Department of Mathematics
http://hdl.handle.net/1956/1065
Sun, 01 Mar 2015 04:34:41 GMT2015-03-01T04:34:41ZOn weak solutions and a covergent numerical scheme for the compressible Navier-Strokes Equations
http://hdl.handle.net/1956/9442
On weak solutions and a covergent numerical scheme for the compressible Navier-Strokes Equations
Svärd, Magnus
Journal article
In this paper, the three-dimensional compressible Navier-Stokes
equations are considered on a periodic domain. We propose a semi-discrete
numerical scheme and derive a priori bounds that ensures that the resulting
system of ordinary di erential equations is solvable for any h > 0. An a
posteriori examination that density remain uniformly bounded away from 0
will establish that a subsequence of the numerical solutions converges to a
weak solution of the compressible Navier-Stokes equations.
Thu, 26 Feb 2015 00:00:00 GMThttp://hdl.handle.net/1956/94422015-02-26T00:00:00ZAssessment of Sequential and Simultaneous Ensemble-based History Matching Methods for Weakly Non-linear Problems
http://hdl.handle.net/1956/9398
Assessment of Sequential and Simultaneous Ensemble-based History Matching Methods for Weakly Non-linear Problems
Fossum, Kristian
Doctoral thesis
<p>The ensemble Kalman filter (EnKF) has, since its introduction in 1994, gained much attention as a tool for sequential data assimilation in many scientific areas. In recent years, the EnKF has been utilized for estimating the poorly known petrophysical parameters in petroleum reservoir models. The ensemble based methodology has inspired several related methods, utilized both in data assimilation and for parameter estimation. All these methods, including the EnKF, can be shown to converge to the correct solution in the case of a Gaussian prior model, Gaussian data error, and linear model dynamics. However, for many problems, where the methods are applied, this is not satisfied. Moreover, several numerical studies have shown that, for such cases, the different methods have different approximation error.</p>
<p>Considering parameter estimation for problems where the model depends on the parameters in a non-linear fashion, this thesis explore the similarities and differences between the EnKF and the alternative methods. Several characteristics are established, and it is shown that each method represents a specific combination of these characteristics. By numerical comparison, it is further shown that a variation of the characteristics produce a variation of the approximation error.</p>
<p>A special emphasis is put on the effect of one characteristic, whether data are assimilated sequentially or simultaneously. Typically, several data types are utilized in the parameter estimation problem. In this thesis, we assume that each data depends on the parameters in a specific non-linear fashion. Considering the assimilation of two weakly non-linear data with different degree of non-linearity, we show, through analytical studies, that the difference between sequential and simultaneous assimilation depends on the combination of data.</p>
<p>Via numerical modelling, we investigate the difference between sequential and simultaneous assimilation on toy models and simplified reservoir problems. Utilizing realistic reservoir data, we show that the assumption of difference in non-linearity for different data holds. Moreover, we demonstrate that, for favourable degree of nonlinearity, it is beneficial to assimilate the data ordered after ascending degree of nonlinearity.</p>
Fri, 13 Feb 2015 00:00:00 GMThttp://hdl.handle.net/1956/93982015-02-13T00:00:00ZDynamic Capillary Effects in the Simulation of Flow and Transport in Porous Media: A New Linearisation Method
http://hdl.handle.net/1956/9374
Dynamic Capillary Effects in the Simulation of Flow and Transport in Porous Media: A New Linearisation Method
Teveldal, Silje Kjønaas
Master thesis
In this thesis mathematical models with and without dynamic capillary effects are developed to model water flow and solute transport through a porous medium. The system of equations are discretised using the finite volume method TPFA in space and the backward Euler method in time. To solve the nonlinear systems appearing at each time step numerically, robust linearisation methods are proposed. These methods do not involve the computation of derivatives. The methods are analysed and have been shown to be linearly convergent and robust. Moreover, the convergence was shown to be independent of mesh size. The influence that the dynamic effects have on flow and transport is studied numerically. Additional numerical experiments were conducted to study the convergence of the linearisation schemes. The numerical results are shown to be in correspondence with the theoretical results.
