Department of Mathematics
http://hdl.handle.net/1956/1065
2019-12-12T11:57:53ZDerived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic
http://hdl.handle.net/1956/20998
Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic
Honigs, Katrina; Lombardi, Luigi; Tirabassi, Sofia
Journal article
We prove that any Fourier–Mukai partner of an abelian surface over an algebraically closed
field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves.
We apply this fact to show that the set of Fourier–Mukai partners of a canonical cover of a
hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater
than three is trivial. These results extend earlier results of Bridgeland–Maciocia and Sosna
to positive characteristic.
2019-01-01T00:00:00ZBayesian and frequentist computation with application to data from the Medical Birth Registry of Norway
http://hdl.handle.net/1956/20830
Bayesian and frequentist computation with application to data from the Medical Birth Registry of Norway
Mannseth, Janne
Doctoral thesis
<p>In chapter 1, the field of statistics is discussed in general terms. Then, Bayes’ theorem is presented together with the posterior distribution. Finally, we consider maximum likelihood estimation and its relation to a two-step inference procedure.</p>
<p>The thesis proceeds to introducing several numerical methods in chapter 2. One of the main methods is Hamiltonian Monte Carlo (HMC). In order to understand HMC, which is a type of Markov chain Monte Carlo (MCMC) algorithm, we first study some theory of Markov chains. Furthermore, MCMC and some examples of its well-known algorithms are encountered. As HMC is based on Hamiltonian dynamics (that originate from the field of physics), a short illustration of the dynamics is given. Furthermore, the system of differential equations used in HMC is discussed, as are several different numerical integration schemes and methods for developing them. A numerical integration scheme is a necessary part of the HMC algorithm. Paper 1 covers HMC and finding more efficient numerical integration schemes to be used in the algorithm.</p>
<p>It is clear that there are situations where the MCMC algorithms perform poorly, especially with regards to time consumption. For certain latent variable models we should consider alternative numerical methods to MCMC. For these models, one could use integrated nested Laplace approximation (INLA). Although INLA has not been used in the model specification here, we are interested in some of the theory it applies. INLA is an efficient alternative for latent Gaussian models and uses some important properties of Gaussian fields (GFs) and Gaussian Markov random fields (GMRFs). From INLA, we here more specifically consider the relation between the computationally demanding covariance matrix and the sparse precision matrix. These two matrices are directly connected to GFs and GMRFs respectively, and we examine some the computational advantages that come with GMRFs compared to GFs. We use the template model builder (TMB) that uses automatic differentiation and Laplace approximation in the process of finding the maximum likelihood estimation of the parameters of the marginal likelihood. The combination in TMB gives the ability to handle complex latent variable models, and this is the methodology that is used for the spatial models in paper 2 and paper 3.</p>
<p>In chapter 3 we look at the field of spatial statistics. This includes its uses and the different types of spatial data that we have. Spatial data are separated by what kind of location information that is available and what type of problem we want to model. Moreover, we discuss the stochastic partial differential equations (SPDE) approach. This is an approach that is able to link GFs and GMRFs using a triangulated grid. This grid, a weak solution to the SPDE and its use with spatial models are examined. To understand the SPDE approach we also consider the Matérn covariance function, which is the covariance function for certain GFs. The SPDE approach is essential for the results of paper 2 and paper 3. The spatial and spatio-temporal models are also explained and some additional results to paper 3 are presented. Finally, we look at health registry data, which is the type of data used in paper 2 and paper 3.</p>
2019-04-25T00:00:00ZModel Construction with Support Vector Machines and Gaussian Processes through Kernel Search
http://hdl.handle.net/1956/20740
Model Construction with Support Vector Machines and Gaussian Processes through Kernel Search
Bentsen, Fredrik Hjorth
Master thesis
2019-08-16T00:00:00ZDrivers of the summer-distribution of Northeast Atlantic mackerel (Scomber scombrus) in the Nordic Seas from 2011 to 2017; a Bayesian hierarchical modelling approach
http://hdl.handle.net/1956/20721
Drivers of the summer-distribution of Northeast Atlantic mackerel (Scomber scombrus) in the Nordic Seas from 2011 to 2017; a Bayesian hierarchical modelling approach
Nikolioudakis, Nikolaos; Skaug, Hans J.; Olafsdottir, Anna H.; Jansen, Teunis; Jacobsen, Jan Arge; Enberg, Katja
Journal article
Identifying factors that are statistically correlated with the geographical distribution dynamics of a species can facilitate our understanding of causal physiological and ecological relationships. Northeast Atlantic (NEA) mackerel is a species of great economic and ecological importance, whose habitat expansion in the last decade has altered the biomass dynamics in the pelagic realm of the Nordic Seas. We highlight drivers that may have regulated the geographical distribution of NEA mackerel during summers, from 2011 to 2017, by fitting Bayesian hierarchical spatiotemporal models on data obtained during the International Ecosystem Summer Survey in the Nordic Seas. Temperature in the upper 50 m of the water column, food availability (approximated by mesozooplankton biomass), a proxy of herring abundance and longitude were the main factors influencing both the catch rates (proxy for fish density) and the occurrence of NEA mackerel. Stock size was not found to directly influence the distribution of the species; however, catch rates in higher latitudes during years of increased stock size were lower. Additionally, we highlight the improved performance of models with spatiotemporal covariance structures, thus providing a useful tool towards elucidating the complex ecological interactions of the pelagic ecosystem of the Nordic Seas.
