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This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.
Applications in various research areas such as signal processing, quantum computing, and computer vision, can be described as constrained optimization tasks on certain subsets of tensor products of vector spaces. In this work, we make use of techniques from Riemannian geometry and analyze optimization tasks on subsets of so-called simple tensors which can be equipped with a differentiable structure. In particular, we introduce a generalized Rayleigh-quotient function on the tensor product of Grassmannians and on the tensor product of Lagrange- Grassmannians. Its optimization enables a unified approach to well-known tasks from different areas of numerical linear algebra, such as: best low-rank approximations of tensors (data compression), computing geometric measures of entanglement (quantum computing) and subspace clustering (image processing). We perform a thorough analysis on the critical points of the generalized Rayleigh-quotient and develop intrinsic numerical methods for its optimization. Explicitly, using the techniques from Riemannian optimization, we present two type of algorithms: a Newton-like and a conjugated gradient algorithm. Their performance is analysed and compared with established methods from the literature.
Optimization problems with composite functions deal with the minimization of the sum
of a smooth function and a convex nonsmooth function. In this thesis several numerical
methods for solving such problems in finite-dimensional spaces are discussed, which are
based on proximity operators.
After some basic results from convex and nonsmooth analysis are summarized, a first-order
method, the proximal gradient method, is presented and its convergence properties are
discussed in detail. Known results from the literature are summarized and supplemented by
additional ones. Subsequently, the main part of the thesis is the derivation of two methods
which, in addition, make use of second-order information and are based on proximal Newton
and proximal quasi-Newton methods, respectively. The difference between the two methods
is that the first one uses a classical line search, while the second one uses a regularization
parameter instead. Both techniques lead to the advantage that, in contrast to many similar
methods, in the respective detailed convergence analysis global convergence to stationary
points can be proved without any restricting precondition. Furthermore, comprehensive
results show the local convergence properties as well as convergence rates of these algorithms,
which are based on rather weak assumptions. Also a method for the solution of the arising
proximal subproblems is investigated.
In addition, the thesis contains an extensive collection of application examples and a detailed
discussion of the related numerical results.
Future broadband wireless networks should be able to support not only best effort traffic but also real-time traffic with strict Quality of Service (QoS) constraints. In addition, their available resources are scare and limit the number of users. To facilitate QoS guarantees and increase the maximum number of concurrent users, wireless networks require careful planning and optimization. In this monograph, we studied three aspects of performance optimization in wireless networks: resource optimization in WLAN infrastructure networks, quality of experience control in wireless mesh networks, and planning and optimization of wireless mesh networks. An adaptive resource management system is required to effectively utilize the limited resources on the air interface and to guarantee QoS for real-time applications. Thereby, both WLAN infrastructure and WLAN mesh networks have to be considered. An a-priori setting of the access parameters is not meaningful due to the contention-based medium access and the high dynamics of the system. Thus, a management system is required which dynamically adjusts the channel access parameters based on the network load. While this is sufficient for wireless infrastructure networks, interferences on neighboring paths and self-interferences have to be considered for wireless mesh networks. In addition, a careful channel allocation and route assignment is needed. Due to the large parameter space, standard optimization techniques fail for optimizing large wireless mesh networks. In this monograph, we reveal that biology-inspired optimization techniques, namely genetic algorithms, are well-suitable for the planning and optimization of wireless mesh networks. Although genetic algorithms generally do not always find the optimal solution, we show that with a good parameter set for the genetic algorithm, the overall throughput of the wireless mesh network can be significantly improved while still sharing the resources fairly among the users.
The Software Defined Networking (SDN) paradigm offers network operators numerous improvements in terms of flexibility, scalability, as well as cost efficiency and vendor independence. However, in order to maximize the benefit from these features, several new challenges in areas such as management and orchestration need to be addressed. This dissertation makes contributions towards three key topics from these areas.
Firstly, we design, implement, and evaluate two multi-objective heuristics for the SDN controller placement problem. Secondly, we develop and apply mechanisms for automated decision making based on the Pareto frontiers that are returned by the multi-objective optimizers. Finally, we investigate and quantify the performance benefits for the SDN control plane that can be achieved by integrating information from external entities such as Network Management Systems (NMSs) into the control loop. Our evaluation results demonstrate the impact of optimizing various parameters of softwarized networks at different levels and are used to derive guidelines for an efficient operation.
At the center of the Internet’s protocol stack stands the Internet Protocol (IP) as a common denominator that enables all communication. To make routing efficient, resilient, and scalable, several aspects must be considered. Care must be taken that traffic is well balanced to make efficient use of the existing network resources, both in failure free operation and in failure scenarios.
