Subduction zones are regions of intense earthquake activity up to great depth. Sources are located inside the subducting lithosphere and, as a consequence, seismic radiation from subduction zone earthquakes is strongly affected by the interior slab structure. The wave field of these intraslab events observed in the forearc region is profoundly influenced by a seismically slow layer atop the slab surface. This several kilometer thick low-velocity channel (wave guide) causes the entrapment of seismic energy producing strong guided wave phases that appear in P onsets in certain regions of the forearc. Observations at the Chile-Peru subduction zone presented here, as well as observations at several other circum-pacific subduction zones show such signals. Guided wave analysis contributes details of immense value regarding the processes near the slab surface, such as layering of subducted lithosphere, source locations of intraslab seismicity and most of all, range and manner of mineralogical phase transitions. <br><br> Seismological data stem from intermediate depth events (depth range 70 km - 300 km) recorded in northern Chile near 21 Grad S during the collaborative research initiative " Deformation Processes in the Andes" (SFB 267). A subset of stations - all located within a slab-parallel transect close to 69 Grad W - show low-frequency first arrivals (2 Hz), sometimes followed by a second high-frequency phase. <br><br> We employ 2-dimensional finite-difference simulations of complete P-SV wave propagation to explore the parameter space of subduction zone wave guides and explain the observations. Key processes underlying the guided wave propagation are studied: Two distinct mechanisms of decoupling of trapped energy from the wave guide are analyzed - a prerequisite to observe the phases at stations located at large distances from the wave guide (up to 100 km). Variations of guided wave effects perpendicular to the strike of the subduction zone are investigated, such as the influence of phases traveling in the fast slab. Further, the merits and limits of guided wave analysis are assessed. Frequency spectra of the guided wave onsets prove to be a robust quantity that captures guided wave characteristics at subduction zones including higher mode excitation. They facilitate the inference of wave guide structure and source positioning: The peak frequency of the guided wave fundamental mode is associated with a certain combination of layer width and velocity contrast. The excitation strength of the guided wave fundamental mode and higher modes is associated with source position and orientation relative to the low-velocity layer. <br><br> The guided wave signals at the Chile-Peru subduction zone are caused by energy that leaks from the subduction zone wave guide. On the one hand, the bend shape of the slab allows for leakage at a depth of 100 km. On the other, equalization of velocities between the wave guide and the host rocks causes further energy leakage at the contact zone between continental and oceanic crust (70 km depth). Guided waves bearing information on deep slab structure can therefore be recorded at specific regions in the forearc. These regions are determined based on slab geometry, and their locations coincide with the observations. <br><br> A number of strong constraints on the structure of the Chile-Peru slab are inferred: The deep wave guide for intraslab events is formed by a layer of 2 km average width that remains seismically slow (7 percent velocity reduction compared to surrounding mantle). This low-velocity layer at the top of the Chile-Peru slab is imaged from a depth of 100 km down to at least 160 km. Intermediate depth events causing the observed phases are located inside the layer or directly beneath it in the slab mantle. The layer is interpreted as partially