<p>This thesis discusses the mathematical formulation and computational treatment of slow strain rate corrosion tests based on nonlinear finite elements methods. The theory is illustrated by a description of classical small strain elastoplasticity theory as implemented in the Comsol Multiphysics 3.2 software package. The possible extension of the theory to finite strain is briefly addressed. Practical simulations and results regarding the evolution of stresses, strains and geometric deformation are also presented and discussed. Experimental data used in simulation where reported by Onchi, Takeo et al. and published in Journal of Nuclear Science and Technology in May 2006.</p>

This thesis discusses the mathematical formulation and computational treatment of slow strain rate corrosion tests based on nonlinear finite elements methods. The theory is illustrated by a description of classical small strain elastoplasticity theory as implemented in the Comsol Multiphysics 3.2 software package. The possible extension of the theory to finite strain is briefly addressed. Practical simulations and results regarding the evolution of stresses, strains and geometric deformation are also presented and discussed. Experimental data used in simulation where reported by Onchi, Takeo et al. and published in Journal of Nuclear Science and Technology in May 2006.

This thesis consists of an introduction and five articles devoted to optimal stopping problems of American type options. In article A, we get general convergence results for the American option rewards for multivariate Markov price processes. These results are used to prove convergence of tree approximations presented in papers A, B, C and E.In article B, we study the problem of optimal reselling for European options. The problem can be transformed to the problem of exercising an American option with two underlying assets. An approximative binomial-trinomial tree algorithm for the reselling model is constructed. In article C, we continue our study of optimal reselling of European options and give the complete solution of the approximation problem. In the article D, we consider general knockout options of American type. A Monte-Carlo method is used to study structure of optimal stopping domains generated by combinations of different pay-off functions and knockout domains.In article E the American option with knock out domains is considered. In order to show convergence of the reward functional the problem is reformulated in such a way that the convergence results in paper A can be applied.

This thesis proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients. The stochastic problem is first transformed into a parametrized one by the use of the Karhunen--Loève expansion. This new problem is then discretized by the discontinuous Galerkin (DG) method. A priori error estimate in the energy norm for the stochastic discontinuous Galerkin solution is derived. In addition, the expected value of the numerical error is theoretically bounded in the energy norm and the L2 norm. In the second approach, the Monte Carlo method is used to generate independent identically distributed realizations of the stochastic coefficients. The resulting deterministic problems are solved by the DG method. Next, estimates are obtained for the error between the average of these approximate solutions and the expected value of the exact solution. The Monte Carlo discontinuous Galerkin method is tested numerically on several examples. Results show that the nonsymmetric DG method is stable independently of meshes and the value of penalty parameter. Symmetric and incomplete DG methods are stable only when the penalty parameter is large enough. Finally, comparisons with the Monte Carlo finite element method and the Monte Carlo discontinuous Galerkin method are presented for several cases.

This thesis demonstrates the advantages of new practical implementations of compressive sensing (CS) algorithms tailored for the graphics processing unit (CPU) using a software platform called Jacket. There exist many applications which utilize CS including medical imaging, signal processing and data acquisition which have benefited from advancements in CS. However, as problems become larger not only do they become more difficult to solve but also more computationally expensive. In light of tins, existing CS algorithms are augmented for practical use on the CPU, reaping performance gains from the highly parallel architecture of the GPU. I discuss the issues associated with this transition and analyze the effects of such a movement, as well as provide results exhibiting advantages of using CPU-based methods.

