Dissertationen zum Thema „Nonlinear field equations“

This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.

This thesis focuses on qualitative and quantitative aspects of some nonlinear PDEs arising in optimal control and differential games, ranging from regularity issues to maximum principles. More precisely, it is concerned with the analysis of some fully nonlinear second order degenerate PDEs over Hörmander vector fields that can be written in Hamilton-Jacobi-Bellman and Isaacs form and those arising in the recent theory of Mean Field Games, where the prototype model is described by a coupled system of PDEs involving a backward Hamilton-Jacobi and a forward Fokker-Planck equation. The thesis is divided in three parts. The first part is devoted to analyze strong maximum principles for fully nonlinear second order degenerate PDEs structured on Hörmander vector fields, having as a particular example fully nonlinear subelliptic PDEs on Carnot groups. These results are achieved by introducing a notion of subunit vector field for these nonlinear degenerate operators in the spirit of the seminal works on linear equations. As a byproduct, we then prove some new strong comparison principles for equations that can be written in Hamilton-Jacobi-Bellman form and Liouville theorems for some second order fully nonlinear degenerate PDEs. The second part of the thesis deals with time-dependent fractional Mean Field Game systems. These equations arise when the dynamics of the average player is described by a stable Lévy process to which corresponds a fractional Laplacian as diffusion operator. More precisely, we establish existence and uniqueness of solutions to such systems of PDEs with regularizing coupling among the equations for every order of the fractional Laplacian $sin(0,1)$. The existence of solutions is addressed via the vanishing viscosity method and we prove that in the subcritical regime the equations are satisfied in classical sense, while if $sleq1/2$ we find weak energy solutions. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We finally show uniqueness of solutions both under the Lasry-Lions monotonicity condition and for short time horizons. The last part focuses on the regularizing effect of evolutive Hamilton-Jacobi equations with Hamiltonian having superlinear growth in the gradient and unbounded right-hand side. In particular, the analysis is performed both for viscous Hamilton-Jacobi equations and its fractional counterpart in the subcritical regime via a duality method. The results are accomplished exploiting the regularity of solutions to Fokker-Planck-type PDEs with rough velocity fields in parabolic Sobolev and Bessel potential spaces respectively.

This thesis discusses various properties of a number of differential equations which we will term "integrable". There are many definitions of this word, but we will confine ourselves to two possible characterisations — either an equation can be transformed by a suitable change of variables to a linear equation, or there exists an infinite number of conserved quantities associated with the equation that commute with each other via some Hamiltonian structure. Both of these definitions rely heavily on the concept of the symmetry of a differential equation, and so Chapters 1 and 2 introduce and explain this idea, based on a geometrical theory of p.d.e.s, and describe the interaction of such methods with variational calculus and Hamiltonian systems. Chapter 3 discusses a somewhat ad hoc method for solving evolution equations involving a series ansatz that reproduces well-known solutions. The method seems to be related to symmetry methods, although the precise connection is unclear. The rest of the thesis is dedicated to the so-called Universal Field Equations and related models. In Chapter 4 we look at the simplest two-dimensional cases, the Bateman and Born-lnfeld equations. By looking at their generalised symmetries and Hamiltonian structures, we can prove that these equations satisfy both the definitions of integrability mentioned above. Chapter Five contains the general argument which demonstrates the linearisability of the Bateman Universal equation by calculation of its generalised symmetries. These symmetries are helpful in analysing and generalising the Lagrangian structure of Universal equations. An example of a linearisable analogue of the Born-lnfeld equation is also included. The chapter concludes with some speculation on Hamiltoian properties.

It is known that there are nonlinear wave equations with localized solitary wave solutions. Some of these solitary waves are stable (with respect to a small perturbation of initial data)and have nonzero spin (nonzero intrinsic angular momentum in the centre of momentum frame). In this paper we consider vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation and investigate the behavior of these spinning solitary waves under the influence of an externally imposed uniform magnetic field. We find that the only stationary spinning solitary wave solutions have spin parallel or antiparallel to the magnetic field direction.

