This thesis investigates inversion techniques used to determine the geoacoustic properties of a shallow-water waveguide. The data used were obtained in the Shallow Water '06 Modal Mapping Experiment in which four buoys drifted over a system of subbottom channels. The method used was perturbative inversion using modal eigenvalues as input parameters, which were found using an autoregressive spectral estimator. This work investigates the differences between a "channel" region and a "no channel" region based on an inferred stratigraphic model. Inversions were performed on data from a single buoy both at individual frequencies and multiple frequencies simultaneously. Since the use of multiple frequencies and a certain set of constraints proved to be an effective method of inversion, the method was applied to data from the other three buoys as well. It is shown that the "channel" and "no channel" regions have significantly different sound speed profiles. / by Rebecca Weeks. / Thesis (M.S.C.S.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

Sound propagation in a waveguide is greatly dependent on the acoustic properties of the boundaries. The effect of these properties can be described by a bottom reflection coefficient RB, and surface reflection coefficient RS. Two methods for estimating reflection coefficients are used in this research. The first, the ratio method, is based on the variations of the Green's function with depth utilizing the ratio of the wavenumber spectra at two depths. The second, the pole method, is based on the wavenumbers of the modal peaks in the spectrum at a particular depth. A method to invert for sound speed and density is also examined. Estimates of RB and RS based on synthetic data by the ratio method were very close to their predicted values, especially for higher frequencies and longer apertures. The pole method returned less precise estimates though with longer apertures, the estimates were better. Using experimental data, results of the pole method as well a geoacoustic inversion technique based on them were mixed. The ratio method was used to estimate RS based on the actual data and returned results close to the predicted phase of p. / by Alexander Conrad. / Vita. / Thesis (M.S.C.S.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

La thèse porte sur l’analyse d’erreur a posteriori pour la résolution numérique de l’équation linéaire des ondes , discrétisée en temps par le schéma de Newmark et en espace par la méthode des éléments finis. Nous adoptons un choix particulier de paramètres pour le schéma de Newmark, notamment β = 1/4, γ = 1/2, qui assure que la méthode est conservative en énergie et d’ordre deux en temps. L’estimation d’erreur a posteriori, d’un ordre optimal en temps et en espace, est élaborée à partir de la discrétisation complète. L’erreur est mesurée dans une norme qui découle naturellement de la physique: H1 en espace et Linf en temps. Nous proposons d’abord un estimateur dit «à 3 points» qui fait intervenir la solution discrète en 3 points successifs du temps à chaque pas de temps. Cet estimateur fait appel à une approximation du Laplacien de la solution discrète qui doit être calculée à chaque pas de temps, en résolvant un problème auxiliaire d'éléments finis. Nous proposons ensuite un estimateur d’erreur alternatif qui permet d’éviter ces calculs supplémentaires: l’estimateur dit «à 5 points» puisqu’il met en jeu le schéma des différences finies d’ordre 4, qui fait intervenir la solution discrète en 5 points successifs du temps à chaque pas de temps. Nous démontrons que nos estimateurs en temps sont d’ordre optimal pour des solutions suffisamment lisses, sur des maillages quasi-uniformes en espace et uniformes en temps, en supposant que les conditions initiales soient discrétisées à l’aide de la projection elliptique. La trouvaille la plus intéressante de cette analyse est le rôle capitale de cette discrétisation : des discrétisations standards pour les conditions initiales, telles que l’interpolation nodale, peuvent être néfastes pour les estimateurs d’erreur en détruisant leur ordre de convergence, bien qu’elles fournissent des solutions numériques tout à fait acceptables. Des expériences numériques prouvent que nos estimateurs d’erreur sont d’ordre optimal en temps comme en espace, même dans les situations non couvertes par la théorie. En outre, notre