Σκοπός της διπλωματικής εργασίας είναι η μελέτη της επίλυσης μερικών διαφορικών εξισώσεων προβλημάτων διάχυσης απο ολοκληρωμένα κυκλώματα ASIC.Κυρίως επικεντρώνεται στην κατασκευή αλγορίθμων για την πλήρη αξιοποίηση των εγγενών χαρακτηριστικών της ASIC αρχιτεκτονικής. / The main subject of this work is the solution of partial non-linear differential equations from ASIC circuits.In order solution to be efficient new algorithms have to be created according to ASIC architecture special characterisitics.

The theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertation we study the approximation of linear partial differential equations on spheres, namely a class of elliptic partial differential equations and the heat equation on the unit sphere. The shifts of a spherical basis function are used to construct the approximate solution. In the elliptic case, both the finite element method and the collocation method are discussed. In the heat equation, only the collocation method is considered. Error estimates in the supremum norms and the Sobolev norms are obtained when certain regularity conditions are imposed on the spherical basis functions.

An investigation of preconditioning techniques is presented for a Newton--Krylov algorithm that is used for the computation of steady, compressible, high Reynolds number flows about airfoils. A second-order centred-difference method is used to discretize the compressible Navier--Stokes (NS) equations that govern the fluid flow. The one-equation Spalart--Allmaras turbulence model is used. The discretized equations are solved using Newton's method and the generalized minimal residual (GMRES) Krylov subspace method is used to approximately solve the linear system. These preconditioning techniques are first applied to the solution of the discretized steady convection-diffusion equation. Various orderings, iterative block incomplete LU (BILU) preconditioning and multigrid preconditioning are explored. The baseline preconditioner is a BILU factorization of a lower-order discretization of the system matrix in the Newton linearization. An ordering based on the minimum discarded fill (MDF) ordering is developed and compared to the widely popular reverse Cuthill--McKee ordering. An evolutionary algorithm is used to investigate and enhance this ordering. For the convection-diffusion equation, the MDF-based ordering performs well and RCM is superior for the NS equations. Experiments for inviscid, laminar and turbulent cases are presented to show the effectiveness of iterative BILU preconditioning in terms of reducing the number of GMRES iterations, and hence the memory requirements of the Newton--Krylov algorithm. Multigrid preconditioning also reduces the number of GMRES iterations. The framework for the iterative BILU and BILU-smoothed multigrid preconditioning algorithms is presented in detail.

An investigation of preconditioning techniques is presented for a Newton--Krylov algorithm that is used for the computation of steady, compressible, high Reynolds number flows about airfoils. A second-order centred-difference method is used to discretize the compressible Navier--Stokes (NS) equations that govern the fluid flow. The one-equation Spalart--Allmaras turbulence model is used. The discretized equations are solved using Newton's method and the generalized minimal residual (GMRES) Krylov subspace method is used to approximately solve the linear system. These preconditioning techniques are first applied to the solution of the discretized steady convection-diffusion equation. Various orderings, iterative block incomplete LU (BILU) preconditioning and multigrid preconditioning are explored. The baseline preconditioner is a BILU factorization of a lower-order discretization of the system matrix in the Newton linearization. An ordering based on the minimum discarded fill (MDF) ordering is developed and compared to the widely popular reverse Cuthill--McKee ordering. An evolutionary algorithm is used to investigate and enhance this ordering. For the convection-diffusion equation, the MDF-based ordering performs well and RCM is superior for the NS equations. Experiments for inviscid, laminar and turbulent cases are presented to show the effectiveness of iterative BILU preconditioning in terms of reducing the number of GMRES iterations, and hence the memory requirements of the Newton--Krylov algorithm. Multigrid preconditioning also reduces the number of GMRES iterations. The framework for the iterative BILU and BILU-smoothed multigrid preconditioning algorithms is presented in detail.

In this paper, we examine a method on how to model and produce Chladni Figures. We walk through how a thin metal plate, when vibrating at certain frequencies, can create various interest patterns. First we discuss the equation for the vertical force exerted on the plate, then we derive a PDE to solve for the nodal lines (lines that remain fixed, while the rest of the plate is oscillating). And, discuss how to create and model these figures, through a finite difference method. There have been several experiments on Chladni Figures, using some sort of vibrating membrane or plate and then either through the use of a speaker or a violin bow, produce frequencies in order to resonate with the membrane. These eigenvalue solutions can been physically observed by putting sand on the plate and vibrating it. We will approximate theses figures, calculate the convergence of the approximation, and relate the generated figures to figures produced in experiments.