In this thesis, we study Schrödinger equations with an external magnetic field. In the first part, we are interested in an eigenvalue problem. We work in an open, bounded and simply connected domain in dimension two. We consider a magnetic potential singular at one point in the domain, and related to the magnetic field being a multiple of a Dirac delta. Those two objects are related to the Bohm-Aharonov effect, in which a charged particle is influenced by the presence of the magnetic potential although it remains in a region where the magnetic field is zero. We consider the Schrödinger magnetic operator appearing in the Schrödinger equation in presence of an external magnetic field. We want to study the spectrum of this operator, and more particularly how it varies when the singular point moves in the domain. We prove some results of continuity and differentiability of the eigenvalues when the singular point moves in the domain or approaches its boundary. Finally, in case of half-integer circulation of the magnetic potential, we study some asymptotic behaviour of the eigenvalues close to their critical points. In the second part, we study nonlinear Schrödinger equations in a cylindrically setting. We are interested in the semiclassical limit of the equation. We prove the existence of a semiclassical solution concentrating on a circle. Moreover, the radius of that circle is determined by the electric potential, but also by the magnetic potential. This result is totally new with respect to the ones before, in which the concentration is driven only by the electric potential.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Traditional light microscopy suffers from the diffraction limit, which limits the spatial resolution to λ/2. The current trend in optical microscopy is the development of techniques to bypass the diffraction limit. Resolutions below 40 nm will make it possible to probe biological systems by imaging the interactions between single molecules and cell membranes. These resolutions will allow for the development of improved drug delivery mechanisms by increasing our understanding of how chemical communication within a cell occurs. The materials sciences would also benefit from these high resolutions. Nanomaterials can be analyzed with Raman spectroscopy for molecular and atomic bond information, or with fluorescence response to determine bulk optical properties with tens of nanometer resolution. Near-field optical microscopy is one of the current techniques, which allows for imaging at resolutions beyond the diffraction limit. Using a combination of a shear force microscope (SFM) and an inverted optical microscope, spectroscopic resolutions below 20 nm have been demonstrated. One technique, in particular, has been named tip enhanced near-field optical microscopy (TENOM). The key to this technique is the use of solid metal probes, which are illuminated in the far field by the excitation wavelength of interest. These probes are custom-designed using finite difference time domain (FDTD) modeling techniques, then fabricated with the use of a focused ion beam (FIB) microscope. The measure of the quality of probe design is based directly on the field enhancement obtainable. The greater the field enhancement of the probe, the more the ratio of near-field to far-field background contribution will increase. The elimination of the far-field signal by a decrease of illumination power will provide the best signal-to-noise ratio in the near-field images. Furthermore, a design that facilitates the delocalization of the near-field imaging from the far-field will be beneficial. Developed is a novel microscope design that employs two-photon non-linear excitation to allow the imaging of the fluorescence from almost any visible fluorophore at resolutions below 30 nm without changing filters or excitation wavelength. The ability of the microscope to image samples at atmospheric pressure, room temperature, and in solution makes it a very promising tool for the biological and materials science communities. The microscope demonstrates the ability to image topographical, optical, and electronic state information for single-molecule identification. A single computer, simple custom control circuits, field programmable gate array (FPGA) data acquisition, and a simplified custom optical system controls the microscope are thoroughly outlined and documented. This versatility enables the end user to custom-design experiments from confocal far-field single molecule imaging to high resolution scanning probe microscopy imaging. Presented are the current capabilities of the microscope, most importantly, high-resolution near-field images of J-aggregates with PIC dye. Single molecules of Rhodamine 6G dye and quantum dots imaged in the far-field are presented to demonstrate the sensitivity of the microscope. A comparison is made with the use of a mode-locked 50 fs pulsed laser source verses a continuous wave laser source on single molecules and J-aggregates in the near-field and far-field. Integration of an intensified CCD camera with a high-resolution monochromator allows for spectral information about the sample. The system will be disseminated as an open system design.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Esta dissertação trata da construção de hierarquias compatíveis com a equação PIV a partir dos modelos: AKNS, dois bósons e dois bósons quadráticos. Também são construidos os problema linear de Jimbo-Miwa dos três modelos e discutimos a hamiltoniana correspondente a equação PIV a partir do formalismo lagrangiano
This dissertation contains the construction of compatible hierarchies with the PIV equation from the models: AKNS, two-boson and quadratic two-boson. Also it is build the Jimbo-Miwa linear problem for the three models and we discuss the hamiltonian corresponding to fouth Painlevé equation from the Lagrangian formalism

Orientador: Abraham Hirsz Zimerman
Banca: Iberê Luiz Caldas
Banca: Roberto André Kraenkel
Resumo: Esta dissertação trata da construção de hierarquias compatíveis com a equação PIV a partir dos modelos: AKNS, dois bósons e dois bósons quadráticos. Também são construidos os problema linear de Jimbo-Miwa dos três modelos e discutimos a hamiltoniana correspondente a equação PIV a partir do formalismo lagrangiano
Abstract: This dissertation contains the construction of compatible hierarchies with the PIV equation from the models: AKNS, two-boson and quadratic two-boson. Also it is build the Jimbo-Miwa linear problem for the three models and we discuss the hamiltonian corresponding to fouth Painlevé equation from the Lagrangian formalism
Mestre

Nous donnons dans cette these deux formulations distinctes dans le super-espace etendu de deux series de hierarchies integrables supersymetriques n = 2 de type korteweg-de vries (kdv). Nous developpons d'une part un formalisme de gelfand-dickey qui utilise l'algebre des operateurs pseudo-differentiels n = 2 preservant la chiralite et une matrice-r non antisymetrique. Nous definissons ainsi une hierarchie de kp n = 2 qui est hamiltonienne par rapport a un crochet de poisson lineaire et a deux crochets quadratiques. Deux series de hierarchies de type kdv n = 2 sont obtenues comme reductions par rapport a l'un ou a l'autre des crochets quadratiques. Nous etudions d'autre part, au niveau bosonique, une extension de la construction de drinfeld-sokolov reposant sur un ensemble de conditions algebriques affaiblies et sur l'existence d'une matrce-r plus generale. La construction de drinfeld-sokolov et sa generalisation peuvent etre etendues au cas des superalgebres de boucles. Nous donnons des conditions suffisantes sur les donnees algebriques pour que les hierarchies integrables ainsi construites soient invariantes sous les transformations de supersymetrie n = 1 ou n = 2. La formulation de ces hierarchies dans le superespace rend la supersymetrie explicite. Cette methode est utilisee pour construire les deux series de hierarchies de type kdv n = 2 obtenues precedemment, ainsi que d'autres hierarchies comme celle de schrodinger non lineaire n = 2.