eclogized lower oceanic crust persisting to depth beyond the volcanic arc. / Subduktionszonen sind bis in große Tiefen von intensiver Erdbebentätigkeit geprägt. Die Erdbebenquellen befinden sich in der subduzierten Lithosphäre (Slab), ihr Wellenfeld wird deshalb stark von der internen Slab-Struktur beeinflusst. Eine Schicht mit reduzierter seismischer Geschwindigkeit im oberen Bereich der Platte kann als Wellenleiter für diese Signale fungieren. In der nur wenige Kilometer dicken Schicht entstehen sogenannte geführte Wellen, die in Teilen des Forearc beobachtet werden. Diese Phasen bergen wertvolle Informationen über die Struktur nahe der Slab-Oberfläche, wie zum Beispiel Dicke der Schichtung, Herdlokationen und vor allem Tiefe und Art mineralogischer Umsetzungen. <br><br> Die Beobachtungen stammen von mitteltiefen Beben (70 km - 300 km) im Untersuchungsgebiet in Nord-Chile und wurden im Rahmen des Sonderforschungsbereich 267 " Deformationsprozesse in den Anden" aufgezeichnet. Stationen in einem Streifen um 69 Grad W, der sich parallel zum Streichen der Subduktionszone erstreckt, zeigen niederfrequente Ersteinsätze, denen teilweise höherfrequente Phasen folgen. <br><br> Mit Hilfe eines 2-dimensionalen Finite-Differenzen-Algorithmus werden die P-SV Wellenausbreitung simuliert, und die Beobachtungen erklärt. Zentrale Fragestellungen zu Wellenleitern in Subduktionszonen werden untersucht: Es werden zwei Mechanismen, die das Auskoppeln seismischer Energie aus dem Wellenleiter ermöglichen beschrieben - eine Grundvoraussetzung für das Auftreten von geführten Wellen in großen Entfernungen vom Wellenleiter (bis zu 100 km). Des weiteren werden Stärken und Grenzen der Analyse von geführten Wellen erörtert. <br><br> Die Spektren der geführten Wellenzüge erweisen sich als robuste Messgröße, um die Charakteristika des Wellenleiters zu bestimmt. Struktur des Wellenleiters und Quellpositionen können so für festgelegte Quell-Empfänger-Geometrien abgeleitet werden. Die Peak-Frequenz der Grundmode wird durch eine Kombination aus Dicke der Schicht und Geschwindigkeitskontrast bestimmt. Die Stärke der Anregung der Grundmode und höherer Moden lässt auf die Lage und Orientierung der Erdbebenquelle relativ zur Schicht schließen. Geschwindigkeitskontrast, Schichtdicke und Quellposition sind von herausragender Bedeutung, um mineralogische Interpretationen des Wellenleiters zu überprüfen. <br><br> Aufbauend auf die Simulationen werden die Beobachtungen interpretiert und Auskunft über die Struktur der Chile-Peru Subduktionszone erhalten: Eine dünne Schicht an der Slab-Oberfläche (durchschnittlich 2 km dick) trägt geringere seismische Geschwindigkeiten als der umgebende Mantel und fungiert als Wellenleiter für intra-platten Ereignisse in Tiefen von 100 bis mindestens 160 km. Ereignisse, die geführte Wellen hervorrufen, liegen in dieser Schicht oder direkt darunter im subduzierten Mantel. Um zu den Stationen in der Forearc-Region zu gelangen, entkoppelt ein Teil der geführten Wellen in einer Tiefe von circa 100 km aus der Niedergeschwindigkeitsschicht. Die Krümmung des Slab erlaubt das Austreten der Wellen und nimmt auch Einfluss auf die Pulsformen. <br><br> Der Wellenleiter in der Chile-Peru Subduktionszone ergibt sich als unregelmäßige Schicht mit reduzierter seismischer Geschwindigkeit, in der geführte Wellen entstehen, in unterschiedlichen Tiefen wieder austreten, und an die freie Oberfläche gelangen. Die Beobachtungsgebiete befinden sich im Forearc und werden durch die Geometrie und Struktur der subduzierten Platte festgelegt. <br><br> Die nur wenige Kilometer dicke, seismisch langsame Schicht an der Oberfläche des Chile-Peru Slab legt nahe, dass die Unterkruste der subduzierten Platte bis in große Tiefen besteht und nicht vollständig eklogitisiert ist. Abgeleitete Schichtdicke, Geschwindigkeitskontrast