2019-03-01T00:00:00ZA Likelihood Ratio and Markov Chain-Based Method to Evaluate Density Forecasting
http://hdl.handle.net/1956/20674
A Likelihood Ratio and Markov Chain-Based Method to Evaluate Density Forecasting
Li, Yushu; Andersson, Lars Jonas
Journal article
In this paper, we propose a likelihood ratio based method to evaluate density forecasts which can jointly evaluate the unconditional forecasted distribution and dependence of the outcomes. Unlike the well‐known Berkowitz test, the proposed method does not requires a parametric specification of time dynamics. We compare our method with the method proposed by several other tests and show that our methodology has very high power against both dependence and incorrect forecasting distributions. Moreover, the loss of power, caused by the non‐parametric nature of the specification of the dynamics, is shown to be small compared to Berkowitz test, even when the parametric form of dynamics is correctly specified in the latter method.
2019-01-01T00:00:00ZUpscaling, analysis, and iterative numerical solution schemes for thermo-poroelasticity
http://hdl.handle.net/1956/20657
Upscaling, analysis, and iterative numerical solution schemes for thermo-poroelasticity
Brun, Mats Kirkesæther
Doctoral thesis
<p>The main objectives of this research project is to provide part of the mathematical
models and simulation technology required to assess large-scale deployment of thermomechanical
subsurface energy storage in the context of intermittent renewable energy.
Within this overarching framework, two main topics are discussed in this dissertation:
Thermo-poroelasticity, i.e., the coupling of geomechanics, flow, and heat within
a porous material, and phase field brittle fracture propagation, i.e., the temporal and
spatial tracking of brittle fracture evolution by creating a diffusive zone around fracture
surfaces through an auxiliary variable known as a phase field. These subjects are highly
relevant for e.g., applications involving injection of heated fluids at high pressures into
the subsurface, where the increased pressure may cause a fracturing of the rock matrix,
in addition to the induced temperature gradient affecting the poroelastic properties of
the surrounding medium.</p>
<p>The part of this dissertation focused on thermo-poroelasticity can again be separated
into three parts; (1) modeling, (2) analysis and (3) numerical implementation. In part
(1), formal upscaling techniques (i.e., homogenization) are employed in order to derive
the constitutive thermo-poroelastic equations from the known equations governing the
physical processes at the pore-scale. Homogenization is a well-known and trusted technique,
which is applicable in situations where the physical processes in question can
be viewed from several scales, and where there is some uniformity or periodicity on
the smaller scale. Within the context of porous media, the two relevant scales are the
pore-scale and the macro-scale. Viewed from the pore scale, a porous medium consists
of solid grains (obeying the laws of solid mechanics), and a fluid which is saturating
the space in between the grains (obeying the laws of fluid mechanics), where the two
processes are coupled at the mutual interface (i.e., at the grain surfaces). At the macroscale,
however, the porous medium is considered as a homogenous material obeying its
own set of physical laws, but which is still somehow implicitly dependent on (or rather,
the culmination of) the processes which are taking place at the pore-scale. Assuming
some uniformity in the distribution of grains (formally, periodicity), the technique of
homogenization allows for deriving the macro-scale model equations from the microscale
model equations.</p>
<p>Part (2) is concerned with analyzing the thermo-poroelastic system derived in part
(1). The existence and uniqueness of a weak solution to this model problem is established
in the fully mixed formulation, under some natural assumptions on the regularity of the source and initial data, as well as imposing some constraints on the effective coefficients.
Since the upscaled thermo-poroelastic system is a nonlinear one, this analysis is
done in two steps: First, a linearized system is analyzed using a standard Galerkin technique
together with the weak compactness properties of the relevant function spaces.
Finally, the previously analyzed linearized system is used to design an iterative scheme
in order to approximate the original nonlinear system. By employing a contraction argument,
the convergence of this iterative scheme is proved, which implies the existence
and uniqueness of a weak solution to the full nonlinear problem.</p>
<p>Part (3) covers the numerical analysis and implementation of the thermo-poroelastic
system derived in part (1). Here, six different iterative algorithms are proposed, all based
on the linearization technique employed in part (2) when analyzing the full nonlinear system,
as well as the stabilized splitting scheme known informally as the ‘𝐿-scheme’. These
six algorithms involve different combinations of coupling / decoupling of the three subproblems
involved (flow, mechanics and heat), i.e., at each iteration either a linearized
system is solved monolithically, two subproblems are solved together decoupled from
the third, or all three subproblems are decoupled. As such, these six algorithms exhaust
all possibilities of coupling / decoupling of the three subproblems. The convergence of
all six algorithms are proved using a contraction argument, similar to the one employed
in part (2), except that only the fully discrete formulation is considered. Several numerical
tests validate the robustness and efficiency of the algorithms. Furthermore, the
performance of all six algorithms are compared with respect to a wide range of different
physical regimes (i.e., with respect to various coupling strengths between the three
subproblems).</p>
<p>Finally, the last part of this dissertation concerns brittle fracture propagation in a
quasi-static elastic medium, where the fracture evolution is tracked by a phase field
variable. In particular, the numerical approximation of such models. Phase field brittle
fracture problems are notoriously difficult to solve, and currently no universally accepted
method exists. Hence, this work involves designing a novel iterative algorithm
for brittle fracture phase field models, analyzing its convergence, and testing it in detail
with several numerical benchmark problems. The proposed algorithm is based on a linearization
as well as stabilization of the model, where the two subproblems (phase field
and mechanics) are solved separately at each iteration, while sharing updated solution
information. Under the natural conditions that the mechanical elastic energy remains
bounded, and that the diffusive zone around crack surfaces must be sufficiently thick,
monotonic convergence of the proposed scheme is proved. These properties are also
confirmed by the extensive numerical tests.</p>
2019-06-28T00:00:00ZUpscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium
http://hdl.handle.net/1956/20656
Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium
Brun, Mats Kirkesæther; Berre, Inga; Nordbotten, Jan Martin; Radu, Florin Adrian
Journal article
We undertake a formal derivation of a linear poro-thermo-elastic system within the
framework of quasi-static deformation. This work is based upon the well-known derivation
of the quasi-static poroelastic equations (also known as the Biot consolidation model) by
homogenization of the fluid-structure interaction at the microscale. We now include energy,
which is coupled to the fluid-structure model by using linear thermoelasticity, with the full
system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar
to the linear poroelastic equations, but with an added conservation of energy equation,
fully coupled to the momentum and mass conservation equations. In the end, we obtain a
system of equations on the macroscale accounting for the effects of mechanical deformation,
heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients
can be explicitly defined in terms of the microstructure of the material. For the heat transfer
we consider two different scaling regimes, one where the Péclet number is small, and
another where it is unity.We also establish the symmetry and positivity for the homogenized
coefficients.