Spinodal instability mechanism and early development of density fluctuations for symmetric nuclear matter at finite temperature are studied. A stochastic extension of Walecka-type relativistic mean-field model including non-linear self-interactions of scalar mesons with NL3 parameter set is employed in the semi-classical approximation. The growth rates of unstable collective modes are investigated below the normal density and at low temperatures. The system exhibits most unstable behavior in longer wave lengths at baryon densities &rho
B = 0.4 &rho
0 , while most unstable behavior occurs in shorter wavelengths at lower baryon densities &rho
B = 0.2 &rho
0 . The unstable response of the system shifts towards longer wavelengths with the increasing temperature at both densities. The early growth of the density correlation functions are calculated, which provide valuable information about the initial size of the condensation and the average speed of condensing fragments. Furthermore, the relativistic results are compared with Skyrme type non-relativistic calculations. Qualitatively similar results are found in both non-relativistic and relativistic descriptions.

Hauptthema der Dissertation ist die Analysis eines nichtlinearen, gekoppelten Systems partieller Differentialgleichungen (PDG), das in der Modellierung der Kristallzüchtung aus der Schmelze mit Magnetfeldern vorkommt. Die zu beschreibenden Phenomäne sind einerseits der im elektromagnetisch geheizten Schmelzofen erfolgende Wärmetransport (Wärmeleitung, -konvektion und -strahlung), und andererseits die Bewegung der Halbleiterschmelze unter dem Einfluss der thermischen Konvektion und der angewendeten elektromagnetischen Kräfte. Das Modell besteht aus den Navier-Stokeschen Gleichungen für eine inkompressible Newtonsche Flüssigkeit, aus der Wärmeleitungsgleichung und aus der elektrotechnischen Näherung des Maxwellschen Systems. Wir erörtern die schwache Formulierung dieses PDG Systems, und wir stellen ein Anfang-Randwertproblem auf, das die Komplexität der Anwendung widerspiegelt. Die Hauptfrage unserer Untersuchung ist die Wohlgestelltheit dieses Problems, sowohl im stationären als auch im zeitabhängigen Fall. Wir zeigen die Existenz schwacher Lösungen in geometrischen Situationen, in welchen unstetige Materialeigenschaften und nichtglatte Trennfläche auftreten dürfen, und für allgemeine Daten. In der Lösung zum zeitabhängigen Problem tritt ein Defektmaß auf, das ausser der Flüssigkeit im Rand der elektrisch leitenden Materialien konzentriert bleibt. Da eine globale Abschätzung der im Strahlungshohlraum ausgestrahlten Wärme auch fehlt, rührt ein Teil dieses Defektmaßes von der nichtlokalen Strahlung her. Die Eindeutigkeit der schwachen Lösung erhalten wir nur unter verstärkten Annahmen: die Kleinheit der gegebenen elektrischen Leistung im stationären Fall, und die Regularität der Lösung im zeitabhängigen Fall. Regularitätseigenschaften wie die Beschränktheit der Temperatur werden, wenn auch nur in vereinfachten Situationen, hergeleitet: glatte Materialtrennfläche und Temperaturunabhängige Koeffiziente im Fall einer stationären Analysis, und entkoppeltes, zeitharmonisches Maxwell für das transiente Problem.
The present PhD thesis is devoted to the analysis of a coupled system of nonlinear partial differential equations (PDE), that arises in the modeling of crystal growth from the melt in magnetic fields. The phenomena described by the model are mainly the heat-transfer processes (by conduction, convection and radiation) taking place in a high-temperatures furnace heated electromagnetically, and the motion of a semiconducting melted material subject to buoyancy and applied electromagnetic forces. The model consists of the Navier-Stokes equations for a newtonian incompressible liquid, coupled to the heat equation and the low-frequency approximation of Maxwell''s equations. We propose a mathematical setting for this PDE system, we derive its weak formulation, and we formulate an (initial) boundary value problem that in the mean reflects the complexity of the real-life application. The well-posedness of this (initial) boundary value problem is the mainmatter of the investigation. We prove the existence of weak solutions allowing for general geometrical situations (discontinuous coefficients, nonsmooth material interfaces) and data, the most important requirement being only that the injected electrical power remains finite. For the time-dependent problem, a defect measure appears in the solution, which apart from the fluid remains concentrated in the boundary of the electrical conductors. In the absence of a global estimate on the radiation emitted in the cavity, a part of the defect measure is due to the nonlocal radiation effects. The uniqueness of the weak solution is obtained only under reinforced assumptions: smallness of the input power in the stationary case, and regularity of the solution in the time-dependent case. Regularity properties, such as the boundedness of temperature are also derived, but only in simplified settings: smooth interfaces and temperature-independent coefficients in the case of a stationary analysis, and, additionally for the transient problem, decoupled time-harmonic Maxwell.