Thu, 25 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/93742014-12-25T00:00:00ZOn the extension giving the truncated Witt vectors
http://hdl.handle.net/1956/9347
On the extension giving the truncated Witt vectors
Skjøtskift, Torgeir
Master thesis
We explore the theory of cohomology of groups and
the classification of group extensions with abelian
kernel. We then look at the group extensions that underlie
the truncated Witt vectors on the truncation set {1,p}
where p is a prime number. It turns out that we
can do without the multiplicative structure on the
source ring A by factoring the extension's representing
cocycle through a map into the p-fold tensor
product of A divided out by the C_p-action.
Mon, 05 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/93472015-01-05T00:00:00ZTensor Induction as Left Kan Extension
http://hdl.handle.net/1956/9319
Tensor Induction as Left Kan Extension
Aye, Kaythi
Master thesis
Tue, 02 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/93192014-12-02T00:00:00ZWeak solutions and convergent numerical schemes of Brenner-Navier-Stokes equations
http://hdl.handle.net/1956/9230
Weak solutions and convergent numerical schemes of Brenner-Navier-Stokes equations
Svärd, Magnus
Research report
Lately, there has been some interest in modifications of the compressible
Navier-Stokes equations to include diffusion of mass. In this paper,
we investigate possible ways to add mass diffusion to the 1-D Navier-Stokes
equations without violating the basic entropy inequality. As a result, we recover
a general form of Brenner's modification of the Navier-Stokes equations.
We consider Brenner's system along with another modification where the viscous
terms collapse to a Laplacian diffusion. For each of the two modifications,
we derive a priori estimates for the PDE, suffciently strong to admit a weak
solution; we propose a numerical scheme and demonstrate that it satisfies the
same a priori estimates. For both modifications, we then demonstrate that the
numerical schemes generate solutions that converge to a weak solution (up to
a subsequence) as the grid is refined.
Tue, 20 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/92302015-01-20T00:00:00ZModeling of oil reservoirs with focus on microbial induced effects
http://hdl.handle.net/1956/9207
Modeling of oil reservoirs with focus on microbial induced effects
Babatunde, Stanley Owulabi
Master thesis
As an abstract to this thesis, we review some literatures in EOR and discussed the processes, strength and weakness of Microbial Enhanced Oil Recovery techniques. A two phase flow model
comprising water and oil via the concept of mean pressure has been formulated using mass conservation equations, Darcy's law and constitutive relations. This resulted in a set of coupled nonlinear parabolic partial
differential equation with primary variables being the mean pressure and water saturation. We discretized these equations in one dimension using a control volume discretization scheme in space and implicit Euler
in time. We employed the IMPES approach which decoupled the primary variables. A model validation test was made by comparison with an analytical solution and with the Couplex-Gas benchmark. The model was used to
investigate two major mechanisms by which the activities of bacterial helps in enhancing the recovery of the residual oil.
Sat, 01 Nov 2014 00:00:00 GMThttp://hdl.handle.net/1956/92072014-11-01T00:00:00ZMathematical modelling of slow drug release from collagen matrices
http://hdl.handle.net/1956/9181
Mathematical modelling of slow drug release from collagen matrices
Erichsen, Birgitte Riisøen
Master thesis
This master's thesis is about controlled drug release, which is a relatively new area of mathematical modelling. In this thesis there have been two major focuses. The first is to further understand the model for drug release from collagen matrices developed earlier by solving it with a different numerical scheme, and the second to develop a new model based on a different geometry. Both models are based on mass conservation and Fick's law, and are therefore possible to compare. The two models have been discretized and implemented, and the results compared to experimental data.
Tue, 23 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/91812014-09-23T00:00:00ZCell-centered finite volume discretizations for deformable porous media
http://hdl.handle.net/1956/9014
Cell-centered finite volume discretizations for deformable porous media
Nordbotten, Jan Martin
Journal article
The development of cell-centered finite volume discretizations for deformation is motivated by the desire for a compatible approach with the discretization of fluid flow in deformable porous media. We express the conservation of momentum in the finite volume sense, and introduce three approximations methods for the cell-face stresses. The discretization method is developed for general grids in one to three spatial dimensions, and leads to a global discrete system of equations for the displacement vector in each cell, after which the stresses are calculated based on a local expression. The method allows for anisotropic, heterogeneous and discontinuous coefficients.