This thesis formulates and analyzes low-order penalty-free discontinuous Galerkin methods for solving the incompressible Stokes and Navier-Stokes equations. Some symmetric and non-symmetric discontinuous Galerkin methods for incompressible Stokes and Navier-Stokes equations require penalizing jump terms for stability and convergence of the methods. These discontinuous Galerkin methods are called interior penalty methods as the penalizing jump terms involve a penalty parameter. It is known that the penalty parameter has to be large enough to prove coercivity of the bilinear form and therefore to obtain existence of the solution for the symmetric case. The momentum equation is satisfied locally on each mesh element, and it depends on the penalty parameter. Setting the penalty parameter equal to zero yields a singular linear system, if piecewise linears are used. To overcome this instability, this thesis discusses an enrichment of the velocity space with locally supported quadratic functions called bubbles. First, the penalty-free non-symmetric discontinuous Galerkin method is analyzed for the Stokes equations. Second, the main contribution of this thesis is the analysis of both symmetric and non-symmetric penalty-free discontinuous Galerkin methods for the incompressible Varier-Stokes equations. Since a direct application of the generalized Lax-Milgram theorem is not possible, the numerical solution is shown to be the solution as a fixed-point of a problem-related map. A priori error estimate is derived.

This work formulates and analyzes a new coupled finite volume (FV) and discontinuous Galerkin (DG) method for convection-diffusion problems. DG methods, though costly, have proved to be accurate for solving convection-diffusion problems and capable of handling discontinuous and tensor coefficients. FV methods have proved to be very efficient but they are only of first order accurate and they become ineffective for tensor coefficient problems. The coupled method takes advantage of both the accuracy of DG methods in the regions containing heterogeneous coefficients and the efficiency of FV methods in other regions. Numerical results demonstrate that this coupled method is able to resolve complicated coefficient problems with a decreased computational cost compared to DG methods. This work can be applied to problems such as the transport of contaminant underground, the CO 2 sequestration and the transport of cells in the body.

The discrete Fourier transform has become an essential tool in the analysis of digital signals. Applications have become widespread since the discovery of the Fast Fourier Transform and the rise of personal computers. The field of digital signal processing is an exciting intersection of mathematics, statistics, and electrical engineering. In this study we aim to gain understanding of the mathematics behind algorithms that can extract chord information from recorded music. We investigate basic music theory, introduce and derive the discrete Fourier transform, and apply Fourier analysis to audio files to extract spectral data.

The concept of rendering an object invisible, once considered unfathomable, can now be deemed achievable using artificial metamaterials. The ability for these advanced structures to refract waves in the negative direction has sparked creativity for future applications. Manipulating electromagnetic waves of all frequencies around an object requires precise and unique parameters, which are calculated from various mathemat- ical laws and equations. We explore the possible interpretations of these parameters and how they are implemented towards the construction of a suitable metamaterial. If carried out correctly, the wave will exit the metamaterial exhibiting the same behavior as when it had entered. Thus, an outside observer will not be able to recognize any abnormal changes in wave frequency or direction. This paper will survey studies and technologies from the past 20 years to arrive at a concise mathematical examination of the possibilities and inherent issues under the umbrella of modern ”cloaking.”

There has been an increasing interest in the field of `smart structures' and `smart materials'. In constructing smart structures, a class of materials called smart materials are often used as sensors and actuators. An example of a smart material is shape memory alloy (SMA). A common actuator configuration uses an SMA wire with a constant load. The non-linear input-output behaviour of SMAs, known as hysteresis, made them difficult to model and control. The research in this thesis examines the effect of PID-controller gain optimization on SMA wire control at different frequencies of operation. A constant-load SMA wire actuator with a PID-controller is used in the study. Heat is applied to the wire using an input electric current. The system is cooled through convection with the surrounding area. The lack of active cooling prevents the system from operating at high frequencies. Three different cost functions are proposed for various applications. The Preisach model is chosen to model the hysteretic behaviour of the SMA wire contraction. Varying material properties such as electrical resistance and heat capacities are modelled to give a more accurate representation of the system's physical behaviour. Simulations show that by optimizing the controller gain values, the bandwidth of the system is improved. An interesting observation is made in the heating cycle of the SMA wire. In order to achieve faster cooling, overshoot is observed at low frequencies. This is a result of the system hysteresis. The system hysteresis allows different input signals to achieve the same output value. Since the rate of cooling is proportional to the temperature above ambient, better cooling is achieved by reaching a higher temperature. The error caused by the overshoot is compensated by the better cooling phase, which is not actively controlled.