Mastergradsoppgaver i matematikk
http://hdl.handle.net/10037/225
2018-10-19T22:50:48ZModelling high intensity laser pulse propagation in air using the modified Korteweg-de Vries equation
http://hdl.handle.net/10037/13417
Rørnes, Bjarne<br />
Ultrafast laser pulse experiments and applications are entering a phase that challenges the validity of mathematical models utilised to model longer pulses in nonlinear optics. This thesis aims to propose a possible mathematical model for high intensity laser pulse propagation in air through a multiple scales expansion of Maxwell’s equations and discuss a method on how to solve the corresponding differential equation, known as the modified Korteweg-de Vries equation in to the small dispersion regime. This equation is solvable using a technique named the scattering transform and due to weak dispersion the equation can be solved asymptotically. The method is based on using the asymptotic WKB approximation for the forward scattering problem and reformulating the inverse scattering as a Riemann-Hilbert problem. Both analytical steps and numerical procedures needed to use the method is discussed and implemented. A full example calculation using a particular initial condition is performed and some challenges using the method for more general initial conditions are discussed.<br />
2018-06-01T00:00:00ZModern climate-economic models and climate policies
http://hdl.handle.net/10037/13076
Grabovskaia, Sofiia<br />
The problem of climate change is one of the most discussed problems nowadays. The global warming has an unquestionable influence on the economic growth of the different countries, and, consequently, on the whole world economics. The climate economics thus is an actual topic to study. Moreover, it is important to predict how the climate will change over the next century and which resulting outcomes are possible.
Climate is changing both because of the natural effects and because of the human activity. Emissions of the greenhouse gases, especially the dioxide of carbon (CO2), are considered as the main cause of climate change. The emissions, obviously, cannot be absolutely stopped right in the moment, because it will stop the economic growth as well.
The main goal of this thesis is to analyze the ways and costs of emission reduction, concepts of the Integrated Assessment Models (IAMs) and damage functions, which are crucial for creating the future emission scenarios. In this thesis we will also explore the modern climate policies and targets of those.<br />
2018-05-15T00:00:00ZOn the Ewald-Oseen scattering formulation for linear and nonlinear transient wave scattering
http://hdl.handle.net/10037/11305
Kuzmina, Anastasiia<br />
In this thesis work we develop and apply EOS formulation to three scattering problems:
two of them are 1D problems and one is 2D. The first chapter comprises EOS
formulations and numerical implementation for 1D scattering problems.
Also in this chapter we use different numerical methods to solve test problem and
choose the most stable and accurate method for solving of the given 1D problems.
In the second chapter we discuss EOS formulation for 2D scattering problem.<br />
2017-05-16T00:00:00ZAlmost affine codes and matroids
http://hdl.handle.net/10037/11301
Diachkov, Konstantin<br />
In this thesis we study various types of block codes, like linear, mutlti-linear, almost affine codes. We also look at how these codes can be described by associated matroids. In addition we look at flags (chains) of codes and see how their behavior can be described using demi-matroids. We also introduce weight polynomials for almost affine codes.<br />
2017-05-15T00:00:00ZA complex contour based perfectly matched layer applied to a pattern generating model equation.
http://hdl.handle.net/10037/10841
Jenssen, Amund<br />
The observable universe consists of several non equilibrium systems that generate spatiotemporal behaviour in the form of various patterns. As the elementary laws of physics and chemistry are unable to explain the pattern forming behaviour of such systems, scientists have turned to desktop experiments and model equations to gain further insight. The model equations that generate numerical solutions similar to real world systems are computationally intensive, and this thesis discusses the possibility of designing a numerical scheme which are to reduce the computation time for a specific model equation.
The design is based on the perfectly matched layer (PML), a mathematical-numerical technique that works as an artificial absorbing layer within the discretized grid boundaries.
The thesis discuss how to impose a PML version of the model equation into the numerical method of lines(MOL) procedure, and various numerical and mathematical techniques are discussed in order to build this scheme.
The numerical simulations for the PML-equation fail to produce the correct spatiotemporal behaviour, and the discussed analysis states that a PML does not apply to the model equation discussed in the thesis.<br />
2017-02-17T00:00:00ZSeparable representations of the Poisson, Helmholtz and complex Helmholtz kernels
http://hdl.handle.net/10037/10840
Bjørgve, Magnar<br />
For high accuracy applications of integral operators in higher dimensions the
complexity of operation and storage usually grows exponentially with dimensions.