The novel finite volume discretization is justified through numerical validation tests, designed to investigate classical challenges in discretization of mechanical equations. In particular our examples explore the stability with respect to the Poisson ratio and spatial discontinuities in the material parameters. For applications, logically Cartesian grids are prevailing, and we also explore the performance on perturbations on such grids, as well as on unstructured grids. For reference, comparison is made in all cases with the lowest-order Lagrangian finite elements, and the finite volume methods proposed herein is comparable for approximating displacement, and is superior for approximating stresses.
Mon, 11 Aug 2014 00:00:00 GMThttp://hdl.handle.net/1956/90142014-08-11T00:00:00ZExplicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositions
http://hdl.handle.net/1956/9006
Explicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositions
Munthe-Kaas, Antonella Zanna
Journal article
We discuss the construction of volume-preserving splitting methods based on a tensor product of single-variable basis functions. The vector field is decomposed as the sum of elementary divergence-free vector fields (EDFVFs), each of them corresponding to a basis function. The theory is a generalization of the monomial basis approach introduced in Xue & Zanna (2013, BIT Numer. Math., 53, 265–281) and has the trigonometric splitting of Quispel & McLaren (2003, J. Comp. Phys., 186, 308–316) and the splitting in shears of McLachlan & Quispel (2004, BIT, 44, 515–538) as special cases. We introduce the concept of diagonalizable EDFVFs and identify the solvable ones as those corresponding to the monomial basis and the exponential basis. In addition to giving a unifying view of some types of volume-preserving splitting methods already known in the literature, the present approach allows us to give a closed-form solution also to other types of vector fields that could not be treated before, namely those corresponding to the mixed tensor product of monomial and exponential (including trigonometric) basis functions.
Sun, 23 Feb 2014 00:00:00 GMThttp://hdl.handle.net/1956/90062014-02-23T00:00:00ZSymmetric spaces and Lie triple systems in numerical analysis of differential equations
http://hdl.handle.net/1956/9002
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
Munthe-Kaas, Hans; Quispel, Reinout; Zanna, Antonella
Journal article
A remarkable number of different numerical algorithms can be understood
and analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant curvature
and their tangents. This theory can be used to unify a range of different topics, such
as polar-type matrix decompositions, splitting methods for computation of the matrix
exponential, composition of selfadjoint numerical integrators and dynamical systems
with symmetries and reversing symmetries. The thread of this paper is the following:
involutive automorphisms on groups induce a factorization at a group level, and a
splitting at the algebra level. In this paper we will give an introduction to the mathematical
theory behind these constructions, and review recent results. Furthermore,
we present a new Yoshida-like technique, for self-adjoint numerical schemes, that
allows to increase the order of preservation of symmetries by two units. The proposed
techniques has the property that all the time-steps are positive.
Sat, 01 Mar 2014 00:00:00 GMThttp://hdl.handle.net/1956/90022014-03-01T00:00:00ZOn the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation
http://hdl.handle.net/1956/8968
On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation
Ali, Alfatih Mohammed A.; Kalisch, Henrik
Journal article
The Korteweg-de Vries (KdV) equation is widely recognized as a simple model
for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of
fluid. While solutions of the KdV equation describe the shape of the free surface, information
about the underlying fluid flow is encoded into the derivation of the equation, and the
present article focuses on the formulation of mass, momentum and energy balance laws in
the context of the KdV approximation. The densities and the associated fluxes appearing in
these balance laws are given in terms of the principal unknown variable η representing the
deflection of the free surface from rest position. The formulae are validated by comparison
with previous work on the steady KdV equation. In particular, the mass flux, total head and
momentum flux in the current context are compared to the quantities Q, R and S used in
the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448–460, 1954) on cnoidal
waves and undular bores.