2018-08-01T00:00:00ZClose-Kin Mark-Recapture Models
http://hdl.handle.net/1956/20644
Close-Kin Mark-Recapture Models
Førland, Brage
Master thesis
Close-Kin Mark-Recapture (CKMR) is a recent extension of the ordinary mark–recapture methods used to estimate animal abundance and other population parameters. Where ordinary
mark–recapture only consider the subsequent identification of the same animal a recapture,
CKMR expands this by also viewing the genetic identification of a relatives as a recapture.
One of the challenges of CKMR models compared to ordinary mark–recapture is that the
recapture probabilities are tightly coupled to the life histories of the animals in questions.
This thesis contains three different contributions to the CKMR literature. Firstly I develop
a CKMR estimator for age structured populations, presented in Ruzzante (2019). Secondly, I
develop theoretical background for half sibling CKMR analysis, and apply kin analysis to data
from the River Etne. Thirdly, it expands on the results from Skaug (2017) and derives several
new results for the case where age of both parent and offspring is unknown.
The first part contains the method development of a parent–offspring CKMR model for
brook trout populations, electrofished yearly in the period 2013-2018. I here develop a moment
estimator for population size for an age structured model, related to the Lincoln–Petersen
estimator. The estimator is applied under two different population assumptions, stable age
structure, and variable recruitment and representative sampling. Special focus is on the small
population situation, where large sample approximations used in previous CKMR studies cannot be assumed. A small sample bias correction for the estimator is developed and validated
using parametric bootstrap simulations. Using the perspective that the parent marks the offspring instead of the commonly used offspring marks juvenile, a simple and general form of
the estimator is derived. Viewing offspring as the marked part of the population also leads to
an expression for the variance of the expected number of parent–offspring pairs in a sample,
which is found to be less than the Poisson variance unless fecundity is very overdispersed.
The second part contains theoretical background and model development for half sibling
CKMR analysis, to examine the conditions under which same cohort siblings are suitable for
CKMR analysis. A half sibling kinship analysis of single year data set of Atlantic salmon from
the River Etne 2013 is performed to check if it is suitable for CKMR.
In the third part, the probability that an individual has a living parent in an age structured
population is discussed in detail. For the case where age information for both parent and
offspring is unavailable, I derive two useful expressions for the probability of a living parent
when mortality is constant, or constant after onset of maturity. With the additional assumption
of constant population size, this probability is shown to be 1/2, similar to what is previously
proved for constant fecundity.
2019-07-09T00:00:00ZEffects of thermally-dependent rock parameters on fracture reactivation
http://hdl.handle.net/1956/20639
Effects of thermally-dependent rock parameters on fracture reactivation
Meier, Marius
Master thesis
The slip tendency is an important tool to measure fracture reactivation in thecontext of subsurface engineering applications. In this thesis, the effects of thermally-dependent rock parameters such as Young’s modulus on the slip tendency are exam-ined by running several two-dimensional simulations in a porous media with a single fracture. The numerical solver used is the MPSA method which is a modified finite volume method used for elasticity problems. First, simulations are done with two different sets of boundary conditions to see which combination of boundary conditions and fracture orientation produces the highest slip tendencies. After the best combi-nation is found, the simulations are done again with varying temperature to see the effect temperature has on the slip tendency. A third series of simulations follow with a temperature gradient acting on the domain which is dependent on the depth of the domain. All the simulation results represented here clearly indicate that temperature variations, varying orientation of the fracture and varying Young’s modulus on the surrounding matrix as a function of the temperature have significant impact on the slip tendency.