analyse a posteriori s’étend au schéma de Newmark d’ordre deux plus général (γ = 1/2). Nous présentons des comparaisons numériques entre notre estimateur à 3 points généralisé et l’estimateur sur des grilles décalées, proposé par Georgoulis et al. Finalement, nous implémentons un algorithme adaptatif en temps et en espace basé sur notre estimateur d’erreur a posteriori à 3 points. Nous concluons par des expériences numériques qui montrent l’efficacité de l’algorithme adaptatif et révèlent l’importance de l’interpolation appropriée de la solution numérique d’un maillage à un autre, surtout vis à vis de l’optimalité de l’estimation d’erreur en temps. / This thesis focuses on the a posteriori error analysis for the linear second-order wave equation discretized by the second order Newmark scheme in time and the finite element method in space. We adopt the particular choice for the parameters in the Newmark scheme, namely β = 1/4, γ = 1/2, since it provides a conservative method with respect to the energy norm. We derive a posteriori error estimates of optimal order in time and space for the fully discrete wave equation. The error is measured in a physically natural norm: H1 in space, Linf in time. Numerical experiments demonstrate that our error estimators are of optimal order in space and time. The resulting estimator in time is referred to as the 3-point estimator since it contains the discrete solution at 3 points in time. The 3-point time error estimator contains the Laplacian of the discrete solution which should be computed via auxiliary finite element problems at each time step. We propose an alternative time error estimator that avoids these additional computations. The resulting estimator is referred to as the 5-point estimator since it contains the fourth order finite differences in time and thus involves the discrete solution at 5 points in time at each time step. We prove that our time estimators are of optimal order at least on sufficiently smooth solutions, quasi-uniform meshes in space and uniform meshes in time. The most interesting finding of this analysis is the crucial importance of the way in which the initial conditions are discretized: a straightforward discretization, such as the nodal interpolation, may ruin the error estimators while providing quite acceptable numerical solution. We also extend the a posteriori error analysis to the general second order Newmark scheme (γ = 1/2) and present numerical comparasion between the general 3-point time error estimator and the staggered grid error estimator proposed by Georgoulis et al. In addition, using obtained a posteriori error bounds, we implement an efficient adaptive algorithm in space and time. We conclude with numerical experiments that show that the manner of interpolation of the numerical solution from one mesh to another plays an important role for optimal behavior of the time error estimator and thus of the whole adaptive algorithm.

Mak Ka Wai Charles. / Thesis submitted in: October 2001. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 89-91). / Text in English; abstracts in English and Chinese. / Mak Ka Wai Charles. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview of the Mathematical Framework --- p.2 / Chapter 1.2 --- System of Interest --- p.7 / Chapter 1.2.1 --- Klein-Gordon equation --- p.7 / Chapter 1.2.2 --- QNM boundary conditions --- p.12 / Chapter 1.3 --- Outline of This Thesis --- p.14 / Chapter 2 --- Green's Function --- p.15 / Chapter 2.1 --- "Formal Expression for G(x,y，w)" --- p.16 / Chapter 2.2 --- "Leaver's Series Solution: An Analytic Expression for g(r, w)" --- p.17 / Chapter 2.3 --- Location of the Cut --- p.22 / Chapter 2.4 --- "Jaffe's Series Solution: An Analytic Expression for f(r,w)" --- p.23 / Chapter 2.5 --- QNMs and Their Locations --- p.26 / Chapter 2.5.1 --- Alternative definitions of QNM --- p.26 / Chapter 2.5.2 --- Methods of searching for QNMs --- p.28 / Chapter 2.5.3 --- Locations of QNMs --- p.29 / Chapter 2.6 --- Green's Function and Eigenspectra --- p.30 / Chapter 3 --- Normalization Function: Analytical Treatment --- p.34 / Chapter 3.1 --- Definition and Properties --- p.34 / Chapter 3.2 --- Analytic Approximations for --- p.36 / Chapter 3.3 --- Polar Perturbations --- p.39 / Chapter 4 --- Normalization Function: Numerical Treatment --- p.42 / Chapter 4.1 --- "Numerical Algorithm for