Wed, 01 Oct 2014 00:00:00 GMThttp://hdl.handle.net/1956/89682014-10-01T00:00:00ZElectrical conductivity of fractured media: A computational study of the self-consistent method
http://hdl.handle.net/1956/8923
Electrical conductivity of fractured media: A computational study of the self-consistent method
Sævik, Pål Næverlid; Berre, Inga; Jakobsen, Morten; Lien, Martha
Conference object
Effective medium theory can be used to link conductivity estimation
methods with prior knowledge about the distribution
of fractures in the investigated geological structure. In the literature,
little work has been presented on assessing the accuracy
of effective medium approximations for dense networks
of finite-sized fractures. We present here a systematic computational
study, comparing the conductivity predictions of
the popular self-consistent method with results from numerical
finite-element simulations. Our results show that the selfconsistent
method is accurate within acceptable error bounds
for a range of parameter values, in some cases even beyond
the percolation limit. We also compare the percolation thresholds
predicted by self-consistent theory with the thresholds obtained
by a numerical percolation algorithm. For the cases we
have studied, the percolation thresholds agree to a remarkable
degree.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/1956/89232012-01-01T00:00:00ZFinite volume hydromechanical simulation in porous media
http://hdl.handle.net/1956/8911
Finite volume hydromechanical simulation in porous media
Nordbotten, Jan Martin
Journal article
Cell-centered finite volume methods are prevailing in numerical simulation of flow in porous media. However, due to the lack of cell-centered finite volume methods for mechanics, coupled flow and deformation is usually treated either by coupled finite-volume-finite element discretizations, or within a finite element setting. The former approach is unfavorable as it introduces two separate grid structures, while the latter approach loses the advantages of finite volume methods for the flow equation. Recently, we proposed a cell-centered finite volume method for elasticity. Herein, we explore the applicability of this novel method to provide a compatible finite volume discretization for coupled hydromechanic flows in porous media. We detail in particular the issue of coupling terms, and show how this is naturally handled. Furthermore, we observe how the cell-centered finite volume framework naturally allows for modeling fractured and fracturing porous media through internal boundary conditions. We support the discussion with a set of numerical examples: the convergence properties of the coupled scheme are first investigated; second, we illustrate the practical applicability of the method both for fractured and heterogeneous media.
Tue, 27 May 2014 00:00:00 GMThttp://hdl.handle.net/1956/89112014-05-27T00:00:00ZControllability on Infinite-Dimensional Manifolds: A Chow-Rashevsky Theorem
http://hdl.handle.net/1956/8886
Controllability on Infinite-Dimensional Manifolds: A Chow-Rashevsky Theorem
Khajeh Salehani, Mahdi; Markina, Irina
Journal article
One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow–Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension. In other words, the classical Chow–Rashevsky theorem, which is in fact a primary theorem in subriemannian geometry, gives a global connectivity property of a subriemannian manifold. In this paper, following the unified approach of Kriegl and Michor (The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, Am. Math. Soc., Providence, 1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Chow–Rashevsky theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable. To indicate an application of our approach to the infinite-dimensional geometric control problems, we conclude the paper with a novel controllability result on the group of orientation-preserving diffeomorphisms of the unit circle.
Tue, 29 Apr 2014 00:00:00 GMThttp://hdl.handle.net/1956/88862014-04-29T00:00:00ZMethods and Tools for Analysis of Symmetric Cryptographic Primitives
http://hdl.handle.net/1956/8828
Methods and Tools for Analysis of Symmetric Cryptographic Primitives
Kazymyrov, Oleksandr
Doctoral thesis
<p>The development of modern cryptography is associated with the emergence of computing machines. Since specialized equipment for protection of sensitive information was initially implemented only in hardware, stream ciphers were widespread. Later, other areas of symmetric and asymmetric cryptography were established with the invention of general-purpose processors. In particular, such symmetric cryptographic primitives as block ciphers, message authentication codes (MACs), authenticated ciphers and others began to develop rapidly. Today various cryptographic algorithms are commonly used in everyday life to protect private data.</p>
<p>Design and analysis of advanced symmetric cryptographic primitives require a lot of time and resources. This is related to many factors, mainly to the cryptanalysis of prospective encryption algorithms under development. Every year new and modified attacks are published, leading to a rapid increase in the quantity of requirements and criteria imposed on cryptoprimitives.</p>