This thesis consists of four papers on the following topics in analysis and probability: analysis on Wiener space, asymptotic properties of orthogonal polynomials, and convergence rates in the central limit theorem. The first paper gives lower bounds on the constants in the Meyer inequality from the Malliavin calculus. It is shown that both constants grow at least like (p-1)-1 or like p when p approaches 1 or ∞ respectively. This agrees with known upper bounds. In the second paper, an extremal problem on Wiener chaos motivates an investigation of the Lp-norms of Hermite polynomials. This is followed up by similar computations for Charlier polynomials in the third paper. In both cases, the Lp-norms present a peculiar behaviour with certain threshold values of p, where the growth rate and the dominating intervals undergo a rapid change. The fourth paper analyzes a connection between probability and numerical analysis. More precisely, known estimates on the convergence rate of finite difference equations are "translated" into results on convergence rates of certain functionals in the central limit theorem. These are also extended, using interpolation of Banach spaces as a main tool. Besov spaces play a central role in the emerging results.

<p>In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We &macr / nd that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis</p>

Laser beam control is important in many applications. Phase modulating spatial light modulators (SLMs) can be used to electronically alter the phase distribution of an optical wave-front and thus change the direction and shape of a laser beam. Physical constraints set limitations to the SLM and an ideal phase distribution can usually not be realised. In order to understand how such components can be used for non-mechanical beam control three nematic liquid crystal (NLC) SLMs have been thoroughly characterised and modelled. The pixel structure and phase quantisation give a discrepancy between ideal and realised phase distributions. The impact on beam steering capability was examined by measurements and simulations of the intensity distribution in the far-field. In two of the studied SLMs the pixel period was shorter than the thickness of the LC layer giving the optical phase shift. This results in a so-called “fringing field”, which was shown to degrade the phase modulation and couple light between polarisation modes. The deformation of the LC was simulated and a finite-difference time-domain (FDTD) algorithm was used to calculate how polarised light propagates through the optically anisotropic SLM. Non-mechanical beam steering and tracking in an optical free-space communication link were demonstrated. Continual optimisation of the steering angle was achieved by feedback from a video camera. The optical properties of the SLM in the time period right after a voltage update were studied. It was shown how light is redistributed between orders during the switching from one blazed grating to another. By appropriate choice of the blazed gratings the effects on the diffraction efficiency can be minimised. The detailed knowledge of the SLM structure and its response to electronic control makes it possible to predict and optimise the device performance in future systems.

In the present thesis, computational methods for theanalysis and design of photonic bandgap structure areconsidered. Many numerical methods have been used to study suchstructures. Among them, the plane wave expansion method is veryoften used. Using this method, we show that inclusions ofelliptic air holes can be used effectively to obtain a largercomplete band gap for two-dimensional (2D) photonic crystals.An optimal design of a 2D photonic crystal is also consideredin the thesis using a combination of the plane wave expansionmethod and the conjugate gradient method. We find that amaximum complete 2D band gap can be obtained by connectingdielectric rods with veins for a photonic crystal with a squarelattice of air holes in GaAs. For some problems, such as defect modes, the plane waveexpansion method is extremely time-consuming. It seems that thefinite-difference time-domain (FDTD) method is promising, sincethe computational time is proportional to the number of thediscretization points in the computation domain (i.e., it is oforderN). A FDTD scheme in a nonorthogonal coordinate systemis presented in the thesis to calculate the band structure of a2D photonic crystal consisting of askew lattice. The algorithmcan easily be used for any complicated inclusion configuration,which can have both the dielectric and metallic constituents.The FDTD method is also applied to calculate the off-plane bandstructures of 2D photonic crystals in the present thesis. Wealso propose a numerical method for computing defect modes in2D crystals (with dielectric or metallic inclusions). Comparedto the FDTD transmission spectra method, our method reduces thecomputation time and memory significantly, and finds as manydefect modes as possible, including those that are not excitedby an incident plane wave in the FDTD transmission spectramethod. The FDTD method has also been applied to calculateguided modes and surface modes in 2D photonic crystals using acombination of the periodic boundary condition and theperfectly matched layer for the boundary treatment. Anefficient FDTD method, in which only real variables are used,is also proposed for the full-wave analysis of guided modes inphotonic crystal fibers. / QC 20100629