2019-07-09T00:00:00ZStatistical approaches for constructing runoff maps - Random forest, linear models and spatial models tested for predictive performance of ungauged catchments in Norway
http://hdl.handle.net/1956/20637
Statistical approaches for constructing runoff maps - Random forest, linear models and spatial models tested for predictive performance of ungauged catchments in Norway
Herud, Ida Jahren
Master thesis
In this thesis we study statistical approaches to tackle predictions of ungauged catchments in Norway. In collaboration with Norwegian Water Resources and Energy Directorate (NVE) we use observations of runoff, catchment characteristics (e.g. elevation and land use) and observations of precipitation for constructing runoff models suitable for runoff maps. The challenge of constructing suitable models for ungauged catchments is due to the lack of observations, and in the ﬁeld of hydrology this problem is known as the problem of ungauged basins (Blöschl et al., 2013). It is common to either use a deterministic hydrological model or asuitable model for transferring observations from a gauged catchment to an ungauged catchment. Our statistical approach for modeling median annual runoff has been done in a three-step procedure where our main focus has been on the models predictive performance also including the uncertainty quantiﬁcation. First, we did an exploratory analysis of observed median annual runoff and catchment characteristics. As a second step we ﬁtted two initial model classes (linear regression models and random forests models) where we observed how the different explanatory variables/features inﬂuenced our predictions. With the main learning’s of our second step we built four different spatial models within the Bayesian framework. From the main learning’s we found that spatial dependency have a large effect on predictive performance, and that gradient basin was the only catchment characteristics that inﬂuenced the models. The model with the best predictive performance was a Bayesian hierarchical model of tree levels where gradient basin was included in addition to a Gaussian random ﬁeld (GRF) and precipitation with a spatially varying coefﬁcient. All models have been carefully evaluated through leave-one-out cross-validation (LOOCV), where each model have been evaluated in terms of predictive performance with the two evaluation metrics; root mean square error (RMSE) and continuous ranked probability score (CRPS). While RMSE describes the difference between observed and predicted runoff, we account for the whole posterior predictive distribution with CPRS, and is thus useful for quantifying the uncertainty of our predictions.
2019-07-09T00:00:00ZMathematical modeling of microbial enhanced oil recovery with focus on bio-plug technology: from the pore to the core scale
http://hdl.handle.net/1956/20573
Mathematical modeling of microbial enhanced oil recovery with focus on bio-plug technology: from the pore to the core scale
Landa-Marbán, David
Doctoral thesis
This dissertation addresses the applications and challenges of both laboratory experiments
and mathematical modeling at different scales, where the main character is
biofilm. Thus, the first part of this work shows biological, chemical and physical concepts
for the laboratory experiments and mathematical terms for the modeling, upscaling
and numerical solutions. The second part contains the research papers.
In our research, we are interested in studying the biofilm to improve the oil extraction.
Most of the biofilm models are based on simplifying assumptions, e.g. impermeability,
a constant biofilm density and accounting for diffusion but neglecting convection for
transport of nutrients. In this work, we propose a pore-scale model for a permeable
multi-component biofilm including a variable biofilm density, detachment and transport
of nutrients due to convection and diffusion. It is through laboratory experiments that
we identify the key processes and variables that need to be considered. Accordingly, we
use experimental determined parameters and compute some of the parameters through
calibration. In addition, we study the sensitivity of the parameters in the mathematical
models.
Pore-scale models are important because they aim to describe physical phenomena
in detail and one can derive core-scale models through upscaling. Then, we can reflect
the effects of the pore-scale processes on the core scale. Upscaling of pore-scale models
allows us to describe the average behavior of a system in an accurate manner with relatively
low computational effort. Then, we upscale this pore-scale model in two different
geometries: a thin channel and a thin tube, in order to derive one-dimensional effective
equations, by investigating the limit as the ratio of the aperture to the length approaches
to zero.
In the core-scale laboratory experiments, biofilm is grown in cylindrical cores. Permeability
and porosity changes over time at different flow rates and nutrient concentrations
are studied. Numerical simulations are performed to compare with the experimental
results. We also present how to extend the model to include chemotaxis and
interfacial tension reduction due to surface active compounds.
Mathematical models for biofilms are based on coupled non-linear partial differential
equations and ordinary differential equations, which may be challenging to solve.
Therefore, it is necessary to use advanced numerical methods and simulations to predict
the behavior on time of the unknowns in these complex systems. We present some of
the common space discretizations, time discretizations and numerical solvers for these models. We also discuss the difficulty of free boundary problems and the numerical
techniques to deal with them. Last but not least, we discuss the challenges of parameter
estimation and the application of sensitivity analysis.
2019-06-14T00:00:00ZEstimating the discretization dependent accuracy of perfusion in coupled capillary flow measurements
http://hdl.handle.net/1956/20552
Estimating the discretization dependent accuracy of perfusion in coupled capillary flow measurements
Hanson, Erik Andreas; Sandmann, Constantin; Malyshev, Alexander; Lundervold, Arvid; Modersitzki, Jan; Hodneland, Erlend
Journal article
One-compartment models are widely used to quantify hemodynamic parameters such as perfusion, blood volume and mean transit time. These parameters are routinely used for clinical diagnosis and monitoring of disease development and are thus of high relevance. However, it is known that common estimation techniques are discretization dependent and values can be erroneous. In this paper we present a new model that enables systematic quantification of discretization errors. Specifically, we introduce a continuous flow model for tracer propagation within the capillary tissue, used to evaluate state-of-the-art one-compartment models. We demonstrate that one-compartment models are capable of recovering perfusion accurately when applied to only one compartment, i.e. the whole region of interest. However, substantial overestimation of perfusion occurs when applied to fractions of a compartment. We further provide values of the estimated overestimation for various discretization levels, and also show that overestimation can be observed in real-life applications. Common practice of using compartment models for fractions of tissue violates model assumptions and careful interpretation is needed when using the computed values for diagnosis and treatment planning.
