The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications. <br /><br /> 'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools. <br /><br /> This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated.

Ein metrischer Graph mit einem selbstadjungierten, negativen Laplace-Operator wird Quantengraph genannt. In dieser Arbeit werden Transporteigenschaften zufälliger Laplace-Operatoren betrachtet. Dazu wird die Multiskalenanalyse (MSA) von euklidischen Räumen auf metrische Graphen angepasst. Eine Überdeckung der metrischen Graphen wird aus gleichmäßig polynomiellem Wachstum und der gleichmäßigen Beschränkung der Kantenlängen gewonnen. Als Hilfsmittel für die MSA werden eine Combes-Thomas-Abschätzung und eine Geometrische Resolventenungleichung bewiesen. Zusammen mit einer Wegner-Abschätzung und der Existenz von verallgemeinerten Eigenfunktionen wird mittels der modifizierten MSA spektrale Lokalisierung (d.h. reines Punktspektrum) mit polynomiell fallenden Eigenfunktionen am unteren Rand des Spektrums für negative Laplace-Operatoren mit zufälligem Potential geschlossen. Dabei sind alle Randbedingungen, die eine nach unten beschränkten Operator liefern, wählbar. / We prove spectral localization for infinite metric graphs with a self-adjoint Laplace operator and a random potential. Therefor we adapt the multiscale analysis (MSA) from the euclidean case to metric graphs. In the MSA a covering of the graph is needed which is obtained from a uniform polynomial growth of the graph. The geometric restrictions of the graph contain a uniform bound on the edge lengths. As boundary conditions we allow all settings which give a lower bounded self-adjoint operator with an associated quadratic form. The result is spectral localization (i.e. pure point spectrum) with polynomially decaying eigenfunctions in a small interval at the ground state energy.

<p>In this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.</p><p>Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.</p><p>To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.</p> / Dissertation

Diffusive mass transport within porous SOFC components is explored using two modeling approaches that can better inform the SOFC electrode design process. These approaches include performance metrics for electrode cross-sectional design and a fractal approach for modeling mass transport within the pore structure of the electrode reaction zone. The performance metrics presented are based on existing analytical models for transport within SOFC electrodes. These metrics include a correction factor for button-cell partial pressure predictions and two forms of dimensionless reactant depletion current density. The performance impacts of multi-dimensional transport phenomena are addressed through the development of design maps that capture the trade-offs inherent in the reduction of mass transport losses within SOFC electrode cross-sections. As a complement to these bulk electrode models, a fractal model is presented for modeling diffusion within the electrochemically active region of an SOFC electrode. The porous electrode is separated into bulk and reaction zone regions, with the bulk electrode modeled in one-dimension based on the dusty-gas formalism. The reaction zone is modeled in detail with a two-dimensional finite element model using a regular Koch pore cross-section as a fractal template for the pore structure. Drawing on concepts from the analysis of porous catalysts, this model leads to a straightforward means of assessing the performance impacts of reaction zone microstructure. Together, the modeling approaches presented provide key insights into the impacts of bulk and microstructural geometry on the performance of porous SOFC components.

We use Darcy's law and conservation of mass to model the flow of a fluid through a porous medium. It is a second order elliptic system with a heterogeneous coefficient. We consider the equations written in mixed form. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period epsilon, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when epsilon is small. Moreover, we present numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces. On the other hand, we also propose to use multiscale mortars as a coarse component to construct a two-level preconditioner for the saddle point linear system arising from the fine scale discretization of the mixed finite element system. The two-level preconditioners are constructed based on the interfaces. We propose a framework to define the interpolation operators for the face based two-level preconditioners for different combination of coarse and fine scale mortar spaces for matching and nonmatching grids. In this dissertation, we show that for quasi-homogeneous problems and matching grids, the condition number of the preconditioned interface operator is bounded by (log(H/h))², which is the same as the traditional two-level preconditioners, for quasi-homogeneous problems. We show several numerical examples to demonstrate that for the strongly heterogeneous porous media, it is often desirable and even necessary to use a higher dimensional coarse mortar space to construct the coarse preconditioner to achieve convergence. We apply our ideas to study slightly compressible single phase and two-phase flow in a porous medium. We find that for the nonlinear single phase problem, the two-level preconditioners could be successfully applied to the symmetrized linear system. For the two-phase problem, using the fine scale, instead of multiscale, velocity solutions from the flow problem can greatly benefit the transport problem. / text