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

Geothermal energy is an important source of clean and renewable energy. This project deals with the study of deep geothermal power plants for the generation of electricity. The design involves the extraction of heat from the Earth and its conversion into electricity. This is performed by allowing fluid deep into the Earth where it gets heated due to the surrounding rock. The fluid gets vaporized and returns to the surface in a heat pipe. Finally, the energy of the fluid is converted into electricity using turbine or organic rankine cycle (ORC). The main feature of the system is the employment of side channels to increase the amount of thermal energy extracted. A finite difference computer model is developed to solve the heat transport equation. The numerical model was employed to evaluate the performance of the design. The major goal was to optimize the output power as a function of parameters such as thermal diffusivity of the rock, depth of the main well, number and length of lateral channels. The sustainable lifetime of the system for a target output power of 2 MW has been calculated for deep geothermal systems with drilling depths of 8000 and 10000 meters, and a financial analysis has been performed to evaluate the economic feasibility of the system for a practical range of geothermal parameters. Results show promising an outlook for deep geothermal systems for practical applications.

The linear elastodynamic two-dimensional anti-plane stress problem, where deformations occur in only one direction is considered for one sided non-planar faults. Fault dynamics are modeled using purely velocity dependent friction laws, and applied on boundaries with complex geometry. Summation-by-parts operators and energy estimates are used to couple a high-order finite difference method with an unstructured finite volume method. The unstructured finite volume method is used near the fault and the high-order finite difference method further away from the fault where no complex geometry is present. Boundary conditions are imposed weakly on characteristic form using the simultaneous approximation term technique, allowing explicit time integration to be used. Numerical computations are performed to verify the accuracy and time stability, of the method.

The absence of appropriate media to cultivate photons efficiently at the micro or nano scale has hindered taking the full advantage of processing information with light. The proposal of such a medium for light, known as photonic crystals (PCs)--multi-dimensional artificially periodic dielectric media--brings the possibility of a revolution in communications and sensing much closer. In such media, one can manipulate light at a scale on the order of the wavelength or even shorter. Applications of PCs other than in communication include bio-sensing because of the peculiar properties of PCs such as the capability of enhance field-matter interaction and control over the group velocity. As a result, PC waveguide (PCW) structures are of interest and it is expected that PC sensors offer the feasibility of multi-analyte and compact sensing schemes as well as the ability of the detection of small absolute analyte quantities (nanoliters) and low-concentration samples (picomoles), which may be advantages over conventional approaches such as fiber optic and slab waveguide sensors. Depending on the nature of the analyte, either dispersive or absorptive sensing schemes may be implemented. Light propagation is controlled fully only with 3D PCs. One of the problems arising due to reducing the dimension to 2D is that PCs become strongly polarization sensitive. In many cases, one wants to implement polarization insensitive devices such that the PC provides a full band gap for all polarizations. To address this problem, a novel type of PC called annular PC is proposed and analyzed. The capability of tuning the TE and TM polarizations independently within the same structure provides great flexibility to produce polarization-independent or polarization-dependent devices as desired. PCW bends are expected to be the essential building blocks of photonic integrated circuits. Sharp corners having small radii of curvature can be obtained. To enhance the low-loss and narrow-band transmission through these bends, PC heterostructures waveguide concept is introduced. We show that in PCWs formed by joining different types of PCs in a single structure, light can flow around extremely sharp bends in ways that are not possible using conventional PCWs based on a single type of PC.

Digital systems have derived performance benefits due to the scaling down of CMOS microprocessor feature sizes towards packing billions of transistors on a chip, or gigascale integration (GSI). This has placed immense bandwidth demands on chip-to-chip and chip-to-board interconnects. The present-day electrical interconnect may limit bandwidth as transmission rates grow. As such, optical interconnects have been proposed as a potential solution. A critical requirement for enabling chip-to-chip and chip-to-board optical interconnection is out-of-plane coupling for directing light between a chip and the board. Any solution for this problem must be compatible with conventional packaging and assembly requirements. This research addresses the prospects for integrating waveguides with mirrors and polymer pillar optical I/O interconnects to provide such a compatible, out-of-plane, chip-to-board packaging solution through the design, analysis, fabrication, and testing of its constituent parts and their ultimate integration.