G. Mathematics of Computing
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- Boltzmann Equation (1)
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Distributed Optimization of Constraint-Coupled Systems via Approximations of the Dual Function
(2024)
This thesis deals with the distributed optimization of constraint-coupled systems. This problem class is often encountered in systems consisting of multiple individual subsystems, which are coupled through shared limited resources. The goal is to optimize each subsystem in a distributed manner while still ensuring that system-wide constraints are satisfied. By introducing dual variables for the system-wide constraints the system-wide problem can be decomposed into individual subproblems. These resulting subproblems can then be coordinated by iteratively adapting the dual variables. This thesis presents two new algorithms that exploit the properties of the dual optimization problem. Both algorithms compute a quadratic surrogate function of the dual function in each iteration, which is optimized to adapt the dual variables. The Quadratically Approximated Dual Ascent (QADA) algorithm computes the surrogate function by solving a regression problem, while the Quasi-Newton Dual Ascent (QNDA) algorithm updates the surrogate function iteratively via a quasi-Newton scheme. Both algorithms employ cutting planes to take the nonsmoothness of the dual function into account. The proposed algorithms are compared to algorithms from the literature on a large number of different benchmark problems, showing superior performance in most cases. In addition to general convex and mixed-integer optimization problems, dual decomposition-based distributed optimization is applied to distributed model predictive control and distributed K-means clustering problems.
Single-phase flows are attracting significant attention in Digital Rock Physics (DRP), primarily for the computation of permeability of rock samples. Despite the active development of algorithms and software for DRP, pore-scale simulations for tight reservoirs — typically characterized by low multiscale porosity and low permeability — remain challenging. The term "multiscale porosity" means that, despite the high imaging resolution, unresolved porosity regions may appear in the image in addition to pure fluid regions. Due to the enormous complexity of pore space geometries, physical processes occurring at different scales, large variations in coefficients, and the extensive size of computational domains, existing numerical algorithms cannot always provide satisfactory results.
Even without unresolved porosity, conventional Stokes solvers designed for computing permeability at higher porosities, in certain cases, tend to stagnate for images of tight rocks. If the Stokes equations are properly discretized, it is known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa algorithm. However, in complex geometries, the Schur complement matrix can become severely ill-conditioned, having a significant portion of non-unit eigenvalues. This makes the established Uzawa preconditioner inefficient. To explain this behavior, we perform spectral analysis of the Pressure Schur Complement formulation for the staggered finite-difference discretization of the Stokes equations. Firstly, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement matrix. Secondly, we demonstrate that its condition number increases with increasing the surface-to-volume ratio of the flow domain. As an alternative to the Uzawa preconditioner, we propose using the diffusive SIMPLE preconditioner for geometries with a large surface-to-volume ratio. We show that the latter is much more efficient and robust for such geometries. Furthermore, we show that the usage of the SIMPLE preconditioner leads to more accurate practical computation of the permeability of tight porous media.
As a central part of the work, a reliable workflow has been developed which includes robust and efficient Stokes-Brinkman and Darcy solvers tailored for low-porosity multiclass samples and is accompanied by a sample classification tool. Extensive studies have been conducted to validate and assess the performance of the workflow. The simulation results illustrate the high accuracy and robustness of the developed flow solvers. Their superior efficiency in computing permeability of tight rocks is demonstrated in comparison with the state-of-the-art commercial solver for DRP.
Additionally, the Navier-Stokes solver for binary images from tight sandstones is discussed.
Recommender systems recommend items (e.g., movies, products, books) to users. In this thesis, we proposed two comprehensive and cluster-induced recommendation-based methods: Orthogonal Inductive Matrix Completion (OMIC) and Burst-induced Multi-armed Bandit (BMAB). Given the presence of side information, the first method is categorized as context-aware. OMIC is the first matrix completion method to approach the problem of incorporating biases, side information terms and a pure low-rank term into a single flexible framework with a well-principled optimization procedure. The second method, BMAB, is context-free. That is, it does not require any side data about users or items. Unlike previous context-free multi-armed bandit approaches, our method considers the temporal dynamics of human communication on the web and treats the problem in a continuous time setting. We built our models' assumptions under solid theoretical foundations. For OMIC, we provided theoretical guarantees in the form of generalization bounds by considering the distribution-free case: no assumptions about the sampling distribution are made. Additionally, we conducted a theoretical analysis of community side information when the sampling distribution is known and an adjusted nuclear norm regularization is applied. We showed that our method requires just a few entries to accurately recover the ratings matrix if the structure of the ground truth closely matches the cluster side information. For BMAB, we provided regret guarantees under mild conditions that demonstrate how the system's stability affects the expected reward. Furthermore, we conducted extensive experiments to validate our proposed methodologies. In a controlled environment, we implemented synthetic data generation techniques capable of replicating the domains for which OMIC and BMAB were designed. As a result, we were able to analyze our algorithms' performance across a broad spectrum of ground truth regimes. Finally, we replicated a real-world scenario by utilizing well-established recommender datasets. After comparing our approaches to several baselines, we observe that they achieved state-of-the-art results in terms of accuracy. Apart from being highly accurate, these methods improve interpretability by describing and quantifying features of the datasets they characterize.