This thesis is presented in two sections. Two different multiscale models are developed in order to increase the computational speed of two well known atomistic algorithms, Molecular Dynamics (MD) and Kinetic Monte Carlo (KMC). In Section I, the MD method is introduced. Following this, a multiscale method of linking an MD simulation of heat conduction to a finite element (FE) simulation is presented. The method is simple to implement into a conventional MD code and is independent of the atomistic model employed. This bridge between the FE and MD simulations works by ensuring that energy is conserved across the FE/MD boundary. The multiscale simulation allows for the investigation of large systems which are beyond the range of MD. The method is tested extensively in the steady state and transient regimes, and is shown to agree with well with large scale MD and FE simulations. Furthermore, the method removes the artificial boundary effects due to the thermostats and hence allows exact temperatures and temperature gradients to be imposed on to an MD simulation. This allows for better study of temperature gradients on crystal defects etc. In Section II, the KMC method is introduced. A continuum model for the KMC method is presented and compared to the standard KMC model of surface diffusion. This method replaces the many discrete back and forth atom jumps performed by a standard KMC algorithm with a single flux that can evolve in time. Elastic strain is then incorporated into both algorithms and used to simulate atom deposition upon a substrate by Molecular Beam Epitaxy. Quantum dot formation due to a mismatch in the lattice spacing between a substrate and a deposited film is readily observed in both models. Furthermore, by depositing alternating layers of substrate and deposit, self-organised quantum dot super-lattices are observed in both models.

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.

No description available.

In this dissertation, Richardson extrapolation and other computational techniques are used to develop a series of high accuracy high efficiency solution techniques for solving partial differential equations (PDEs). A Richardson extrapolation-based sixth-order method with multiple coarse grid (MCG) updating strategy is developed for 2D and 3D steady-state equations on uniform grids. Richardson extrapolation is applied to explicitly obtain a sixth-order solution on the coarse grid from two fourth-order solutions with different related scale grids. The MCG updating strategy directly computes a sixth-order solution on the fine grid by using various combinations of multiple coarse grids. A multiscale multigrid (MSMG) method is used to solve the linear systems resulting from fourth-order compact (FOC) discretizations. Numerical investigations show that the proposed methods compute high accuracy solutions and have better computational efficiency and scalability than the existing Richardson extrapolation-based sixth order method with iterative operator based interpolation. Completed Richardson extrapolation is explored to compute sixth-order solutions on the entire fine grid. The correction between the fourth-order solution and the extrapolated sixth-order solution rather than the extrapolated sixth-order solution is involved in the interpolation process to compute sixth-order solutions for all fine grid points. The completed Richardson extrapolation does not involve significant computational cost, thus it can reach high accuracy and high efficiency goals at the same time. There are three different techniques worked with Richardson extrapolation for computing fine grid sixth-order solutions, which are the iterative operator based interpolation, the MCG updating strategy and the completed Richardson extrapolation. In order to compare the accuracy of these Richardson extrapolation-based sixth-order methods, truncation error analysis is conducted on solving a 2D Poisson equation. Numerical comparisons are also carried out to verify the theoretical analysis. Richardson extrapolation-based high accuracy high efficiency computation is extended to solve unsteady-state equations. A higher-order alternating direction implicit (ADI) method with completed Richardson extrapolation is developed for solving unsteady 2D convection-diffusion equations. The completed Richardson extrapolation is used to improve the accuracy of the solution obtained from a high-order ADI method in spatial and temporal domains simultaneously. Stability analysis is given to show the effects of Richardson extrapolation on stable numerical solutions from the underlying ADI method.