In order to personalize or tailor treatments to maximize impact among different subgroups, there is need to model not only the main effects of intervention but also the variation in intervention impact by baseline individual level risk characteristics. To this end a suitable statistical model will allow researchers to answer a major research question: who benefits or is harmed by this intervention program? Commonly in social and psychological research, the baseline risk may be unobservable and have to be estimated from observed indicators that are measured with errors; also it may have nonlinear relationship with the outcome. Most of the existing nonlinear structural equation models (SEM’s) developed to address such problems employ polynomial or fully parametric nonlinear functions to define the structural equations. These methods are limited because they require functional forms to be specified beforehand and even if the models include higher order polynomials there may be problems when the focus of interest relates to the function over its whole domain. To develop a more flexible statistical modeling technique for assessing complex relationships between a proximal/distal outcome and 1) baseline characteristics measured with errors, and 2) baseline-treatment interaction; such that the shapes of these relationships are data driven and there is no need for the shapes to be determined a priori. In the ALV model structure the nonlinear components of the regression equations are represented as generalized additive model (GAM), or generalized additive mixed-effects model (GAMM). Replication study results show that the ALV model estimates of underlying relationships in the data are sufficiently close to the true pattern. The ALV modeling technique allows researchers to assess how an intervention affects individuals differently as a function of baseline risk that is itself measured with error, and uncover complex relationships in the data that might otherwise be missed. Although the ALV approach is computationally intensive, it relieves its users from the need to decide functional forms before the model is run. It can be extended to examine complex nonlinearity between growth factors and distal outcomes in a longitudinal study.

Dans cette thèse, tout d’abord, nous faisons l’Analyse Mathématique du modèle exact du chauffage radiatif d’un corps semi-transparent $\Omega$ par une source radiative noire qui l’entoure. Il s’agit donc d’étudier le couplage d’un système d’Equations de Transfert Radiatif avec condition au bord de réflectivité indépendantes avec une équation de conduction de la chaleur non linéaire avec condition limite non linéaire de type Robin. Nous prouvons l’existence et l’unicité de la solution et nous démontrons des bornes uniformes sur la solution et les intensités radiatives dans chaque bande de longueurs d’ondes pour laquelle le corps est semi-transparent, en fonction de bornes sur les données, Deuxièmement, nous considérons le problème du contrôle optimal de la température absolue à l’intérieur du corps semi-transparent $\Omega$ en agissant sur la température absolue de la source radiative noire qui l’entoure. À cet égard, nous introduisons la fonctionnelle coût appropriée et l’ensemble des contrôles admissibles $T_{S}$, pour lesquels nous prouvons l’existence de contrôles optimaux. En introduisant l’espace des états et l’équation d’état, une condition nécessaire de premier ordre pour qu’un contrôle $T_{S}$ : t ! $T_{S}$ (t) soit optimal, est alors dérivée sous la forme d’une inéquation variationnelle en utilisant le théorème des fonctions implicites et le problème adjoint. Ensuite, nous considérons le problème de l’existence et de l’unicité d’une solution faible des équations de la thermoviscoélasticité dans une formulation mixte de type Hellinger- Reissner, la nouveauté par rapport au travail de M.E. Rognes et R. Winther (M3AS, 2010) étant ici l’apparition de la viscosité dans certains coefficients de l’équation constitutive, viscosité qui dépend dans ce contexte de la température absolue T(x, t) et donc en particulier du temps t. Enfin, nous considérons dans ce cadre le problème du contrôle optimal de la déformation du corps semi-transparent $\Omega$, en agissant sur la température absolue de la source radiative noire qui l’entoure. Nous prouvons l’existence d’un contrôle optimal et nous calculons la dérivée Fréchet de la fonctionnelle coût réduite
This thesis begins with a rigorous mathematical analysis of the radiative heating of a semi-transparent body made of glass, by a black radiative source surrounding it. This requires the study of the coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution, and give also uniform bounds on the solution i.e. on the absolute temperature distribution inside the body and on the radiative intensities. Now, we consider the temperature $T_{S}$ of the black radiative source S surrounding the semi-transparent body $\Omega$ as the control variable. We adjust the absolute temperature distribution (x, t) 7! T(x, t) inside the semi-transparent body near a desired temperature distribution Td(·, ·) during the time interval of radiative heating ]0, tf [ by acting on $T_{S}$. In this respect, we introduce the appropriate cost functional and the set of admissible controls $T_{S}$, for which we prove the existence of optimal controls. Introducing the State Space and the State Equation, a first order necessary condition for a control $T_{S}$ : t 7! $T_{S}$ (t) to be optimal is then derived in the form of a Variational Inequality by using the Implicit Function Theorem and the adjoint problem. We come now to the goal problem which is the deformation of the semi-transparent body $\Omega$ by heating it with a black radiative source surrounding it. We introduce a weak mixed formulation of this thermoviscoelasticity problem and study the existence and uniqueness of its solution, the novelty here with respect to the work of M.E. Rognes et R. Winther (M3AS, 2010) being the apparition of the viscosity in some of the coefficients of the constitutive equation, viscosity which depends on the absolute temperature T(x, t) and thus in particular on the time t. Finally, we state in this setting the related optimal control problem of the deformation of the semi-transparent body $\Omega$, by acting on the absolute temperature of the black radiative source surrounding it. We prove the existence of an optimal control and we compute the Fréchet derivative of the associated reduced cost functional