g(x,w)" --- p.42 / Chapter 4.1.1 --- Method --- p.42 / Chapter 4.1.2 --- Equation governing R(z) --- p.45 / Chapter 4.1.3 --- "Equations governing A(x, z) and B(x, z)" --- p.45 / Chapter 4.2 --- "Numerical Algorithm for g(x, ´ؤw)" --- p.49 / Chapter 4.3 --- Numerical Result of q(γ) --- p.50 / Chapter 4.4 --- Comparison of Numerical Result with Analytic Approximations --- p.56 / Chapter 5 --- "Branch Cut Strength of G(x, y, w)" --- p.58 / Chapter 5.1 --- "Relation between q(γ) and ΔG(x,y, ´ؤiγ)" --- p.58 / Chapter 5.2 --- Proof of the Power Law --- p.60 / Chapter 5.3 --- "Numerical Results for ΔG(x, y, ´ؤiγ)" --- p.63 / Chapter 5.4 --- Study of a Physically Important Limit --- p.65 / Chapter 5.4.1 --- Limiting x and y --- p.65 / Chapter 5.4.2 --- Poles on the unphysical sheet --- p.69 / Chapter 5.4.3 --- Zerilli potential --- p.77 / Chapter 6 --- Conclusion --- p.81 / Chapter A --- Tortoise Coordinate --- p.84 / Chapter B --- Solution of the Generalized Coulomb Wave Equation --- p.86 / Chapter C --- Derivation of (5.1) --- p.88 / Bibliography --- p.89

In this thesis, we consider four different stochastic partial differential equations. Firstly, we study stochastic Helmholtz equation driven by an additive white noise, in a bounded convex domain with smooth boundary of Rd (d = 2, 3). And then with the help of the perfectly matched layers technique, we also consider the stochastic scattering problem of Helmholtz type. The second part of this thesis is to investigate the time harmonic case for stochastic Maxwell's equations driven by an color noise in a simple medium, and then we expand the results to the stochastic Maxwell's equations in case of dispersive media in Rd (d = 2, 3). Thirdly, we study stochastic parabolic partial differential equation driven by space-time color noise, where the domain O is a bounded domain in R2 with boundary ∂O of class C2+alpha for 0 < alpha < 1/2. In the last part, we discuss the stochastic wave equation (SWE) driven by nonlinear noise in 1D case, where the noise 626x6t W(x, t) is the space-time mixed second-order derivative of the Brownian sheet. / Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions. / One of the goals of this thesis is to derive error estimates for numerical solutions of the above four kinds SPDEs. The difficulty in the error analysis in finite element methods and general numerical approximations for a SPDE is the lack of regularity of its solution. To overcome such a difficulty, we follow the approach of [4] by first discretizing the noise and then applying standard finite element methods and discontinuous Galerkin methods to the stochastic Helmholtz equation and Maxwell equations with discretized noise; standard finite element method to the stochastic parabolic equation with discretized color noise; Galerkin method to the stochastic wave equation with discretized white noise, and we obtain error estimates are comparable to the error estimates of finite difference schemes. / We shall focus on some SPDEs where randomness only affects the right-hand sides of the equations. To solve the four types of SPDEs using, for example, the Monte Carlo method, one needs many solvers for the deterministic problem with multiple right-hand sides. We present several efficient deterministic solvers such as flexible CG method and block flexible GMRES method, which are absolutely essential in computing statistical quantities. / Zhang, Kai. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3552. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 144-155). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Under adequate conditions on $g$, we show the density in $L^2((0,\pi),(0,2\pi))$ of the set of functions $p$ for which \begin{equation*} u_{tt}(x,t)-u_{xx}(x,t)= g(u(x,t)) + p(x,t) \end{equation*} has a weak solution subject to \begin{equation*} \begin{aligned} u(x,t)&=u(x,t+2\pi)\\ u(0,t)&=u(\pi,t)=0. \end{aligned} \end{equation*} To achieve this, we prove a Saddle Point Principle by means of a refined variant of the deformation lemma of Rabinowitz. Generally, inf-sup techniques allow the characterization of critical values by taking the minimum of the maximae on some particular class of sets. In this version of the Saddle Point Principle, we introduce sufficient conditions for the existence of a saddle-structure which is not restricted to finite-dimensional subspaces.