Ein Beitrag zur Zustandsschätzung in Niederspannungsnetzen mit niedrigredundanter Messwertaufnahme
(2020)
Durch den wachsenden Anteil an Erzeugungsanlagen und leistungsstarken Verbrauchern aus dem Verkehr- und Wärmesektor kommen Niederspannungsnetze immer näher an ihre Betriebsgrenzen. Da für die Niederspannungsnetze bisher keine Messwerterfassung vorgesehen war, können Netzbetreiber Grenzverletzungen nicht erkennen. Um dieses zu ändern, werden deutsche Anschlussnutzer in Zukunft flächendeckend mit modernen Messeinrichtungen oder intelligenten Messsystemen (auch als Smart Meter bezeichnet) ausgestattet sein. Diese sind in der Lage über eine Kommunikationseinheit, das Smart-Meter-Gateway, Messdaten an die Netzbetreiber zu senden. Werden Messdaten aber als personenbezogene Netzzustandsdaten deklariert, so ist aus Datenschutzgründen eine Erhebung dieser Daten weitgehend untersagt.
Ziel dieser Arbeit ist es eine Zustandsschätzung zu entwickeln, die auch bei niedrigredundanter Messwertaufnahme für den Netzbetrieb von Niederspannungsnetzen anwendbare Ergebnisse liefert. Neben geeigneten Algorithmen zur Zustandsschätzung ist dazu die Generierung von Ersatzwerten im Fokus.
Die Untersuchungen und Erkenntnisse dieser Arbeit tragen dazu bei, den Verteilnetzbetreibern bei den maßgeblichen Entscheidungen in Bezug auf die Zustandsschätzung in Niederspannungsnetzen zu unterstützen. Erst wenn Niederspannungsnetze mit Hilfe der Zustandsschätzung beobachtbar sind, können darauf aufbauende Konzepte zur Regelung entwickelt werden, um die Energiewende zu unterstützen.
Many loads acting on a vehicle depend on the condition and quality of roads
traveled as well as on the driving style of the motorist. Thus, during vehicle development,
good knowledge on these further operations conditions is advantageous.
For that purpose, usage models for different kinds of vehicles are considered. Based
on these mathematical descriptions, representative routes for multiple user
types can be simulated in a predefined geographical region. The obtained individual
driving schedules consist of coordinates of starting and target points and can
thus be routed on the true road network. Additionally, different factors, like the
topography, can be evaluated along the track.
Available statistics resulting from travel survey are integrated to guarantee reasonable
trip length. Population figures are used to estimate the number of vehicles in
contained administrative units. The creation of thousands of those geo-referenced
trips then allows the determination of realistic measures of the durability loads.
Private as well as commercial use of vehicles is modeled. For the former, commuters
are modeled as the main user group conducting daily drives to work and
additional leisure time a shopping trip during workweek. For the latter, taxis as
example for users of passenger cars are considered. The model of light-duty commercial
vehicles is split into two types of driving patterns, stars and tours, and in
the common traffic classes of long-distance, local and city traffic.
Algorithms to simulate reasonable target points based on geographical and statistical
data are presented in detail. Examples for the evaluation of routes based
on topographical factors and speed profiles comparing the influence of the driving
style are included.
We present a numerical scheme to simulate a moving rigid body with arbitrary shape suspended in a rarefied gas micro flows, in view of applications to complex computations of moving structures in micro or vacuum systems. The rarefied gas is simulated by solving the Boltzmann equation using a DSMC particle method. The motion of the rigid body is governed by the Newton-Euler equations, where the force and the torque on the rigid body is computed from the momentum transfer of the gas molecules colliding with the body. The resulting motion of the rigid body affects in turn again the gas flow in the surroundings. This means that a two-way coupling has been modeled. We validate the scheme by performing various numerical experiments in 1-, 2- and 3-dimensional computational domains. We have presented 1-dimensional actuator problem, 2-dimensional cavity driven flow problem, Brownian diffusion of a spherical particle both with translational and rotational motions, and finally thermophoresis on a spherical particles. We compare the numerical results obtained from the numerical simulations with the existing theories in each test examples.
Pedestrian Flow Models
(2014)
There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries.