Nichtüberlappende Gebietszerlegungsmethoden für lineare und quasilineare (monotone und nichtmonotone) Probleme

In dieser Arbeit werden nichtüberlappende Gebietszerlegungsmethoden einerseits hinsichtlich der zu lösenden Problemklassen verallgemeinert und andererseits in bisher nicht untersuchten Kontexten betrachtet. Dabei stehen funktionalanalytische Untersuchungen zur Wohldefiniertheit, eindeutigen Lösbarkeit und Konvergenz im Vordergrund. Im ersten Teil werden lineare elliptische Dirichlet-Randwertprobleme behandelt, wobei neben Problemen mit dominantem Hauptteil auch solche mit singulärer Störung desselben, wie konvektions- oder reaktionsdominante Probleme zugelassen sind. Der zweite Teil befasst sich mit (gleichmäßig) monotonen koerziven quasilinearen elliptischen Dirichlet-Randwertproblemen. In beiden Fällen wird das Lipschitz-Gebiet in endlich viele Lipschitz-Teilgebiete zerlegt, wobei insbesondere Kreuzungspunkte und Teilgebiete ohne Außenrand zugelassen sind. Anschließend werden Transmissionsprobleme mit frei wählbaren $L^{\infty}$-Parameterfunktionen hergeleitet, wobei die Konormalenableitungen als Funktionale auf geeigneten Funktionenräumen über den Teilrändern ($H_{00}^{1/2}(\Gamma)$) interpretiert werden. Die iterative Lösung dieser Transmissionsprobleme mit einem Ansatz von Deng führt auf eine Substrukturierungsmethode mit Robin-artigen Transmissionsbedingungen, bei der eine Auswertung der Konormalenableitungen aufgrund einer geschickten Aufdatierung der Robin-Daten nicht notwendig ist (insbesondere ist die bekannte Robin-Robin-Methode von Lions als Spezialfall enthalten). Die Konvergenz bezüglich einer partitionierten $H^1$-Norm wird für beide Problemklassen gezeigt. Dabei werden keine über $H^1$ hinausgehende Regularitätsforderungen an die Lösungen gestellt und die Gebiete müssen keine zusätzlichen Glattheitsvoraussetzungen erfüllen. Im letzten Kapitel werden nichtmonotone koerzive quasilineare Probleme untersucht, wobei das Zugrunde liegende Gebiet nur in zwei Lipschitz-Teilgebiete zerlegt sein soll. Das zugehörige nichtlineare Transmissionsproblem wird durch Kirchhoff-Transformation in lineare Teilprobleme mit nichtlinearen Kopplungsbedingungen überführt. Ein optimierungsbasierter Lösungsansatz, welcher einen geeigneten Abstand der rücktransformierten Dirichlet-Daten der linearen Teilprobleme auf den Teilrändern minimiert, führt auf ein optimales Kontrollproblem. Die dabei entstehenden regularisierten freien Minimierungsprobleme werden mit Hilfe eines Gradientenverfahrens unter minimalen Glattheitsforderungen an die Nichtlinearitäten gelöst. Unter zusätzlichen Glattheitsvoraussetzungen an die Nichtlinearitäten und weiteren technischen Voraussetzungen an die Lösung des quasilinearen Ausgangsproblems, kann zudem die quadratische Konvergenz des Newton-Verfahrens gesichert werden.

Handlungsinduktionen durch die visuelle Wahrnehmung von Linear- und Drehbewegungen im Sport

