A Rayleigh Quotient Fixed Point Method for Criticality Eigenvalue Problems in Neutron Transport

The alpha- and k-effective eigenproblems describe the criticality and fundamental neutron flux mode of a nuclear system. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. The k-effective eigenvalue problem has generally been solved using power iteration. For problems with dominance ratios close to one, however, power iteration can be intractably slow. We present the Rayleigh Quotient Fixed Point (RQFP) methods, nonlinear fixed-point methods that are applied to the primitive discretizations of the neutron transport eigenvalue equations. We prove that the discretized eigenvalue equations form a primitive linear system for one-dimensional slab geometry where the unique, positive angular flux eigenvectors are guaranteed to exist from the Perron-Frobenius Theorem for primitive matrices. These methods are capable of solving subcritical and supercritical alpha- and k-effective eigenvalue problems. The derived eigenvalue updates are proven to be optimal in the least squares sense and positive eigenvector updates are guaranteed from the properties of primitive matrices. We consider infinite-medium, one-dimensional slabs and spheres, two-dimensional cylinders, and three-dimensional quarter core benchmark problems and show the ability of the Rayleigh Quotient Fixed Point method to obtain the fundamental eigenvalue and eigenvector of these systems, even when the discretized eigenvalue equations no longer form a primitive system. We also consider the use of Anderson acceleration to accelerate the convergence of the Rayleigh Quotient Fixed Point methods.We demonstrate that for alpha-eigenvalue problems, the Rayleigh Quotient Fixed Point method substantially reduces the number of iterations required for convergence when compared to traditional alpha-eigenvalue methods such as the critical search method. For a wide variety of problems, the RQFP method for alpha-eigenvalue problems reduces the number of iterations required by up to factors of 50 and converges problems that other methods are unable to converge. For $k$-effective problems, the RQFP method provides moderate reductions of iterations required for convergence when compared to power iteration. In particular, the RQFP method does well for infinite-medium problems or problems where the eigenvector is highly localized. We also demonstrate acceleration of the RQFP method by Anderson acceleration. For slowly converging alpha-eigenvalue problems solved using the RQFP method, Anderson acceleration can provide acceleration of the linear fixed-point method convergence by a factor of up to ten.By looking at the linear algebraic structure of the discretized neutron transport eigenvalue problems, the RQFP method guarantees the existence of the positive angular flux eigenvector and its corresponding eigenvalue. By examining a wide variety of problems of interest to nuclear engineers, we show that the RQFP method is robust, easily implementable in neutron transport codes, and an efficient solution method for eigenvalue problems in neutron transport

Properties of dirty bosons in disordered optical lattices

The study of the disordered Bose-Hubbard model is key to understanding the interplay of disorder and interactions. Despite many studies with uniform diagonal disorder, few have inquired into experimental realizations with an additional correlated off-diagonal disorder. The presence of a trap and finite temperature effects in experiments lead to multiple do- mains of the Superfluid, Mott-Insulator/normal and the Bose-Glass phase. Previous studies using approximate theories produced results that are not in accordance with experiments. Stochastic Series Expansion is a finite temperature technique that can solve Bosonic lattice Hamiltonians exactly for large systems. Here, studies are performed for an extensive range of parameters using disorder distributions that are similar to experiments. Insights are first acquired by studying trap-free situations. Constant density calculations show that, although the qualitative features of the phase diagram remain robust between speckle disorder and uniform box disorder, there are quantitative differences. Studies of the Bose-glass phase ex- plicitly show that it is composed of superfluid puddles that are stable to finite temperature effects for large temperature ranges. Finite temperature behavior of a strongly correlated sys- tem reveals that at unit filling, the transition temperature of the superfluid is increased due to the addition of disorder. Inquires are then extended to discern the properties of trapped systems. Extensive calculations show that domain-like structures that develop can be rig- orously demarcated using the single-particle eigenstates extracted from the single-particle density matrix. Observables are calculated for the system at the single-site and global scales, showing that intermediate length scales provide the correct description of the physics of the domains in these systems. These techniques are used to conclusively show the possibility of the re-entrant superfluid that should be accessible to experiments. The temperature de- pendence of the re-entrant domain is explicitly calculated to be within experimental limits provided interactions are not too large. Comparisons with the local density approximation show reasonable agreement at low disorder strengths. At large disorder strengths there can be quantitative errors and can also result in qualitative errors. The phase diagram due to speckle disorder is presented for a range of values that are readily accessible to experiments. It is quantitatively shown that the effects of off-diagonal disorder are minimal. The superfluid remains unaffected despite large disorder in the tunneling term. Full scale ab initio calcu- lations of the largest trapped disordered systems to date are performed in order to identify the superfluid-Bose-glass phase boundary in collaboration with experiments. Results show remarkable agreement, but there are open questions with regards to the possibility of glassy dynamics