Nous étudions l’analyse asymptotique en relativité générale sous deux aspects: le peeling et le scattering (diffusion) conforme. Le peeling est construit pour les champs scalaires linéaire et non-linéaires et pour les champs de Dirac en espace-temps de Kerr (qui est non-stationnaire et à symétrie simplement axiale), généralisant les travaux de L. Mason et J-P. Nicolas (2009, 2012). La méthode des champs de vecteurs (estimations d’énergie géométriques) et la technique de compactification conforme sont développées. Elles nous permettent de formuler les définitions du peeling à tous ordres et d’obtenir les données initiales optimales qui assurent ces comportements. Une théorie de la diffusion conforme pour les équations de champs sans masse de spîn n/2 dans l’espace-temps de Minkowski est construite.En effectuant les compactifications conformes (complète et partielle), l’espace-temps est complété en ajoutant une frontière constituée de deux hypersurfaces isotropes représentant respectivement les points limites passés et futurs des géodésiques de type lumière. Le comportement asymptotique des champs s’obtient en résolvant le problème de Cauchy pour l’équation rééchelonnée et en considérant les traces des solutions sur ces bords. L’inversibilité des opérateurs de trace, qui associent le comportement asymptotique passé ou futur aux données initiales, s’obtient en résolvant le problème de Goursat sur le bord conforme. L’opérateur de diffusion conforme est alors obtenu par composition de l’opérateur de trace futur avec l’inverse de l’opérateur de trace passé
This work explores two aspects of asymptotic analysis in general relativity: peeling and conformal scattering.On the one hand, the peeling is constructed for linear and nonlinear scalar fields as well as Dirac fields on Kerr spacetime, which is non-stationary and merely axially symmetric. This generalizes the work of L. Mason and J-P. Nicolas (2009, 2012). The vector field method (geometric energy estimates) and the conformal technique are developed. They allow us to formulate the definition of the peeling at all orders and to obtain the optimal space of initial data which guarantees these behaviours. On the other hand, a conformal scattering theory for the spin-n/2 zero rest-mass equations on Minkowski spacetime is constructed. Using the conformal compactifications (full and partial), the spacetime is completed with two null hypersurfaces representing respectively the past and future end points of null geodesics. The asymptotic behaviour of fields is then obtained by solving the Cauchy problem for the rescaled equation and considering the traces of the solutions on these hypersurfaces. The invertibility of the trace operators, that to the initial data associate the future or past asymptotic behaviours, is obtained by solving the Goursat problem on the conformal boundary. The conformal scattering operator is then obtained by composing the future trace operator with the inverse of the past trace operator

Die Hauptthemen dieser Arbeit sind einerseits eine tiefgehende Analyse von nichtlinearen differential-algebraischen Gleichungen (DAEs) vom Index 2, die aus der modifizierten Knotenanalyse (MNA) von elektrischen Schaltkreisen hervorgehen, und andererseits die Entwicklung von Konvergenzkriterien für Waveform Relaxationsmethoden zum Lösen gekoppelter Probleme. Ein Schwerpunkt in beiden genannten Themen ist die Beziehung zwischen der Topologie eines Schaltkreises und mathematischen Eigenschaften der zugehörigen DAE. Der Analyse-Teil umfasst eine detaillierte Beschreibung einer Normalform für Schaltkreis DAEs vom Index 2 und Abschätzungen, die für die Sensitivität des Schaltkreises bezüglich seiner Input-Quellen folgen. Es wird gezeigt, wie diese Abschätzungen wesentlich von der topologischen Position der Input-Quellen im Schaltkreis abhängen. Die zunehmend komplexen Schaltkreise in technologischen Geräten erfordern oftmals eine Modellierung als gekoppeltes System. Waveform relaxation (WR) empfiehlt sich zur Lösung solch gekoppelter Probleme, da sie auf die Subprobleme angepasste Lösungsmethoden und Schrittweiten ermöglicht. Es ist bekannt, dass WR zwar bei Anwendung auf gewöhnliche Differentialgleichungen konvergiert, falls diese eine Lipschitz-Bedingung erfüllen, selbiges jedoch bei DAEs nicht ohne Hinzunahme eines Kontraktivitätskriteriums sichergestellt werden kann. Wir beschreiben allgemeine Konvergenzkriterien für WR auf DAEs vom Index 2. Für den Fall von Schaltkreisen, die entweder mit anderen Schaltkreisen oder mit elektromagnetischen Feldern verkoppelt sind, leiten wir außerdem hinreichende topologische Konvergenzkriterien her, die anhand von Beispielen veranschaulicht werden. Weiterhin werden die Konvergenzraten des Jacobi WR Verfahrens und des Gauss-Seidel WR Verfahrens verglichen. Simulationen von einfachen Beispielsystemen zeigen drastische Unterschiede des WR-Konvergenzverhaltens, abhängig davon, ob die Konvergenzbedingungen erfüllt sind oder nicht.
The main topics of this thesis are firstly a thorough analysis of nonlinear differential-algebraic equations (DAEs) of index 2 which arise from the modified nodal analysis (MNA) for electrical circuits and secondly the derivation of convergence criteria for waveform relaxation (WR) methods on coupled problems. In both topics, a particular focus is put on the relations between a circuit's topology and the mathematical properties of the corresponding DAE. The analysis encompasses a detailed description of a normal form for circuit DAEs of index 2 and consequences for the sensitivity of the circuit with respect to its input source terms. More precisely, we provide bounds which describe how strongly changes in the input sources of the circuit affect its behaviour. Crucial constants in these bounds are determined in terms of the topological position of the input sources in the circuit. The increasingly complex electrical circuits in technological devices often call for coupled systems modelling. Allowing for each subsystem to be solved by dedicated numerical solvers and time scales, WR is an adequate method in this setting. It is well-known that while WR converges on ordinary differential equations if a Lipschitz condition is satisfied, an additional convergence criterion is required to guarantee convergence on DAEs. We present general convergence criteria for WR on higher index DAEs. Furthermore, based on our results of the analysis part, we derive topological convergence criteria for coupled circuit/circuit problems and field/circuit problems. Examples illustrate how to practically check if the criteria are satisfied. If a sufficient convergence criterion holds, we specify at which rate of convergence the Jacobi and Gauss-Seidel WR methods converge. Simulations of simple benchmark systems illustrate the drastically different convergence behaviour of WR depending on whether or not the circuit topological convergence conditions are satisfied.

