Multiconductor line with non homogeneous dielectric is modeled and analyzed. Model of the multiconductor line is presented. The finite element method and Matlab PDE Tools software were used for analysis and calculation of characteristic impedances. The program for characteristic impedance calculation was developed. Calculation results are presented and analyzed. They confirm that the finite element method and the presented program can be used for calculations of characteristic impedances of multiconductor lines with non homogeneous dielectric.

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

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In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.

A trend in parallel computer architecture is that systems with a large shared memory are becoming more and more popular. A shared memory system can be either a uniform memory architecture (UMA) or a cache coherent non-uniform memory architecture (cc-NUMA). In the present thesis, the performance of parallel PDE solvers on cc-NUMA computers is studied. In particular, we consider the shared namespace programming model, represented by OpenMP. Since the main memory is physically, or geographically distributed over several multi-processor nodes, the latency for local memory accesses is smaller than for remote accesses. Therefore, the geographical locality of the data becomes important. The focus of the present thesis is to study multithreaded PDE solvers on cc-NUMA systems, in particular their memory access pattern with respect to geographical locality. The questions posed are: (1) How large is the influence on performance of the non-uniformity of the memory system? (2) How should a program be written in order to reduce this influence? (3) Is it possible to introduce optimizations in the computer system for this purpose? The main conclusion is that geographical locality is important for performance on cc-NUMA systems. This is shown experimentally for a broad range of PDE solvers as well as theoretically using a model involving characteristics of computer systems and applications. Geographical locality can be achieved through migration directives that are inserted by the programmer or — possibly in the future — automatically by the compiler. On some systems, it can also be accomplished by means of transparent, hardware initiated migration and replication. However, a necessary condition that must be fulfilled if migration is to be effective is that the memory access pattern must not be "speckled", i.e. as few threads as possible shall make accesses to each memory page. We also conclude that OpenMP is competitive with MPI on cc-NUMA systems if care is taken to get a favourable data distribution.

Η παρούσα εργασία ασχολείται με μεθόδους εύρεσης κυματικών λύσεων καθώς και λύσεων οδευόντων κυμάτων επί σειράς πολύ γνωστών μερικών διαφορικών εξισώσεων καθώς και με θεωρήματα μελέτης της ύπαρξης και της μοναδικότητας, ευστάθειας, ασυμπτωτικής συμπεριφοράς και μονοτονίας των ανωτέρω λύσεων. Θα περιοριστούμε σε μερικές ansatze μεθόδους εύρεσης κυματικών λύσεων, καθώς και στην ύπαρξη και μοναδικότητα ειδικών κατηγοριών κυματικών λύσεων. / --

<p>Thin liquid films are often studied by reducing the Navier-Stokes equations</p><p>using Reynolds lubrication theory, which leverages a small aspect ratio</p><p>to yield simplified governing equations. In this dissertation a plate</p><p>coating application, in which polydimethylsiloxane coats a silicon substrate,</p><p>is studied using this approach. Thermal Marangoni stress</p><p>drives fluid motion against the resistance of gravity, with the parameter</p><p>regime being chosen such that these stresses lead to a stable advancing front.</p><p>Additional localized thermal Marangoni stress is used to control the thin film;</p><p>in particular, coating thickness is modulated through the intensity of such</p><p>localized forcing. As thermal effects are central to film dynamics, the dissertation</p><p>focuses specifically on the effect that incorporating temperature dependence</p><p>into viscosity, surface tension, and density has on film dynamics and control.</p><p>Incorporating temperature dependence into viscosity, in particular,</p><p>leads to qualitative changes in film dynamics.</p><p>A mathematical model is developed in which the temperature dependence</p><p>of viscosity and surface tension is carefully taken into account.</p><p>This model is then</p><p>studied through numerical computation of solutions, qualitative analysis,</p><p>and asymptotic analysis. A thorough comparison is made between the</p><p>behavior of solutions to the temperature-independent and</p><p>temperature-dependent models. It is shown that using</p><p>localized thermal Marangoni stress as a control mechanism is feasible</p><p>in both models. Among constant steady-state solutions</p><p>there is a unique such solution in the temperature-dependent model,</p><p>but not in the temperature-independent model, a feature that</p><p>better reflects the known dynamics of the physical system.</p><p>The interaction of boundary conditions with finite domain size is shown</p><p>to generate both periodic and finite-time blow-up solutions, with</p><p>qualitative differences in solution behavior between models.</p><p>This interaction also accounts for the fact that locally perturbed solutions,</p><p>which arise when localized thermal Marangoni forcing is too weak</p><p>to effectively control thin film thickness, exist only for a discrete</p><p>set of boundary heights.</p><p>Modulating the intensity of localized thermal Marangoni forcing is</p><p>an effective means of modulating the thickness of a thin film</p><p>for a plate coating application; however, such control must be initiated before</p><p>the film reaches the full thickness it would reach in the absence of</p><p>such localized forcing. This conclusion holds for both the temperature-independent</p><p>and temperature-dependent mathematical models; furthermore, incorporating</p><p>temperature dependence into viscosity causes qualitative changes in solution</p><p>behavior that better align with known features of the underlying physical system.</p> / Dissertation