Finding the optimal routing in a network is an NP-complete problem. Therefore, routing optimization is usually performed using heuristics. This dissertation shows that a routing optimized with one objective function is often not good when looking at other objective functions. It can even be worse than unoptimized routing with respect to that objective function. After looking at failure-free routing and traffic distribution in different failure scenarios, the analysis is extended to include the loop-free alternate (LFA) IP fast reroute mechanism. Different application scenarios of LFAs are examined and a special focus is set on the fact that LFAs usually cannot protect all traffic in a network even against single link failures. Thus, the routing optimization for LFAs is targeted on both link utilization and failure coverage. Finally, the pre-congestion notification mechanism PCN for network admission control and overload protection is analyzed and optimized. Different design options for implementing the protocol are compared, before algorithms are developed for the calculation and optimization of protocol parameters and PCN-based routing.
The second part of the thesis tackles a routing problem that can only be resolved on a global scale. The scalability of the Internet is at risk since a major and intensifying growth of the interdomain routing tables has been observed. Several protocols and architectures are analyzed that can be used to make interdomain routing more scalable. The most promising approach is the locator/identifier (Loc/ID) split architecture which separates routing from host identification. This way, changes in connectivity, mobility of end hosts, or traffic-engineering activities are hidden from the routing in the core of the Internet and the routing tables can be kept much smaller. All of the currently proposed Loc/ID split approaches have their downsides. In particular, the fact that most architectures use the ID for routing outside the Internet’s core is a poor design, which inhibits many of the possible features of a new routing architecture. To better understand the problems and to provide a solution for a scalable routing design that implements a true Loc/ID split, the new GLI-Split protocol is developed in this thesis, which provides separation of global and local routing and uses an ID that is independent from any routing decisions.
Besides GLI-Split, several other new routing architectures implementing Loc/ID split have been proposed for the Internet. Most of them assume that a mapping system is queried for EID-to-RLOC mappings by an intermediate node at the border of an edge network. When the mapping system is queried by an intermediate node, packets are already on their way towards their destination, and therefore, the mapping system must be fast, scalable, secure, resilient, and should be able to relay packets without locators to nodes that can forward them to the correct destination. The dissertation develops a classification for all proposed mapping system architectures and shows their similarities and differences. Finally, the fast two-level mapping system FIRMS is developed. It includes security and resilience features as well as a relay service for initial packets of a flow when intermediate nodes encounter a cache miss for the EID-to-RLOC mapping.
Grundvoraussetzung einer erfolgreichen zementlosen Endoprothesenverankerungen ist eine hohe Primärstabilität durch formschlüssige Implantation. Nach erfolgreicher Primärfixation entscheidet über die Langzeitfunktion neben der Verschleißsituation insbesondere die funktionelle Spannungsverteilung an der Implantatoberfläche und im angrenzenden Knochen. Um optimale Bedingungen im Bereich der Grenzfläche zwischen Werkstoff und Biosystem zu schaffen, ist die präzise Präparation des Knochens entscheidend. Hierbei spielt neben dem Operateur und der Beschaffenheit des Knochens das verwendete System zur Zerspanung, bestehend aus der Säge und dem Sägeblatt, eine wichtige Rolle. Oszillierende Sägen werden neben der Gipsbehandlung ausschließlich bei der Knochenbearbeitung verwendet, wobei Fragestellungen zu Leistungs- und Qualitätssteigerung auftreten. Ansatzpunkt ist unter anderem das Sägeblatt. Die Modellvielfalt der auf dem Markt erhältlichen Sägeblätter erschwert die richtige Auswahl. Bei verschiedenen Modellen und deren unterschiedlichen Ausführungen ist es schwierig, Vergleiche anzustellen und Verbesserungen einzuführen. Die Charakteristik eines Sägeblattes ist durch die Schärfe beschrieben, um eine schnelle Zerspanung des Knochens zu gewährleisten, und durch die Steifigkeit des Blattes, um eine möglichst geringe Abweichung aus der Sägelinie sicherzustellen. Die Schärfe des Sägeblattes ist von der Zahnform und -geometrie abhängig. Die Steifigkeit ist abhängig von Material, Geometrie, Ausführung und Eigenschwingung/Eigenform des Blattes. Modifikationen an diesem System führen auch zu einer veränderten Eigenform des Sägeblattes. In der folgender Arbeit wurden verschiedene Ausführungen von Sägeblättern hinsichtlich ihrer Eigenform charakterisiert. In einem praktischen Versuch wurde versucht, den Einfluss des Sägeblattes auf die Vibration im Knochen von dem der Säge abzugrenzen. Ziel der Studie war daher die Berechnung der Eigenform der Sägeblätter, die experimentelle Überprüfung der Eigenform und die Abgrenzung des Einflusses von Sägeblatt und Säge auf die Vibration beim Sägevorgang. 9 verschiedene Sägeblätter unterschiedlicher Form und Geometrie wurden untersucht. Zunächst wurde eine Eigenformberechnung mittels der Finiten Elemente Methode durchgeführt. Anschließend erfolgte die Eigenformbestimmung im Shakerversuch. Um den Einfluss der Sägeblätter auf die Beschleunigung im Knochen von dem der Säge abzugrenzen erfolgte die Beschleunigungsmessung in einem Sägeprüfstand. Bezüglich des Eigenschwingungsverhalten konnten die Sägeblätter drei Gruppen zugeordnet werden. Als optimal erweist sich anhand der Eigenformberechnung und -bestimmung ein Sägeblatt mit 2 Löchern und einer Prägung von 136 bar. Hier zeigte sich eine Reduzierung der Maximal-Amplitude von 15,3 % und der Minimal-Amplitude von 23,5 %.