by Chi-chung Lee.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1993.
Includes bibliographical references (leaves 45-47).
INTRODUCTION --- p.1
Chapter CHAPTER 1 --- RADIAL SYMMETRY OF GROUND STATES --- p.7
Chapter CHAPTER 2 --- EXISTENCE OF A GROUND STATE --- p.14
Chapter CHAPTER 3 --- UNIQUENESS OF GROUND STATE I --- p.23
Chapter CHAPTER 4 --- UNIQUENESS OF GROUND STATE II --- p.35
REFERENCES --- p.45

We show that Einstein’s gravitational field equations for the Friedmann- Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne- Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F.Williams. In particular we cast much of these authors’ works into a single framework via straightforward derivations of the EMP and NLS equations from a simple linear combination of the relevant Einstein equations. By rewriting the resulting expression in terms of the volume expansion factor and performing a change of variables, we obtain an uncoupled EMP or NLS equation that is independent of the imposition of additional conservation equations. Since the correspondences shown here present an alternative route for obtaining exact solutions to Einstein’s equations, we reconstruct many known exact solutions via their EMP or NLS counterparts and show by numerical analysis the stability properties of many solutions.

The Multiplicative Schwarz Preconditioned Inexact Newton (MSPIN) algorithm is presented as a complement to Additive Schwarz Preconditioned Inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. The ASPIN framework, as an option for the outermost solver, successfully handles strong nonlinearities in computational fluid dynamics, but is barely explored for the highly nonlinear models of complex multiphase flow with capillarity, heterogeneity, and complex geometry. In this dissertation, the fully implicit ASPIN method is demonstrated for a finite volume discretization based on incompressible two-phase reservoir simulators in the presence of capillary forces and gravity. Numerical experiments show that the number of global nonlinear iterations is not only scalable with respect to the number of processors, but also significantly reduced compared with the standard inexact Newton method with a backtracking technique. Moreover, the ASPIN method, in contrast with the IMPES method, saves overall execution time because of the savings in timestep size. We consider the additive and multiplicative types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Moreover, we provide the convergence analysis of the MSPIN algorithm. Under suitable assumptions, it is shown that MSPIN is locally convergent, and desired superlinear or even quadratic convergence can be obtained when the forcing terms are picked suitably. Numerical experiments show that MSPIN can be significantly more robust than Newton methods based on global linearizations, and that MSPIN can be more robust than ASPIN, and maintain fast convergence even for challenging problems, such as high-Reynolds number Navier-Stokes equations.