<p>Most of this thesis is devoted to analysis and improvement of cryptographic attacks and corresponding criteria for basic components. Almost all modern cryptoprimitives use nonlinear mappings for protection against advanced attacks. In connection with that a new method was proposed for the generation of random substitutions (S-boxes) with extreme cryptographic indicators that can be used in the next-generation ciphers to provide high and ultra-high security levels. In addition, several criteria imposed on S-boxes used in block ciphers were analyzed and their significance for block ciphers was proven. It is worth mentioning a practical method of testing two vectorial Boolean functions and a universal tool for checking properties of arbitrary binary nonlinear components presented in papers gathered in this thesis.</p>
<p>Another part of the thesis is dedicated to the cryptanalysis of hash functions as well as block and stream ciphers. To be more precise, an algebraic attack based on a binary decision diagram (BDD) was performed on the reduced Data Encryption Standard (DES), a scaled-down version of Advanced Encryption Standard (AES) and extended affine (EA) equivalence problem. Moreover, an algebraic approach was used to reconstruct an initial representation of the current Russian hash standard GOST 34.11-2012. Finally, a backward states tree method has been used to analyze stream ciphers based on the combination principle of linear and nonlinear feedback registers.</p>
Mon, 01 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/88282014-12-01T00:00:00ZClassical and Stochastic Slit Löwner Evolution
http://hdl.handle.net/1956/8746
Classical and Stochastic Slit Löwner Evolution
Ivanov, Georgy
Doctoral thesis
Fri, 10 Oct 2014 00:00:00 GMThttp://hdl.handle.net/1956/87462014-10-10T00:00:00ZImproving the error rates of the Begg and Mazumdar test for publication bias in fixed effects meta-analysis
http://hdl.handle.net/1956/8611
Improving the error rates of the Begg and Mazumdar test for publication bias in fixed effects meta-analysis
Gjerdevik, Miriam; Heuch, Ivar
Journal article
<p>Background: The rank correlation test introduced by Begg and Mazumdar is extensively used in meta-analysis to test for publication bias in clinical and epidemiological studies. It is based on correlating the standardized treatment effect with the variance of the treatment effect using Kendall’s tau as the measure of association. To our knowledge, the operational characteristics regarding the significance level of the test have not, however, been fully assessed.</p>
<p>Methods: We propose an alternative rank correlation test to improve the error rates of the original Begg and Mazumdar test. This test is based on the simulated distribution of the estimated measure of association, conditional on sampling variances. Furthermore, Spearman’s rho is suggested as an alternative rank correlation coefficient. The attained level and power of the tests are studied by simulations of meta-analyses assuming the fixed effects model.</p>
<p>Results: The significance levels of the original Begg and Mazumdar test often deviate considerably from the nominal level, the null hypothesis being rejected too infrequently. It is proven mathematically that the assumptions for using the rank correlation test are not strictly satisfied. The pairs of variables fail to be independent, and there is a correlation between the standardized effect sizes and sampling variances under the null hypothesis of no publication bias. In the meta-analysis setting, the adverse consequences of a false negative test are more profound than the disadvantages of a false positive test. Our alternative test improves the error rates in fixed effects meta-analysis. Its significance level equals the nominal value, and the Type II error rate is reduced. In small data sets Spearman’s rho should be preferred to Kendall’s tau as the measure of association.</p>
<p>Conclusions: As the attained significance levels of the test introduced by Begg and Mazumdar often deviate greatly from the nominal level, modified rank correlation tests, improving the error rates, should be preferred when testing for publication bias assuming fixed effects meta-analysis.</p>
Mon, 22 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/86112014-09-22T00:00:00ZA study in Univalence
http://hdl.handle.net/1956/8566
A study in Univalence
Husebø, Anders Knarvik
Master thesis
In this master thesis we want to study the newly discovered homotopy type theory, and its
models within mathematics. We look at models in simplicial sets, simplicial symmetric monoids, and a new category which could be called multi pointed simplicial sets. We also describe dependent type theory from the informatical point of view, and some implications of it.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/85662014-06-02T00:00:00ZPortfolio Optimization with PCC-GARCH-CVaR model
http://hdl.handle.net/1956/8555
Portfolio Optimization with PCC-GARCH-CVaR model
Xi, Linda Mon
Master thesis
This thesis investigates the Conditional Value-at-Risk (CVaR) portfolio optimization approach combined with a univariate GARCH model and pair-copula constructions (PCC) to determine the optimal asset allocation for a portfolio.
The methodology focuses on minimizing CVaR as the risk measure in replacement of variance used in the traditional optimization framework of Markowitz. GARCH model provides a tool for predicting and analyzing the time-varying volatility financial assets are exposed to, while copulas allow us to model the non-linear dependence structure and margins separately.
We compare the performance of the CVaR optimized portfolio with other investment strategies such as Constant-Mix and Buy-and-Hold. Although the selection of strategy depends on the investor risk profile, it is empirically shown that the proposed CVaR optimized portfolio outperforms the other two investment strategies based on the accumulated wealth in the long run.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/85552014-06-02T00:00:00Z