2018-07-20T00:00:00ZWell-posedness of the fully coupled quasi-static thermo-poroelastic equations with nonlinear convective transport
http://hdl.handle.net/1956/19773
Well-posedness of the fully coupled quasi-static thermo-poroelastic equations with nonlinear convective transport
Brun, Mats Kirkesæther; Radu, Florin Adrian; Ahmed, Elyes; Nordbotten, Jan Martin
Journal article
This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biot's model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in these formulations are the fluid pressure, temperature and elastic displacement as well as the Darcy flux, heat flux and total stress. The well-posedness of a linearized formulation is addressed first through the use of a Galerkin method and suitable a priori estimates. This is used next to study the well-posedness of an iterative solution procedure for the full nonlinear problem. A convergence proof for this algorithm is then inferred by a contraction of successive difference functions of the iterates using suitable norms.
2019-03-01T00:00:00ZIs chronic low back pain a risk factor for diabetes? The Nord-Trøndelag Health Study
http://hdl.handle.net/1956/19729
Is chronic low back pain a risk factor for diabetes? The Nord-Trøndelag Health Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Sørgjerd, Elin Pettersen; Åsvold, Bjørn Olav; Zwart, John-Anker
Journal article
<p>Objective: The purpose of this study was to examine the risk of diabetes associated with the presence or absence of chronic low back pain, considering both cross-sectional and cohort data.</p>
<p>Research design and methods: Analyses were based on the Norwegian HUNT2 and HUNT3 surveys of Nord-Trøndelag County. The prevalence of diabetes was compared in groups with and without chronic low back pain among 45 157 participants aged 30–69 years. Associations between low back pain at baseline and risk of diabetes were examined in an 11-year follow-up of 30 380 individuals with no baseline diagnosis of diabetes. The comorbidity between diabetes and low back pain was assessed at the end of follow-up. All analyses were carried out considering generalized linear models incorporating adjustment for other relevant risk factors.</p>
<p>Results: Cross-sectional analyses did not reveal any association between low back pain and diabetes. With adjustment for age, body mass index, physical activity and smoking, the cohort study of women showed a significant association between low back pain at baseline and risk of diabetes (RR 1.30; 95%  CI 1.09 to 1.54, p=0.003). The association differed between age groups (p=0.015), with a stronger association in relatively young women. In men, no association was found in the whole age range (RR 1.02; 95%  CI 0.86 to 1.21, p=0.82). No association was observed between diabetes and chronic low back pain at the end of follow-up.</p>
<p>Conclusion: Among younger women, those with chronic low back pain may have an increased risk of diabetes.</p>
2018-01-01T00:00:00ZSimplified energy landscape for modularity using total variation
http://hdl.handle.net/1956/19714
Simplified energy landscape for modularity using total variation
Boyd, Zachary; Bae, Egil; Tai, Xue-Cheng; Bertozzi, Andrea L.
Journal article
Networks capture pairwise interactions between entities and are frequently used in applications such as social networks, food networks, and protein interaction networks, to name a few. Communities, cohesive groups of nodes, often form in these applications, and identifying them gives insight into the overall organization of the network. One common quality function used to identify community structure is modularity. In Hu et al. [SIAM J. Appl. Math., 73 (2013), pp. 2224--2246], it was shown that modularity optimization is equivalent to minimizing a particular nonconvex total variation (TV) based functional over a discrete domain. They solve this problem---assuming the number of communities is known---using a Merriman--Bence--Osher (MBO) scheme. We show that modularity optimization is equivalent to minimizing a convex TV-based functional over a discrete domain---again, assuming the number of communities is known. Furthermore, we show that modularity has no convex relaxation satisfying certain natural conditions. We therefore find a manageable nonconvex approximation using a Ginzburg--Landau functional, which provably converges to the correct energy in the limit of a certain parameter. We then derive an MBO algorithm that has fewer hand-tuned parameters than in Hu et al. and that is seven times faster at solving the associated diffusion equation due to the fact that the underlying discretization is unconditionally stable. Our numerical tests include a hyperspectral video whose associated graph has $2.9\times10^7$ edges, which is roughly 37 times larger than what was handled in the paper of Hu et al.
2018-01-01T00:00:00ZUser's guide to Express: A tool for building knowledge-based systems for statistical data analysis
http://hdl.handle.net/1956/19587
User's guide to Express: A tool for building knowledge-based systems for statistical data analysis
Aarseth, Jan H.; Heuch, Ivar
Research report
1996-07-01T00:00:00ZSegmentation of data traces with applications to dipmeter oil well measurements
http://hdl.handle.net/1956/19586
Segmentation of data traces with applications to dipmeter oil well measurements
Karlsen, Hans; Tjøstheim, Dag
Research report
1988-02-01T00:00:00ZMultivariate compound Poisson distributions and infinite divisibility
http://hdl.handle.net/1956/19583
Multivariate compound Poisson distributions and infinite divisibility
Sundt, Bjørn
Research report
1999-03-01T00:00:00ZRecursions for Convolutions of discrete uniform Distributions revisited
http://hdl.handle.net/1956/19585
Recursions for Convolutions of discrete uniform Distributions revisited
Sundt, Bjørn
Research report
1997-09-01T00:00:00ZOn multivariate Vernic recursions
http://hdl.handle.net/1956/19584
On multivariate Vernic recursions
Sundt, Bjørn
Research report
1999-07-01T00:00:00ZLogistrule: A knowledge-based system for logistic regression
http://hdl.handle.net/1956/19581
Logistrule: A knowledge-based system for logistic regression
Aarseth, Jan H.; Heuch, Ivar
Research report
1996-07-01T00:00:00ZMultiple bilinear time series models
http://hdl.handle.net/1956/19582
Multiple bilinear time series models
Stensholt, Boonchai K.; Tjøstheim, Dag
Research report
1985-06-01T00:00:00ZInterfaces in a knowledge-based statistical system, as exemplified by Express
http://hdl.handle.net/1956/19580
Interfaces in a knowledge-based statistical system, as exemplified by Express
Aarseth, Jan H.; Heuch, Ivar
Research report
1996-07-01T00:00:00ZIntegral conditions for Skorohod stochastic differential equations
http://hdl.handle.net/1956/19579
Integral conditions for Skorohod stochastic differential equations
Gjessing, Håkon K.