Parallel Algorithms for Time Dependent Density Functional Theory in Real-space and Real-time

Density functional theory (DFT) and time dependent density functional theory (TDDFT) have had great success solving for ground state and excited states properties of molecules, solids and nanostructures. However, these problems are particularly hard to scale. Both the size of the discrete system and the number of needed eigenstates increase with the number of electrons. A complete parallel framework for DFT and TDDFT calculations applied to molecules and nanostructures is presented in this dissertation. This includes the development of custom numerical algorithms for eigenvalue problems and linear systems. New functionality in the FEAST eigenvalue solver presents an additional level of distributed memory parallelism and is used for the ground state DFT calculation, allowing larger molecules to be simulated. A parallel domain decomposition linear system solver has also been implemented. This approach uses a Schur complement technique and a combination of direct sparse solvers to outperform black box distributed memory solvers in both performance and scalability. All other aspects of the code have been rewritten to operate in the domain decomposition framework and have been parallelized using both MPI and OpenMP. Numerical experiments demonstrate that our all-electron code can be applied to systems containing up to a few thousand atoms

Dynamisch-netzwerkweite Lichtsignalanlagenoptimierung

Nowadays, many cities in the world are suffering from problems like congestion, pollution, and traffic accidents which are caused by vehicular traffic. The correct scheduling of traffic lights can help to alleviate these problems by improving the flow of vehicles through the cities. The main aim of this dissertation is to build an approach to find the good traffic signal plans for a large area. The two major features of the approach developed in this thesis are real-time and system-wide. Since traffic flow changes with the time of day, the real-time computation of the traffic signal plans can improve the operation efficiency of traffic lights compared with fixed signal plans which is an old but still often used technology in the world. The proposed approach is the serial optimization with a hierarchical control framework. The upper level is the level for macro control strategies including a network partition strategy and a network signal coordination strategy. The network partition strategy means that the urban network is partitioned into smaller sub networks based on the network's topological graph and intersections' priority order. The priority order is computed by the sorting model of priority order (SMoPO), which offers the opportunity for the higher priority intersection to be coordinated earlier and to obtain more benefit. The network signal coordination strategy is developed to determine which intersections form a coordination pair and which traffic streams need to be coordinated. This strategy converts the optimization problem into a much simpler one. The number of operations and the computation time to solve this optimization problem drops largely. The lower level is the level for micro parameters calculation, in which a method for the computation of the optimal relative offsets is proposed which is based on cyclic flow profiles. All of the developed strategies were programed and interfaced with the microscopic simulation tool "SUMO". To verify the success and the dynamic feasibility of strategies, the computation speed tests were done in three-by-three to sixty-by-sixty grid nets to demonstrate the real-time feasibility of the approach. After that, microsimulation studies have