Deformation of a Gaussian beam is observed when it propagates through a plasma. Self-focusing of the beam may be observed when the intensity of the laser increases the index of refraction of plasma gas. Due to the difficulties in solving the nonlinear partial differential equation in Maxwell's wave equation, a numerical technique has been developed in favor of the traditional analytical method. Result of numerical solution shows consistency with the analytical method. This further suggests the validity of the numerical technique employed. A three dimensional graphics package was used to depict the numerical data obtained from the calculation. Plots from the data further show the deformation of the Gaussian beam as it propagates through the plasma gas.

The most common tool for solving spacetime problems using finite elements is based on semidiscretization: discretizing in space by a finite element method and then advancing in time by a numerical scheme. Contrary to this standard procedure, in this dissertation we consider formulations where time is another coordinate of the domain. Therefore, spacetime problems can be studied as boundary value problems, where initial conditions are considered as part of the spacetime boundary conditions. When seeking solutions to these problems, it is natural to ask what are the correct spaces of functions to choose, to obtain wellposedness. This motivates the study of an abstract theory for unbounded partial differential operators associated with a general boundary value problem on a bounded domain. A framework for choosing the spaces is introduced, and conditions for the solvability of weak formulations are provided. We apply this framework to study wave problems on tents and to study wellposed discontinuous Petrov-Galerkin (DPG) formulations for the Schrödinger and wave equations. Several numerical issues are also discussed.

This thesis is concerned with the development of numerical methods using finite difference techniques for the discretization of initial value problems (IVPs) and initial boundary value problems (IBVPs) of certain hyperbolic systems which are first order in time and second order in space. This type of system appears in some formulations of Einstein equations, such as ADM, BSSN, NOR, and the generalized harmonic formulation. For IVP, the stability method proposed in [14] is extended from second and fourth order centered schemes, to 2n-order accuracy, including also the case when some first order derivatives are approximated with off-centered finite difference operators (FDO) and dissipation is added to the right-hand sides of the equations. For the model problem of the wave equation, special attention is paid to the analysis of Courant limits and numerical speeds. Although off-centered FDOs have larger truncation errors than centered FDOs, it is shown that in certain situations, off-centering by just one point can be beneficial for the overall accuracy of the numerical scheme. The wave equation is also analyzed in respect to its initial boundary value problem. All three types of boundaries - outflow, inflow and completely inflow that can appear in this case, are investigated. Using the ghost-point method, 2n-accurate (n = 1, 4) numerical prescriptions are prescribed for each type of boundary. The inflow boundary is also approached using the SAT-SBP method. In the end of the thesis, a 1-D variant of BSSN formulation is derived and some of its IBVPs are considered. The boundary procedures, based on the ghost-point method, are intended to preserve the interior 2n-accuracy. Numerical tests show that this is the case if sufficient dissipation is added to the rhs of the equations. / Diese Doktorarbeit beschäftigt sich mit der Entwicklung numerischer Verfahren für die Diskretisierung des Anfangswertproblems und des Anfangs-Randwertproblems unter Einsatz von finite-Differenzen-Techniken für bestimmte hyperbolischer Systeme erster Ordnung in der Zeit und zweiter Ordnung im Raum. Diese Art von Systemen erscheinen in einigen Formulierungen der Einstein'schen-Feldgleichungen, wie zB. den ADM, BSSN oder NOR Formulierungen, oder der sogenanten verallgemeinerten harmonischen Darstellung. Im Hinblick auf das Anfangswertproblem untersuche ich zunächst tiefgehend die mathematischen Eigenschaften von finite-Differenzen-Operatoren (FDO) erster und zweiter Ordnung mit 2n-facher