Die vorliegende Arbeit beschäftigt sich mit den Einflüssen visuell wahrgenommener Bewegungsmerkmale auf die Handlungssteuerung eines Beobachters. Im speziellen geht es darum, wie die Bewegungsrichtung und die Bewegungsgeschwindigkeit als aufgabenirrelevante Reize die Ausführung von motorischen Reaktionen auf Farbreize beeinflussen und dabei schnellere bzw. verzögerte Reaktionszeiten bewirken. Bisherige Studien dazu waren auf lineare Bewegungen (von rechts nach links und umgekehrt) und sehr einfache Reizumgebungen (Bewegungen einfacher geometrischer Symbole, Punktwolken, Lichtpunktläufer etc.) begrenzt (z.B. Ehrenstein, 1994; Bosbach, 2004, Wittfoth, Buck, Fahle & Herrmann, 2006). In der vorliegenden Dissertation wurde die Gültigkeit dieser Befunde für Dreh- und Tiefenbewegungen sowie komplexe Bewegungsformen (menschliche Bewegungsabläufe im Sport) erweitert, theoretisch aufgearbeitet sowie in einer Serie von sechs Reaktionszeitexperimenten mittels Simon-Paradigma empirisch überprüft. Allen Experimenten war gemeinsam, dass Versuchspersonen an einem Computermonitor auf einen Farbwechsel innerhalb des dynamischen visuellen Reizes durch einen Tastendruck (links, rechts, proximal oder distal positionierte Taste) reagieren sollten, wobei die Geschwindigkeit und die Richtung der Bewegungen für die Reaktionen irrelevant waren. Zum Einfluss von Drehbewegungen bei geometrischen Symbolen (Exp. 1 und 1a) sowie bei menschlichen Drehbewegungen (Exp. 2) zeigen die Ergebnisse, dass Probanden signifikant schneller reagieren, wenn die Richtungsinformationen einer Drehbewegung kompatibel zu den räumlichen Merkmalen der geforderten Tastenreaktion sind. Der Komplexitätsgrad des visuellen Ereignisses spielt dabei keine Rolle. Für die kognitive Verarbeitung des Bewegungsreizes stellt nicht der Drehsinn, sondern die relative Bewegungsrichtung oberhalb und unterhalb der Drehachse das entscheidende räumliche Kriterium dar. Zum Einfluss räumlicher Tiefenbewegungen einer Kugel (Exp. 3) und einer gehenden Person (Exp. 4) belegen unsere Befunde, dass Probanden signifikant schneller reagieren, wenn sich der Reiz auf den Beobachter zu bewegt und ein proximaler gegenüber einem distalen Tastendruck gefordert ist sowie umgekehrt. Auch hier spielt der Komplexitätsgrad des visuellen Ereignisses keine Rolle. In beiden Experimenten führt die Wahrnehmung der Bewegungsrichtung zu einer Handlungsinduktion, die im kompatiblen Fall eine schnelle und im inkompatiblen Fall eine verzögerte Handlungsausführung bewirkt. In den Experimenten 5 und 6 wurden die Einflüsse von wahrgenommenen menschlichen Laufbewegungen (freies Laufen vs. Laufbandlaufen) untersucht, die mit und ohne eine Positionsveränderung erfolgten. Dabei zeigte sich, dass unabhängig von der Positionsveränderung die Laufgeschwindigkeit zu keiner Modulation des richtungsbasierten Simon Effekts führt. Zusammenfassend lassen sich die Studienergebnisse gut in effektbasierte Konzepte zur Handlungssteuerung (z.B. die Theorie der Ereigniskodierung von Hommel et al., 2001) einordnen. Weitere Untersuchungen sind nötig, um diese Ergebnisse auf großmotorische Reaktionen und Displays, die stärker an visuell wahrnehmbaren Ereignissen des Sports angelehnt sind, zu übertragen.

Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

Implementing Distributed Systems Using Linear Naming

Linear graph reduction is a simple computational model in which the cost of naming things is explicitly represented. The key idea is the notion of "linearity". A name is linear if it is only used once, so with linear naming you cannot create more than one outstanding reference to an entity. As a result, linear naming is cheap to support and easy to reason about. Programs can be translated into the linear graph reduction model such that linear names in the program are implemented directly as linear names in the model. Nonlinear names are supported by constructing them out of linear names. The translation thus exposes those places where the program uses names in expensive, nonlinear ways. Two applications demonstrate the utility of using linear graph reduction: First, in the area of distributed computing, linear naming makes it easy to support cheap cross-network references and highly portable data structures, Linear naming also facilitates demand driven migration of tasks and data around the network without requiring explicit guidance from the programmer. Second, linear graph reduction reveals a new characterization of the phenomenon of state. Systems in which state appears are those which depend on certain -global- system properties. State is not a localizable phenomenon, which suggests that our usual object oriented metaphor for state is flawed.

Model-Based Matching by Linear Combinations of Prototypes

We describe a method for modeling object classes (such as faces) using 2D example images and an algorithm for matching a model to a novel image. The object class models are "learned'' from example images that we call prototypes. In addition to the images, the pixelwise correspondences between a reference prototype and each of the other prototypes must also be provided. Thus a model consists of a linear combination of prototypical shapes and textures. A stochastic gradient descent algorithm is used to match a model to a novel image by minimizing the error between the model and the novel image. Example models are shown as well as example matches to novel images. The robustness of the matching algorithm is also evaluated. The technique can be used for a number of applications including the computation of correspondence between novel images of a certain known class, object recognition, image synthesis and image compression.

Learning Linear, Sparse, Factorial Codes

In previous work (Olshausen & Field 1996), an algorithm was described for learning linear sparse codes which, when trained on natural images, produces a set of basis functions that are spatially localized, oriented, and bandpass (i.e., wavelet-like). This note shows how the algorithm may be interpreted within a maximum-likelihood framework. Several useful insights emerge from this connection: it makes explicit the relation to statistical independence (i.e., factorial coding), it shows a formal relationship to the algorithm of Bell and Sejnowski (1995), and it suggests how to adapt parameters that were previously fixed.