One method that has proven successful for handling these difficulties are the
separation of the integral kernels as linear combinations of products of
one-dimensional kernels, commonly referred to as separation of variables.
In this thesis we optimize the existing separable forms of the Poisson and
complex Helmholtz kernels used in the program package MRCPP. We then find a new
separable representation of the (non-complex) Helmholtz kernel.<br />
2017-02-15T00:00:00ZInsulating the Vacuum. Calculating the Casimir force using the boundary integral method with von Neumann boundary conditions
http://hdl.handle.net/10037/9778
Utheim, Marius<br />
In 2012, a new method for calculating the Casimir force between compact objects was developed, expressing the force in terms of a boundary integral equation. The case of perfectly conducting objects with Dirichlet boundary conditions in two dimensions was treated by Isak Kilen. The method was later extended to three dimensions by Karl Øyvind Mikalsen.
The contribution of this thesis will be to develop the method in two dimensions for the case when the objects are perfectly insulating, meaning von Neumann boundary conditions. A formula for the Casimir force in terms of a boundary integral problem is derived and shown to correctly predict the force between two parallel plates, except for a missing factor of 2 that was also observed for Dirichlet boundary conditions. The developed formula contains a coefficient that is dependent on the regularization scheme used, and it is not clear whether this coefficient is geometry-independent.<br />
2016-08-15T00:00:00ZModelling laser-matter interactions using resonant states
http://hdl.handle.net/10037/9300
Juhász, Dávid<br />
Studying how light interacts with materials has become important for many technological applications from optical communication to developing of new materials. Therefore scientists have always tried to improve their understanding of these effects. The primary goal has always been to microscopically describe the pertinent processes. This paper provides a brief introduction into the interactions of atoms with laser fields. Precisely this interaction, photoelectric effect and the blackbody radiation were those findings which started off the development of quantum mechanics. This theory allowed better description of atoms and it will be used in this work to handle the problem we are confronting. We will consider two of the simplest potentials and let the atom interact with a strong laser pulse in these potentials. From this interaction the so called resonant states will arise. The goal of this thesis is to investigate to what extent and in what meaning these resonant states form a complete set of functions and consequently can be used for expansion of atomic states of physical importance.<br />
2016-05-12T00:00:00ZHigh frequency financial time series prediction: machine learning approach
http://hdl.handle.net/10037/9255
Zankova, Ekaterina<br />
Machine learning is a rapidly evolving subfield of computer science. It has enormous amount of applications. One of the application domains is financial data analysis. Machine learning was usually applied for analysis and forecasting of daily financial time series. Availability of high frequency financial data became another challenge with its own specifics and difficulties. Regressors, being a significant part of machine learning field, have been selected as study subjects for this project. The purpose of this research is to apply machine learning techniques for predicting high frequency financial time series. Experiments are conducted using several regressors which are evaluated with respect to prediction quality and computation cost. The obtained results were analysed in order to reveal parameter combination for particular regressor that yields the best results according to chosen performance criteria.<br />
2016-05-13T00:00:00ZModelling the evolution of ideal, infinite domain patterns, on a finite domain using a Perfectly Matched Layer
http://hdl.handle.net/10037/9192
Antrushin, Andrey<br />
The Swift-Hohenberg equation is an evolution equation which can produce a Pattern, or a pattern-like picture, to be more precise. For example, it could be used to model some simple natural patterns, like stripes and rolls that one may observe in a Rayleigh-Benard convection experiment.
But for any pattern formation obtained by an evolution equation to look ideal, we have to consider this equation analytically on the infinite domain. If one wants to calculate and present the results numerically, the problem has to be discretized, the number of steps turns out to be finite then, and at some point the lateral boundaries appear. These boundaries cause a backward reflection and destroy the pattern eventually.
The idea of using a Perfectly Matched Layer in order to obtain some reflectionless boundaries was suggested. It should let us model the evolution of an ideal, infinite domain pattern, on an actual finite domain. From a set of numerical methods the most suitable ones will be chosen. The MATLAB environment is used to write a code to visualise the results.
Any investigations that could concern the fact of applying a Perfectly Matched Layer to an evolution Swift-Hohenberg equation seem to be absolutely new and yet untouched. This Thesis might be considered as a first step, as an introduction to the problem and its possible solutions.<br />
2016-01-28T00:00:00Z