In the future Internet, the people-centric communication paradigm will be complemented by a ubiquitous communication among people and devices, or even a communication between devices. This comes along with the need for a more flexible, cheap, widely available Internet access. Two types of wireless networks are considered most appropriate for attaining those goals. While wireless sensor networks (WSNs) enhance the Internet’s reach by providing data about the properties of the environment, wireless mesh networks (WMNs) extend the Internet access possibilities beyond the wired backbone. This monograph contains four chapters which present modeling and optimization methods for WSNs and WMNs. Minimizing energy consumptions is the most important goal of WSN optimization and the literature consequently provides countless energy consumption models. The first part of the monograph studies to what extent the used energy consumption model influences the outcome of analytical WSN optimizations. These considerations enable the second contribution, namely overcoming the problems on the way to a standardized energy-efficient WSN communication stack based on IEEE 802.15.4 and ZigBee. For WMNs both problems are of minor interest whereas the network performance has a higher weight. The third part of the work, therefore, presents algorithms for calculating the max-min fair network throughput in WMNs with multiple link rates and Internet gateway. The last contribution of the monograph investigates the impact of the LRA concept which proposes to systematically assign more robust link rates than actually necessary, thereby allowing to exploit the trade-off between spatial reuse and per-link throughput. A systematical study shows that a network-wide slightly more conservative LRA than necessary increases the throughput of a WMN where max-min fairness is guaranteed. It moreover turns out that LRA is suitable for increasing the performance of a contention-based WMN and is a valuable optimization tool.
Today's Internet is no longer only controlled by a single stakeholder, e.g. a standard body or a telecommunications company.
Rather, the interests of a multitude of stakeholders, e.g. application developers, hardware vendors, cloud operators, and network operators, collide during the development and operation of applications in the Internet.
Each of these stakeholders considers different KPIs to be important and attempts to optimise scenarios in its favour.
This results in different, often opposing views and can cause problems for the complete network ecosystem.
One example of such a scenario are Signalling Storms in the mobile Internet, with one of the largest occurring in Japan in 2012 due to the release and high popularity of a free instant messaging application.
The network traffic generated by the application caused a high number of connections to the Internet being established and terminated.
This resulted in a similarly high number of signalling messages in the mobile network, causing overload and a loss of service for 2.5 million users over 4 hours.
While the network operator suffers the largest impact of this signalling overload, it does not control the application.
Thus, the network operator can not change the application traffic characteristics to generate less network signalling traffic.
The stakeholders who could prevent, or at least reduce, such behaviour, i.e. application developers or hardware vendors, have no direct benefit from modifying their products in such a way.
This results in a clash of interests which negatively impacts the network performance for all participants.
The goal of this monograph is to provide an overview over the complex structures of stakeholder relationships in today's Internet applications in mobile networks.
To this end, we study different scenarios where such interests clash and suggest methods where tradeoffs can be optimised for all participants.
If such an optimisation is not possible or attempts at it might lead to adverse effects, we discuss the reasons.
Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces
(2018)
This thesis is concerned with a class of general-purpose algorithms for constrained minimization problems, variational inequalities, and quasi-variational inequalities in Banach spaces.
A substantial amount of background material from Banach space theory, convex analysis, variational analysis, and optimization theory is presented, including some results which are refinements of those existing in the literature. This basis is used to formulate an augmented Lagrangian algorithm with multiplier safeguarding for the solution of constrained optimization problems in Banach spaces. The method is analyzed in terms of local and global convergence, and many popular problem classes such as nonlinear programming, semidefinite programming, and function space optimization are shown to be included as special cases of the general setting.
The algorithmic framework is then extended to variational and quasi-variational inequalities, which include, by extension, Nash and generalized Nash equilibrium problems. For these problem classes, the convergence is analyzed in detail. The thesis then presents a rich collection of application examples for all problem classes, including implementation details and numerical results.