been performed to evaluate the performance of the strategies. The first case study was a hypothetic eight-by-eight grid net with varied traffic demands and link lengths, and the results revealed that the strategy was effective when the intersections were not oversaturated. The others were two real networks in Braunschweig City, whose input data was from the Project AIM (Application platform Intelligent Mobility). The simulation results showed the delay time was decreased on average in both cases compared to Webster's model.Der motorisierte Individualverkehr führt in fast allen großen Städten der Welt zu Staus, Umweltverschmutzung und Unfällen. Eine gute Anpassung der Steuerprogramme von Lichtsignalanlagen (LSA) an die jeweilige Verkehrssituation kann dazu beitragen, den Verkehrsablauf flüssiger, weniger umweltbelastend und sicherer zu gestalten. Das Hauptziel dieser Arbeit ist die Entwicklung eines Verfahrens, welches diese Anpassung auch für gro?e Städte mit einer Vielzahl von LSAs ermöglicht. Diese Arbeit folgt dabei zwei Zielvorgaben. Das zu entwickelnde System soll realzeitfähig sein und auch große Städte mit einigen tausend LSA versorgen können. Die Realzeitkomponente ist von Bedeutung, weil die verkehrliche Nachfrage sehr starken und teilweise nicht vorhersehbaren Schwankungen unterliegt. Von einem solchen adaptiven Verfahren kann erwartet werden, dass es effizienter ist als die Festzeitsteuerungen, die noch immer in vielen Teilen der Welt benutzt werden. In dieser Arbeit wurde zu diesem Zweck ein serielles Optimierungsverfahren entwickelt, das in ein hierarchisches Steuerungsrahmenwerk eingebunden ist. Die übergeordnete Ebene enthält ein Verfahren, mit dessen Hilfe ein Netzwerk in Teilnetze zerlegt werden kann. Dieses Verfahren basiert auf dem topologischen Graphen des Netzwerkes und der Prioritätenfolge der Kreuzungen des Netzwerks. Die Prioritätenfolge wird durch das in dieser Arbeit entwickelte Verfahren Sorting Model of Priority Order (SMoPO) berechnet, mit dessen Hilfe festgelegt wird, wann im Laufe des Optimierungsprozesses welche Kreuzung optimiert wird - Kreuzungen mit einer höheren Priorität zuerst, die weniger wichtigen später. Darüber hinaus wurde für diese Ebene ein Koordinierungsverfahren entwickelt, mit dem bestimmt werden kann, welche Paare von Kreuzungen jeweils zu koordinieren sind. Dieser Ansatz reduziert die Komplexität dieses Optimierungsproblems dramatisch, weil es eine Unterteilung eines großen Problems in viele kleinere ermöglicht. Die untere Ebene dieses Rahmenwerkes ist die Berechnung der Mikroparameter der einzelnen Kreuzung. Die optimalen Offsets werden mit Hilfe einer Methode berechnet, die auf zyklischen Flussprofilen basiert. Alle in der Arbeit entwickelten Verfahren wurden in Computerprogrammen umgesetzt und mit einer Schnittstelle zu dem mikroskopischen Verkehrssimulationstool "SUMO" versehen. Um die Realzeitfähigkeit der Strategien zu überprüfen, wurden Geschwindigkeitstests für Drei-mal-Drei bis Sechzig-mal-Sechzig Quadratgitternetze durchgeführt. Anschließend wurden mehrere Fallstudien mit SUMO durchgeführt, um die Qualität des neuen Verfahrens zu evaluieren. Die erste Fallstudie war ein künstliches Acht-mal-Acht Quadratgitternetz mit variierender Verkehrsnachfrage und Kantenlängen. Die Ergebnisse haben gezeigt, dass die Strategie effektiv ist, wenn die Kreuzungen nicht übersättigt sind. Die anderen Fallbeispiele waren zwei echte Netze aus dem Braunschweiger Stadtgebiet mit Inputdaten aus dem Projekt AIM (Anwendungsplattform Intelligente Mobilität). Auch hier konnte die neue Strategie die Verlustzeiten in beiden Fällen im Vergleich zum Webster-Modell verringern

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science