Research report
1994-06-01T00:00:00ZAssessing uncertainty in knowledge-based systems for data analysis by simulation
http://hdl.handle.net/1956/19577
Assessing uncertainty in knowledge-based systems for data analysis by simulation
Aarseth, Jan H.; Heuch, Ivar
Research report
1996-08-01T00:00:00ZEstimation of AR parameters in time series with suddenly changing structure
http://hdl.handle.net/1956/19578
Estimation of AR parameters in time series with suddenly changing structure
Johnsen, Gyrid; Tjøstheim, Dag
Research report
1988-07-01T00:00:00ZSome Computing Aspects of Projective Geometry I. Basic functions, algorithms and procedures.
http://hdl.handle.net/1956/19575
Some Computing Aspects of Projective Geometry I. Basic functions, algorithms and procedures.
Holme, Audun
Research report
1980-09-01T00:00:00ZSub-bases of pleasant h-bases
http://hdl.handle.net/1956/19576
Sub-bases of pleasant h-bases
Selmer, Ernst S.
Research report
1987-02-01T00:00:00ZPolynomials homologically supported on determinantal loci
http://hdl.handle.net/1956/19574
Polynomials homologically supported on determinantal loci
Pragacz, P.; Ratajski, J.
Research report
1991-10-01T00:00:00ZOn the postage stamp problem with three stamp denominations, III
http://hdl.handle.net/1956/19573
On the postage stamp problem with three stamp denominations, III
Selmer, Ernst S.
Research report
1984-01-01T00:00:00ZOn regular Frobenius bases
http://hdl.handle.net/1956/19571
On regular Frobenius bases
Selmer, Ernst S.
Research report
1987-03-01T00:00:00ZOn Shellsort and the Frobenius problem
http://hdl.handle.net/1956/19572
On Shellsort and the Frobenius problem
Selmer, Ernst S.
Research report
1987-02-01T00:00:00ZOn Extremal Bases for the h-range Problem, II
http://hdl.handle.net/1956/19570
On Extremal Bases for the h-range Problem, II
Kirfel, Christoph
Research report
1990-02-01T00:00:00ZOn Extremal Bases for the h-range Problem, I
http://hdl.handle.net/1956/19569
On Extremal Bases for the h-range Problem, I
Kirfel, Christoph
Research report
1989-11-01T00:00:00ZOn cubic factors of certain trinomials
http://hdl.handle.net/1956/19568
On cubic factors of certain trinomials
Tverberg, Helge
Research report
1982-10-01T00:00:00ZDie Struktur der Regulären Darstellungen
http://hdl.handle.net/1956/19565
Die Struktur der Regulären Darstellungen
Kirfel, Christoph
Research report
1987-05-01T00:00:00ZGeometric Transversals for Families of disjoint Translates in the Plane
http://hdl.handle.net/1956/19567
Geometric Transversals for Families of disjoint Translates in the Plane
Holmsen, Andreas
Research report
2000-04-01T00:00:00ZDivided differences and ideals generated by symmetric polynomials
http://hdl.handle.net/1956/19566
Divided differences and ideals generated by symmetric polynomials
Lascoux, A.; Pragacz, P.
Research report
1991-04-01T00:00:00ZA numerical study of algebraic space curves.
http://hdl.handle.net/1956/19564
A numerical study of algebraic space curves.
Holme, Audun
Research report
1981-12-01T00:00:00ZVilhelm Bjerknes og Bergensskolen. Forelesning holdt på Matematisk institutt, Universitetet i Bergen 22.04.1998
http://hdl.handle.net/1956/19563
Vilhelm Bjerknes og Bergensskolen. Forelesning holdt på Matematisk institutt, Universitetet i Bergen 22.04.1998
Utaaker, Kåre
Research report
1998-10-01T00:00:00ZUsing a σ-coordinate numerical ocean model for simulating the circulation at Ormen Lange
http://hdl.handle.net/1956/19562
Using a σ-coordinate numerical ocean model for simulating the circulation at Ormen Lange
Eliassen, Inge K.; Berntsen, Jarle
Research report
2000-01-01T00:00:00ZUsers guide for a modesplit σ-coordinate numerical Ocean model : Version 1.0
http://hdl.handle.net/1956/19561
Users guide for a modesplit σ-coordinate numerical Ocean model : Version 1.0
Berntsen, Jarle
Research report
2000-01-01T00:00:00ZUnconditionally stable Methods for Hamilton-Jacobi Equations
http://hdl.handle.net/1956/19560
Unconditionally stable Methods for Hamilton-Jacobi Equations
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-05-01T00:00:00ZTrapping of dilute ion components in wells and double wells in higher equatorial magnetic regions: A kinetic theory including collisions, varying background and additional fields
http://hdl.handle.net/1956/19559
Trapping of dilute ion components in wells and double wells in higher equatorial magnetic regions: A kinetic theory including collisions, varying background and additional fields
Øien, Alf H.