Genaugigkeit. Anschließend erweitere ich eine in der Literatur beschriebene Methode zur Stabilitätsanalyse für Systeme mit zentrierten FDOs in zweiter und vierter Genauigkeitsordung auf Systeme mit gemischten zentrierten und nicht zentrierten Ableitungsoperatoren 2n-facher Genauigkeit, eingeschlossen zusätzlicher Dämpfungsterme, wie sie bei numerischen Simulationen der allgemeinen Relativitätstheorie üblich sind. Bei der Untersuchung der einfachen Wellengleichung als Fallbeispiel wird besonderes Augenmerk auf die Analyse der Courant-Grenzen und numerischen Geschwindigkeiten gelegt. Obwohl unzentrierte, diskrete Ableitungsoperatoren größere Diskretisierungs-Fehler besitzen als zentrierte Ableitungsoperatoren, wird gezeigt, daß man in bestimmten Situationen eine Dezentrierung des numerischen Moleküls von nur einem Punkt bezüglich des zentrierten FDO eine höhere Genauigkeit des numerischen Systems erzielen kann. Die Wellen-Gleichung in einer Dimension wurde ebenfalls im Hinblick auf das Anfangswertproblem untersucht. In Abhängigkeit des Wertes des sogenannten Shift-Vektors, müssen entweder zwei (vollständig eingehende Welle), eine (eingehende Welle) oder keine Randbedingung (ausgehende Welle) definiert werden. In dieser Arbeit wurden alle drei Fälle mit Hilfe der 'Ghost-point-methode' numerisch simuliert und untersucht, und zwar auf eine Weise, daß alle diese Algorithmen stabil sind und eine 2n-Genauigkeit besitzen. In der 'ghost-point-methode' werden die Evolutionsgleichungen bis zum letzen Punkt im Gitter diskretisiert unter Verwendung von zentrierten FDOs und die zusätzlichen Punkte die am Rand benötigt werden ('Ghost-points') werden unter Benutzung von Randwertbedingungen und Extrapolationen abgeschätzt. Für den Zufluß-Randwert wurde zusätzlich noch eine andere Implementierung entwickelt, welche auf der sogenannten SBP-SAT (Summation by parts-simulatanous approximation term) basiert. In dieser Methode werden die diskreten Ableitungen durch Operatoren angenähert, welche die 'Summation-by-parts' Regeln erfüllen. Die Randwertbedingungen selber werden in zusätzlichen Termen integriert, welche zu den Evolutionsgleichnungen der Punkte nahe des Randes hinzuaddiert werden und zwar auf eine Weise, daß die 'summation-by-parts' Eigenschaften erhalten bleiben. Am Ende dieser Arbeit wurde noch eine eindimensionale (kugelsymmetrische) Version der BSSN Formulierung abgeleitet und einige physikalisch relevanten Anfangs-Randwertprobleme werden diskutiert. Die Randwert-Algorithmen, welche für diesen Fall ausgearbeitet wurden, basieren auf der 'Ghost-point-Methode' and erfüllen die innere 2n-Genauigkeit solange genügend Reibung in den Gleichungen zugefügt wird.

I propose two Galerkin methods based on the transfer-of-approximation property for static and dynamic acoustic boundary value problems in seismic applications. For problems with heterogeneous coefficients, the polynomial finite element spaces are no longer optimal unless special meshing techniques are employed. The transfer-of-approximation property provides a general framework to construct the optimal approximation subspace on regular grids. The transfer-of-approximation finite element method is theoretically attractive for that it works for both scalar and vectorial elliptic problems. However the numerical cost is prohibitive. To compute each transfer-of-approximation finite element basis, a problem as hard as the original one has to be solved. Furthermore due to the difficulty of basis localization, the resulting stiffness and mass matrices are dense. The 2D harmonic coordinate finite element method (HCFEM) achieves optimal second-order convergence for static and dynamic acoustic boundary value problems with variable coefficients at the cost of solving two auxiliary elliptic boundary value problems. Unlike the conventional FEM, no special domain partitions, adapted to discontinuity surfaces in coe cients, are required in HCFEM to obtain the optimal convergence rate. The resulting sti ness and mass matrices are constructed in a systematic procedure, and have the same sparsity pattern as those in the standard finite element method. Mass-lumping in HCFEM maintains the optimal order of convergence, due to the smoothness property of acoustic solutions in harmonic coordinates, and overcomes the numerical obstacle of inverting the mass matrix every time update, results in an efficient, explicit time step.