Model-Based Matching of Line Drawings by Linear Combinations of Prototypes

We describe a technique for finding pixelwise correspondences between two images by using models of objects of the same class to guide the search. The object models are 'learned' from example images (also called prototypes) of an object class. The models consist of a linear combination ofsprototypes. The flow fields giving pixelwise correspondences between a base prototype and each of the other prototypes must be given. A novel image of an object of the same class is matched to a model by minimizing an error between the novel image and the current guess for the closest modelsimage. Currently, the algorithm applies to line drawings of objects. An extension to real grey level images is discussed.

On Convergence Properties of the EM Algorithm for Gaussian Mixtures

"Expectation-Maximization'' (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix $P$, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of $P$ and provide new results analyzing the effect that $P$ has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models.

On the Convergence of Stochastic Iterative Dynamic Programming Algorithms

Recent developments in the area of reinforcement learning have yielded a number of new algorithms for the prediction and control of Markovian environments. These algorithms, including the TD(lambda) algorithm of Sutton (1988) and the Q-learning algorithm of Watkins (1989), can be motivated heuristically as approximations to dynamic programming (DP). In this paper we provide a rigorous proof of convergence of these DP-based learning algorithms by relating them to the powerful techniques of stochastic approximation theory via a new convergence theorem. The theorem establishes a general class of convergent algorithms to which both TD(lambda) and Q-learning belong.

On Degeneracy of Linear Reconstruction from Three Views: Linear Line Complex and Applications

This paper investigates the linear degeneracies of projective structure estimation from point and line features across three views. We show that the rank of the linear system of equations for recovering the trilinear tensor of three views reduces to 23 (instead of 26) in the case when the scene is a Linear Line Complex (set of lines in space intersecting at a common line) and is 21 when the scene is planar. The LLC situation is only linearly degenerate, and we show that one can obtain a unique solution when the admissibility constraints of the tensor are accounted for. The line configuration described by an LLC, rather than being some obscure case, is in fact quite typical. It includes, as a particular example, the case of a camera moving down a hallway in an office environment or down an urban street. Furthermore, an LLC situation may occur as an artifact such as in direct estimation from spatio-temporal derivatives of image brightness. Therefore, an investigation into degeneracies and their remedy is important also in practice.

Technological convergence: a strategic perspective

The information and communication technologies (ICT) sectors are in a process of technological convergence. Determinant factors in this process are the liberalisation of the telecommunications markets and technological change. Many firms are engaged in a process of mergers and alliances to position themselves in this new framework. Technological and demand uncertainties are very important. Our objective in this paper is to study the economic determinants of the strategies of the firms. With this aim, we review some key technological and demand aspects. We shed some light on the strategic motivations of the firms by establishing a parallel with the evolution of the retailing sector

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

Non-linear mechanical behavior of the elastomer polydimethylsiloxane (PDMS) used in the manufacture of microfluidic devices

Polydimethylsiloxane (PDMS) is the elastomer of choice to create a variety of microfluidic devices by soft lithography techniques (eg., [1], [2], [3], [4]). Accurate and reliable design, manufacture, and operation of microfluidic devices made from PDMS, require a detailed characterization of the deformation and failure behavior of the material. This paper discusses progress in a recently-initiated research project towards this goal. We have conducted large-deformation tension and compression experiments on traditional macroscale specimens, as well as microscale tension experiments on thin-film (≈ 50µm thickness) specimens of PDMS with varying ratios of monomer:curing agent (5:1, 10:1, 20:1). We find that the stress-stretch response of these materials shows significant variability, even for nominally identically prepared specimens. A non-linear, large-deformation rubber-elasticity model [5], [6] is applied to represent the behavior of PDMS. The constitutive model has been implemented in a finite-element program [7] to aid the design of microfluidic devices made from this material. As a first attempt towards the goal of estimating the non-linear material parameters for PDMS from indentation experiments, we have conducted micro-indentation experiments using a spherical indenter-tip, and carried out corresponding numerical simulations to verify how well the numerically-predicted P(load-h(depth of indentation) curves compare with the corresponding experimental measurements. The results are encouraging, and show the possibility of estimating the material parameters for PDMS from relatively simple micro-indentation experiments, and corresponding numerical simulations.

Convex linear combination processes for compositions

Aitchison and Bacon-Shone (1999) considered convex linear combinations of compositions. In other words, they investigated compositions of compositions, where the mixing composition follows a logistic Normal distribution (or a perturbation process) and the compositions being mixed follow a logistic Normal distribution. In this paper, I investigate the extension to situations where the mixing composition varies with a number of dimensions. Examples would be where the mixing proportions vary with time or distance or a combination of the two. Practical situations include a river where the mixing proportions vary along the river, or across a lake and possibly with a time trend. This is illustrated with a dataset similar to that used in the Aitchison and Bacon-Shone paper, which looked at how pollution in a loch depended on the pollution in the three rivers that feed the loch. Here, I explicitly model the variation in the linear combination across the loch, assuming that the mean of the logistic Normal distribution depends on the river flows and relative distance from the source origins