Research report
2001-08-01T00:00:00ZTheoretical Investigation of Heat and Streaming generated by high Intensity Ultrasound
http://hdl.handle.net/1956/19557
Theoretical Investigation of Heat and Streaming generated by high Intensity Ultrasound
Tjøtta, Sigve
Research report
1998-07-01T00:00:00ZTime-of-Flight Marching + Transport Collapse: An Alternative to Streamlines for Two-Phase Media Flow with Capillary Forces.
http://hdl.handle.net/1956/19558
Time-of-Flight Marching + Transport Collapse: An Alternative to Streamlines for Two-Phase Media Flow with Capillary Forces.
Berre, Inga; Dahle, Helge K.; Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas; Natvig, Jostein Roald
Research report
2002-01-01T00:00:00ZThe effects of trapped and untrapped particles on an electrostatic wave packet
http://hdl.handle.net/1956/19555
The effects of trapped and untrapped particles on an electrostatic wave packet
Espedal, Magne S.
Research report
1973-12-01T00:00:00ZThe propagation of discontinuities for linear hyperbolic partial differential equations
http://hdl.handle.net/1956/19556
The propagation of discontinuities for linear hyperbolic partial differential equations
Eckhoff, Knut S.
Research report
1974-08-01T00:00:00ZThe Corrected Operator Splitting Approach Applied to a Nonlinear Advection-Diffusion Problem
http://hdl.handle.net/1956/19554
The Corrected Operator Splitting Approach Applied to a Nonlinear Advection-Diffusion Problem
Karlsen, Kenneth Hvistendahl; Brusdal, K.; Dahle, Helge K.; Evje, Steinar; Lie, Knut-Andreas
Research report
1997-01-01T00:00:00ZSur le transport de masse produit par des oscillations en milieu compressible, dissipatif et inhomogène
http://hdl.handle.net/1956/19553
Sur le transport de masse produit par des oscillations en milieu compressible, dissipatif et inhomogène
Tjøtta, Jacqueline Naze; Tjøtta, Sigve
Research report
1972-12-01T00:00:00ZStellarator - tokamak configurations
http://hdl.handle.net/1956/19552
Stellarator - tokamak configurations
Berge, Gerhard; Freidberg, Jeffrey P.
Research report
1990-01-01T00:00:00ZSolution of a stationary Fokker-Planck equation.
http://hdl.handle.net/1956/19550
Solution of a stationary Fokker-Planck equation.
Leversen, Tore; Tjøtta, Jacqueline Naze
Research report
1971-06-01T00:00:00ZSpace decompostition Methods For nonsymmetric parabolic Problems
http://hdl.handle.net/1956/19551
Space decompostition Methods For nonsymmetric parabolic Problems
Rognes, Øystein; Tai, Xue-Cheng
Research report
1999-11-01T00:00:00ZRate of Convergence of a Space Decomposition method and Applications to Linear and Nonlinear Elliptic Problems
http://hdl.handle.net/1956/19548
Rate of Convergence of a Space Decomposition method and Applications to Linear and Nonlinear Elliptic Problems
Espedal, Magne S.; Tai, Xue-Cheng
Research report
1996-09-01T00:00:00ZSolution of a non-strictly hyperbolic system modelling non-isothermal two-phase flow in a porous medium
http://hdl.handle.net/1956/19549
Solution of a non-strictly hyperbolic system modelling non-isothermal two-phase flow in a porous medium
Barkve, Tor
Research report
1987-04-01T00:00:00ZRate of Convergence for Subspace Correction Methods for nonlinear Variational Inequalities
http://hdl.handle.net/1956/19547
Rate of Convergence for Subspace Correction Methods for nonlinear Variational Inequalities
Tai, Xue-Cheng
Research report
1999-09-01T00:00:00ZRate of Convergence for some constraint Dcomposition methods for nonlinear variational inequalities.
http://hdl.handle.net/1956/19546
Rate of Convergence for some constraint Dcomposition methods for nonlinear variational inequalities.
Tai, Xue-Cheng
Research report
2000-10-01T00:00:00ZPortfolio Optimization in a Lévy Market with Intertemporal Substitution and Transaction Costs
http://hdl.handle.net/1956/19544
Portfolio Optimization in a Lévy Market with Intertemporal Substitution and Transaction Costs
Benth, Fred Espen; Karlsen, Kenneth Hvistendahl; Reikvam, Kristin
Research report
2000-05-01T00:00:00ZProgagation of Shallow Water Inertia-Gravity Waves with Fractional Step Methods
http://hdl.handle.net/1956/19545
Progagation of Shallow Water Inertia-Gravity Waves with Fractional Step Methods
Berntsen, Jarle; Espelid, Terje 0.
Research report
1995-07-01T00:00:00ZPerturbation about neutral solutions occurring in shear flows in stratified, incompressible and inviscid fluids.
http://hdl.handle.net/1956/19543
Perturbation about neutral solutions occurring in shear flows in stratified, incompressible and inviscid fluids.
Engevik, Leif Eirik
Research report
1973-06-01T00:00:00ZParameter estimation with the augmented Lagrangian method for a parabolic equation.
http://hdl.handle.net/1956/19541
Parameter estimation with the augmented Lagrangian method for a parabolic equation.