Studies of trace gas fluxes have advanced the understanding of bulk interactions between the atmosphere and ecosystems. Micrometeorological instrumentation is currently unable to resolve vertical scalar sources and sinks within plant canopies. Inverted analytical Lagrangian equations provide a non-intrusive method to calculate source distributions. These equations are based on Taylor's (1921) description of scalar dispersion, which requires a measure of the degree of correlation between turbulent motions, defined by the Lagrangian length scale (L). Inverse Lagrangian (IL) analyses can be unstable, and the uncertainty in L leads to uncertainty in source predictions. A review of the literature on studies using IL analysis with various scalars in a multitude of canopy types found that parameterizations where L reduces to zero at the ground produce better results in the IL analysis than those that increase closer to the ground, but no individual L parameterization gives better results than any other does. The review also found that the relationship between L and the measurable Eulerian length scale (Le) may be more complex in plant canopies than the linear scaling investigated in boundary layer flows. The magnitude and profile shape of L was investigated within a corn and a forest canopy using field measurements to constrain an analytical Lagrangian equation. Measurements of net CO2 flux, soil-to-atmosphere CO2 flux, and in-canopy profiles of CO2 concentrations provided the information required to solve for L in a global optimization algorithm for half hour intervals. For dates when the corn was a strong CO2 sink, and for the majority of dates for the forest, the optimization frequently located L profiles that follow a convex shape. A constrained optimization then smoothed the profile shape to a sigmoidal equation. Inputting the optimized L profiles in the forward and inverse Lagrangian equations leads to strong correlations between measured and calculated concentrations (corn canopy: C_{calc} = 1.00C_{meas} +52.41 mumol m^{-3}, r^2 = 0.996; forest canopy: C_{calc} = 0.98C_{meas} +276.5 mumol m^{-3}, r^2 = 0.99) and fluxes (corn canopy: F_{soil} = 0.67F_{calc} - 0.12 mumol m^{-2}s^{-1}, r^2 = 0.71, F_{net} = 1.17F_{calc} + 1.97mumol m^{-2}s^{-1}, r^2 = 0.85; forest canopy: F_{soil} = 0.72F_{calc} - 1.92 mumol m^{-2}s^{-1}, r^2 = 0.18, F_{net} = 1.24F_{calc} + 0.65 mumol m^{-2}s^{-1}, r^2 = 0.88). In the corn canopy, coefficients of the sigmoidal equation were specific to each half hour and did not scale with any measured variable. Coefficients of the optimized L equation in the forest canopy scaled weakly with variables related to the stability above the canopy. Plausible L profiles for both canopies were associated with negative bulk Richardson number values.
Funding from NSERC.

No campo "Projeto ou Bolsa de Financiamento" a referência à minha bolsa foi colocada manualmente não tendo sido selecionada na lista apresentada uma vez que não aparece.
Tese no âmbito do Programa Interuniversitário de Doutoramento em Matemática, orientada pelo Professor Doutor José Augusto Mendes Ferreira e pela Doutora Maria Teresa de Abrunhosa Barata e apresentada ao Departamento de Matemática da Faculdade de Ciências e Tecnologia da Universidade de Coimbra.
The computation of the magnetic field in the umbra- the darkest part of a sunspot, was the main motivation of this work. After several simplifications of the physical scenario to concretize this ambitious goal, it was realized that one should start by constructing stable and convergent numerical methods for nonlinear parabolic equations. In this thesis, numerical methods for initial boundary value problems (IBVPs) of nonlinear parabolic equations with Dirichlet or Neumann boundary conditions are proposed, and their stability and convergence analysis is established. These methods, defined in nonuniform partitions, can be seen simultaneously as finite difference methods (FDMs) and piecewise linear finite element methods (FEMs). Error estimates showing that the semi-discretization errors are second-order convergent with respect to a discrete version of the usual H1-norm are established. These error estimates show that the proposed methods lead to a second-order approximation for the solution and for its gradient. In the scope of the FDM, these error estimates are supraconvergent. This means that although they present spatial truncation error with first-order with respect to the norm ∥ · ∥∞ , the corresponding error is second-order convergent with respect to a discrete version of the H1-norm. On the other hand, in the finite element community, these error estimates can be seen as superconvergent estimates. In fact, piecewise linear FEMs lead, for linear elliptic equations, to first-order approximations with respect to the usual H1-norm, and second-order approximations with respect to the usual L2-norm. Although this fact, the second-order convergence is concluded with respect to a discrete version of the H1-norm. It should be pointed out that for differential problems with Dirichlet boundary conditions in two-dimensional domains or Neumann boundary conditions in the one-dimensional case, the error estimates are constructed assuming solution in C4. For differential problems with Dirichlet boundary conditions in the one-dimensional case, lower smoothness assumptions are imposed. The application of the developed methods to simulate strong and vertical magnetic fields is also an objective of the present work. Given this scenario, the numerical simulation is considered only the vertical component of the magnetic fields and one horizontal component. Dirichlet boundary conditions are assumed in a rectangular domain and are defined using numerical data, as well as the initial condition. The velocity field is also assumed to be known and obtained from numerical data too. As the quality of the magnitude of the magnetic field deteriorates with time due to the convective-dominated regime, a stabilization improvement is considered. Due to limitations on computational time, data handling, and availability of sunspot simulation, the numerical experiment is performed on a Network region where, on a shorter spatial scale, the condition is very similar to the one presented on the umbra of the sunspots. Another goal of the present thesis is the automatic detection and geometric definition of sunspots, including the limits of umbra and penumbra, in solar images. An image processing algorithm based on mathematical morphology is proposed, and its performance to detect and segment sunspots is analyzed. For this purpose, the Geophysical and Astronomical Observatory of the University of Coimbra database was used. In the near future, those results will be used to define the computational domain for the sunspots magnetic field evolution.
A evolução do campo magnético na umbra- a parte mais escura de uma mancha solar, foi a motivação central para este trabalho. Depois de várias simplificações no cenário físico, iniciou-se o estudo de métodos numéricos para equações de derivadas parciais parabólicas não lineares. Nesta tese são propostos métodos numéricos para problemas não lineares parabólicos com condições inicial e de fronteira do tipo Dirichlet ou Neumann e é estabelecida a sua análise numérica no que diz respeito à estabilidade e convergência. Os métodos propostos, definidos em partições não uniformes, podem ser vistos simultaneamente como métodos de diferenças finitas e métodos de elementos finitos segmentados lineares. São ainda construídas estimativas de segunda ordem para o erro associado à discretização espacial, relativamente a uma versão discreta da norma H1. Estas estimativas mostram que os métodos propostos permitem construir aproximações de segunda ordem para a solução bem como para o seu gradiente. No âmbito dos métodos de diferenças finitas, as estimativas de erro estabelecidas são estimativas supraconvergentes, isto é, o erro de truncatura associado à discretização espacial é de primeira ordem relativamente à norma ∥.∥∞ e o correspondente erro global é de segunda ordem relativamente a uma norma que pode ser vista como uma versão discreta da norma usual de H1. Na comunidade dos métodos de elementos finitos, estes resultados podem ser vistos como resultados de superconvergência. De facto, embora baseado no método de elementos finitos segmentado linear que apresenta, para equações elípticas lineares, ordem 2 relativamente à norma usual de L2 e ordem 1 relativamente à norma de H1, conclui-se ordem 2 relativamente a uma norma que pode ser vista como uma discretização da norma usual de H1. É de salientar que, quando o problema diferencial é complementado com condições de Dirichlet em domínios bidimensionais ou condições de Neumann num intervalo, as estimativas de erro são construídas assumindo que as soluções analíticas estão em C4. Por outro lado, para problemas com condições de Dirichlet no caso unidimensional são impostas condições de regularidade mais fracas. A aplicação dos métodos desenvolvidos na simulação do campo magnético na umbra é também um dos objetivos da presente trabalho. Neste contexto são consideradas na simulação numérica a componente vertical e uma componente horizontal. No que diz respeito às condições de fronteira, são assumidas condições de Dirichlet definidas a partir de dados numéricos, assim como a condição inicial. Supõe-se que o campo de velocidades é também conhecido. Atendendo a que o problema em questão é dominado pela convecção, observa-se que a qualidade do campo magnético se deteriora no tempo. Com o objetivo de contornar esta patologia numérica, é implementado um método de estabilização. Outro objetivo da presente tese é a detecção automática e definição geométrica das manchas solares, incluindo os limites da umbra e da penumbra, em imagens do sol. Um algoritmo de processamento de imagens baseado em morfologia matemática é proposto e seu desempenho na detecção e segmentação de manchas solares é analisado. Para o efeito, utiliza-se a base de dados do Observatório Geofísico e Astronómico da Universidade de Coimbra. Os resultados obtidos serão utilizados, num trabalho futuro, para definir o domínio computacional para o estudo da evolução do campo magnético de manchas solares.