Aspects of the asymptotic behaviour of cell-growth models described by partial differential equations, and systems of partial differential equations, are considered. The models considered describe the evolution of the size-distribution or age-distribution of a population of cells undergoing growth and division. First, the relationship between the behaviour, with and without dispersion, of a single-compartment size-distribution model of cell-growth with fixed-size cell division (where cells can only divide at a single, critical size) is considered. In this model dispersion accounts for stochastic variation in the growth process of each individual cell. Existence, uniqueness and the asymptotic stability of the solution is shown for a size-distribution model of cell-growth with dispersion and fixed-size cell division. The conditions for the analysis to hold for a more general class of division behaviours are also discussed. A class of nonlocal ordinary differential equations is studied, which contains as a subset the nonlocal ordinary differential equations describing the steady size-distributions of a single-compartment model of cell-growth. Existence of solutions to these equations is found to be implied by the existence of 'upper' and 'lower' solutions, which also provide bounds for the solution. A multi-compartment, age-distribution model of cell-growth is studied, which describes the evolution of the age-distribution of cells in different phases of cell-growth. The stability of the model when periodic solutions exist is examined. Sufficient conditions are given for the existence of stable steady age-distributions, as well as for stable periodic solutions. Finally, a multi-compartment age-size distribution model of cell-growth is studied, which describes the evolution of the age-size distribution of cells in different phases of cell-growth. Sufficient conditions are given for the existence of steady age-size distributions. An outline of the analysis required to prove stability of the steady age-size distributions of the model is also given. The analysis is based on ideas introduced in the previous chapters.

<p>Problems in stochastic homogenization theory typically deal with approximating differential operators with rapidly oscillatory random coefficients by operators with homogenized deterministic coefficients. Even though the convergence of these operators in multiple scales is well-studied in the existing literature in the form of a Law of Large Numbers, very little is known about their rate of convergence or their large deviations.</p><p>In the first part of this thesis, we we establish analytic results for the Gaussian correction in homogenization of an elliptic differential equation with random diffusion in randomly layered media. We also derive a Central Limit Theorem for a diffusion in a weakly random media.</p><p>In the second part of this thesis devise a technique for obtaining large deviation results for homogenization problems in random media. We consider the special cases of an elliptic equation with random potential, the random diffusion problem in randomly layered media and a reaction-diffusion equation with highly oscillatory reaction term.</p> / Dissertation

This paper discusses how a boundary-based surface fitting approach can be utilised to smoothly reconstruct a given human face where the scan data corresponding to the face is provided. In particular, the paper discusses how a solution to the Biharmonic equation can be used to set up the corresponding boundary value problem. We show how a compact explicit solution method can be utilised for efficiently solving the chosen Biharmonic equation. Thus, given the raw scan data of a 3D face, we extract a series of profile curves from the data which can then be utilised as boundary conditions to solve the Biharmonic equation. The resulting solution provides us a continuous single surface patch describing the original face.

The discussion in this paper focuses on how boundary based smooth shape design can be carried out. For this we treat surface generation as a mathematical boundary-value problem. In particular, we utilize elliptic Partial Differential Equations (PDEs) of arbitrary order. Using the methodology outlined here a designer can therefore generate the geometry of shapes satisfying an arbitrary set of boundary conditions. The boundary conditions for the chosen PDE can be specified as curves in 3-space defining the profile geometry of the shape. We show how a compact analytic solution for the chosen arbitrary order PDE can be formulated enabling complex shapes to be designed and manipulated in real time. This solution scheme, although analytic, satisfies exactly, even in the case of general boundary conditions, where the resulting surface has a closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples.

One of the central difficulties in many models of glacier and ice sheet flow lies in the prescription of boundary conditions at the bed. Often, processes which occur there dominate the evolution of the ice mass as they control the speed at which the ice is able to slide over the bed. In part I of this thesis, we study two complications to classical models of glacier and ice sheet sliding. First, we focus on the effect of cavity formation on the sliding of a glacier over an undeformable, impermeable bed. Our results do not support the widely used sliding law $u_b = C\tau_b^pN^{-q}$, but indicate that $\tau_b/N$ actually decreases with $u_b/N$ at high values of the latter, as suggested previously for simple periodic beds by Fowler (1986). The second problem studied is that of an ice stream whose motion is controlled by bed obstacles with wavelengths comparable to the thickness of ice. By contrast with classical sliding theory for ice of constant viscosity,the bulk flow velocity does not depend linearly on the driving stress. Indeed, the bulk flow velocity may even be a multi-valued function of driving stress and ice thickness. In the second part of the thesis, attention is turned to the formation of drumlins. The viscous till model of Hindmarsh (1998) and Fowler (2000) is analysed in some detail. It is shown that the model does not predict the formation of three-dimensional drumlins, but only that of two-dimensional features, which may be interpreted as Rogen moraines. A non-linear model allows the simulation of the predicted bedforms at finite amplitude. Results obtained indicate that the growth of bedforms invariably leads to cavitation. A model for travelling waves in the presence of cavitation is also developed, which shows that such travelling waves can indeed exist. Their shape is, however, unlike that of real bedforms, with a steep downstream face and no internal stratification. These results indicate that Hindmarsh and Fowler's model is probably not successful at describing the processes which lead to the formation of streamlined subglacial bedforms.