This thesis discusses a novel method for modelling facial action units. It presents facial action units model based on boundary value problems for accurate representation of human facial expression in three-dimensions. In particular, a solution to a fourth order elliptic Partial Differential Equation (PDE) subject to suitable boundary conditions is utilized, where the chosen boundary curves are based on muscles movement defined by Facial Action Coding System (FACS). This study involved three stages: modelling faces, manipulating faces and application to simple facial animation. In the first stage, PDE method is used in modelling and generating a smooth 3D face. The PDE formulation using small sets of parameters contributes to the efficiency of human face representation. In the manipulation stage, a generic PDE face of neutral expression is manipulated to a face with expression using PDE descriptors that uniquely represents an action unit. A combination of the PDE descriptor results in a generic PDE face having an expression, which successfully modelled four basic expressions: happy, sad, fear and disgust. An example of application is given using simple animation technique called blendshapes. This technique uses generic PDE face in animating basic expressions.

Cette thèse est dédiée à la description et à l'analyse mathématique de phénomènes d'hétérogénéités et de congestion dans les modèles de la mécanique des fluides.On montre un lien rigoureux entre des modèles de congestion douce de type Navier-Stokes compressible qui intègrent des forces de répulsion à très courte portée entre composants élémentaires; et des modèles de congestion dure de type compressible/incompressible décrivant les transitions entre zones libres et zones congestionnées.On s'intéresse ensuite à la modélisation macroscopique de mélanges formés par des particules solides immergées dans un fluide.On apporte dans ce cadre une première réponse mathématique à la question de la transition entre les régimes de suspensions dictés par les interactions hydrodynamiques et les régimes granulaires dictés par les contacts entre les particules solides.On met par cette démarche en évidence le rôle crucial joué par les effets de mémoire dans le régime granulaire.Cette approche permet également un nouveau point de vue pour l'étude mathématique des fluides avec viscosité dépendant de la pression.On s'intéresse enfin à la modélisation microscopique et macroscopique du trafic routier.Des schémas numériques originaux sont proposés afin de reproduire des phénomènes de persistance d'embouteillages. / This thesis is dedicated to the description and the mathematical analysis of heterogeneities and congestion phenomena in fluid mechanics models.A rigorous link between soft congestion models, based on the compressible Navier--Stokes equations which take into account short--range repulsive forces between elementary components; and hard congestion models which describe the transitions between free/compressible zones and congested/incompressible zones.We are interested then in the macroscopic modelling of mixtures composed solid particles immersed in a fluid.We provide a first mathematical answer to the question of the transition between the suspension regime dictated by hydrodynamical interactions and the granular regime dictated by the contacts between the solid particles.The method highlights the crucial role played by the memory effects in the granular regime.This approach enables also a new point of view concerning fluids with pressure-dependent viscosities.We finally deal with the microscopic and the macroscopic modelling of vehicular traffic.Original numerical schemes are proposed to robustly reproduce persistent traffic jams.