Nearfield Acoustical Holography (NAH) is an acoustic field visualization technique that can be used to reconstruct three-dimensional (3-D) acoustic fields by projecting two-dimensional (2-D) data measured on a hologram surface. However, linear NAH algorithms developed and improved by many researchers can result in significant reconstruction errors when they are applied to reconstruct 3-D acoustic fields that are radiated from a high-level noise source and include significant nonlinear components. Here, planar, nonlinear acoustical holography procedures are developed that can be used to reconstruct 3-D, nonlinear acoustic fields radiated from a high-level noise source based on 2-D acoustic pressure data measured on a hologram surface. The first nonlinear acoustic holography procedure is derived for reconstructing steady-state acoustic pressure fields by applying perturbation and renormalization methods to nonlinear, dissipative, pressure-based Westervelt Wave Equation (WWE). The nonlinear acoustic pressure fields radiated from a high-level pulsating sphere and an infinite-size, vibrating panel are used to validate this procedure. Although the WWE-based algorithm is successfully validated by those two numerical simulations, it still has several limitations: (1) Only the fundamental frequency and its second harmonic nonlinear components can be reconstructed; (2) the application of this algorithm is limited to mono-frequency source cases; (3) the effects of bent wave rays caused by transverse particle velocities are not included; (4) only acoustic pressure fields can be reconstructed. In order to address the limitations of the steady-state, WWE-based procedure, a transient, planar, nonlinear acoustic holography algorithm is developed that can be used to reconstruct 3-D nonlinear acoustic pressure and particle velocity fields. This procedure is based on Kuznetsov Wave Equation (KWE) that is directly solved by using temporal and spatial Fourier Transforms. When compared to the WWE-based procedure, the KWE-based procedure can be applied to multi-frequency source cases where each frequency component can contain both linear and nonlinear components. The effects of nonlinear bent wave rays can be also considered by using this algorithm. The KWE-based procedure is validated by conducting an experiment with a compression driver and four numerical simulations. The numerical and experimental results show that holographically-projected acoustic fields match well with directly-calculated and directly-measured fields.