Magnus-type exponential integrators for large systems of ordinary differential equations defined by a time-dependent skew-Hermitian matrix. We construct and analyze defect-based local error estimators as the basis for adaptive stepsize selection. The resulting procedures provide a posteriori information on the local error and hence enable the accurate, efficient, and reliable time integration of the model equations. The theoretical results are illustrated on two numerical examples

Solvent-dependent facile synthesis of diaryl selenides and biphenols employing selenium dioxide

Biphenols are important structure motifs for ligand systems in organic catalysis and are therefore included in the category of so-called “privileged ligands”. We have developed a new synthetic pathway to construct these structures by the use of selenium dioxide, a stable, powerful, and commercially available oxidizer. Our new, and easy to perform protocol gives rise to biphenols and diaryl selenides depending on the solvent employed. Oxidative treatment of phenols in acetic acid yields the corresponding biphenols, whereas conversion in pyridine results in the preferred formation of diaryl selenides. As a consequence, we were able to isolate a broad scope of novel diaryl selenides, which could act as pincer-like ligands with further applications in organic synthesis or as ligands in transition metal catalysis

Beer Brewing and the Environmental Engineer: Tapping into Experiential Learning

Second to water, beer may perhaps be the next most desirable beverage in the lives of countless environmental engineering students. But do they fully understand or appreciate the engineering and scientific principles behind beer making? While considerable effort has been put forth in academia to teach and explain the critical environmental process of fermentation, too many students are limited to examples and explanations contained within a course textbook. The United States Military Academy is committed to providing experiential learning opportunities that reach beyond traditional classroom instruction. Our Environmental Biological Systems Course (EV396) offers an opportunity for environmental engineers to achieve a deeper, more practical understanding and appreciation for biological systems within our environment. As part of the experiential learning process, EV396 requires students to successfully brew beer in a laboratory setting to enhance their understanding of microbial metabolic processes, disinfection principles, and aseptic techniques. This paper aims to highlight and explain the linkage between the complex process of alcoholic fermentation involved in beer brewing to the environmental engineering practice. Indeed, environmental engineers often face challenges where they must design and operate biological systems and apply engineering concepts like those integral to brewing beer, including conventional wastewater management, microbial fuel cells, hazardous waste treatment and remediation, slow sand filtration, and disinfection. As part of this fermentation laboratory experience, students select the style of beer they wish to brew and exercise the engineered techniques required to brew a safe and refreshing product. Additionally, students are required to submit a detailed report demonstrating their ability to identify and evaluate key physiochemical and biochemical engineering processes. Calculations involve fermentation efficacy, specific gravity and yield, theoretical and actual ethanol content, and scaling from bench experiments to commercial production. The laboratory familiarizes students with engineering concepts, including substrates that serve as carbon and energy sources, methods for creating anaerobic reactors, and solid-liquid separation processes. Using the 5-point Likert scale, with 5 indicating greatest achievement, student laboratory performance scores are consistently greater than 3 and many are above 4, indicating effective learning, application, and understanding. Historical assessment and evaluation of how well this experiential learning laboratory supports course objectives and ABET Student Outcomes and Program Criteria are discussed in detail