Nilssen, Trygve K.; Tai, Xue-Cheng
Research report
2001-01-01T00:00:00ZPermeability estimation with the Augmented Lagrangian method for a nonlinear diffusion equation.
http://hdl.handle.net/1956/19542
Permeability estimation with the Augmented Lagrangian method for a nonlinear diffusion equation.
Nilssen, Trygve K.; Mannseth, Trond; Tai, Xue-Cheng
Research report
2001-01-01T00:00:00ZParallel Function Decomposition Methods and Numerical Applications
http://hdl.handle.net/1956/19540
Parallel Function Decomposition Methods and Numerical Applications
Tai, Xue-Cheng
Research report
1997-09-01T00:00:00ZParallel function decomposition and space decomposition methods - part II. Applications to splitting and domain decomposition
http://hdl.handle.net/1956/19539
Parallel function decomposition and space decomposition methods - part II. Applications to splitting and domain decomposition
Tai, Xue-Cheng
Research report
1994-04-01T00:00:00ZOperator splitting Methods for degenerate Convection-diffusion Equations II: Numerical Examples with Emphasis on Reservoir Simulation and Sedimentation
http://hdl.handle.net/1956/19536
Operator splitting Methods for degenerate Convection-diffusion Equations II: Numerical Examples with Emphasis on Reservoir Simulation and Sedimentation
Holden, Helge; Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas
Research report
1999-12-01T00:00:00ZOptimal Portfolio management Rules in a Non-Gaussian Market with Durability and intertemporal Substitution
http://hdl.handle.net/1956/19537
Optimal Portfolio management Rules in a Non-Gaussian Market with Durability and intertemporal Substitution
Benth, Fred Espen; Karlsen, Kenneth Hvistendahl; Reikvam, Kristin
Research report
2000-05-01T00:00:00ZOverlapping domain Decomposition and multigrid Methods of inverse Problems
http://hdl.handle.net/1956/19538
Overlapping domain Decomposition and multigrid Methods of inverse Problems
Tai, Xue-Cheng; Frøyen, Johnny; Espedal, Magne S.; Chan, Tony F.
Research report
1997-11-01T00:00:00ZOn the Uniqueness and Stability of Entropy Solutions of nonlinear degenerate parabolic Equations with rough Coefficients
http://hdl.handle.net/1956/19533
On the Uniqueness and Stability of Entropy Solutions of nonlinear degenerate parabolic Equations with rough Coefficients
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-05-01T00:00:00ZOn the use of the Richtmyer procedure to compute a finite amplitude sound beam from a piston source
http://hdl.handle.net/1956/19534
On the use of the Richtmyer procedure to compute a finite amplitude sound beam from a piston source
Berntsen, Jarle
Research report
1987-04-03T00:00:00ZOperator Splitting Methods for Degenerate Convection-Diffusion Equations I: Convergence and Entropy Estimates
http://hdl.handle.net/1956/19535
Operator Splitting Methods for Degenerate Convection-Diffusion Equations I: Convergence and Entropy Estimates
Holden, Helge; Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas
Research report
1999-10-01T00:00:00ZOn the uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic equations.
http://hdl.handle.net/1956/19532
On the uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic equations.
Holden, Helge; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2002-04-01T00:00:00ZOn the stability of plane inviscid Couette flow
http://hdl.handle.net/1956/19531
On the stability of plane inviscid Couette flow
Engevik, Leif Eirik
Research report
1966-11-01T00:00:00ZOn the Stability of a Magnetized Plasma with a Continuous Density Gradient in a Uniform External Force Field
http://hdl.handle.net/1956/19530
On the Stability of a Magnetized Plasma with a Continuous Density Gradient in a Uniform External Force Field
Berge, Gerhard
Research report
1964-06-01T00:00:00ZOn the Solution of the Solitary Wave Problem
http://hdl.handle.net/1956/19529
On the Solution of the Solitary Wave Problem
Engevik, Leif Eirik
Research report
1997-07-01T00:00:00ZOn the numerical approximation of derivatives by a modified Fourier collocation method
http://hdl.handle.net/1956/19528
On the numerical approximation of derivatives by a modified Fourier collocation method
Eckhoff, Knut S.; Wasberg, Carl Erik
Research report
1995-07-01T00:00:00ZOn the mass transport induced by time-dependent oscillations of finite amplitude in a nonhomogeneous fluid. II: General results for a liquid
http://hdl.handle.net/1956/19527
On the mass transport induced by time-dependent oscillations of finite amplitude in a nonhomogeneous fluid. II: General results for a liquid
Tjøtta, Jacqueline Naze; Tjøtta, Sigve
Research report
1973-08-01T00:00:00ZOn the mass transport induced by time-dependent oscillations of finite amplitude in a nonhomogeneous fluid. I: General results for a perfect gas
http://hdl.handle.net/1956/19526
On the mass transport induced by time-dependent oscillations of finite amplitude in a nonhomogeneous fluid. I: General results for a perfect gas
Tjøtta, Jacqueline Naze; Tjøtta, Sigve
Research report
1973-05-01T00:00:00ZOn the existence of optimal controls for a singular stochastic control problem in finance.
http://hdl.handle.net/1956/19525
On the existence of optimal controls for a singular stochastic control problem in finance.
Benth, Fred Espen; Karlsen, Kenneth Hvistendahl; Reikvam, Kristin
Research report
2000-11-01T00:00:00ZOn the convergence Rate of Operator splitting for weakly coupled Systems of Hamilton-Jacobi Equations.
http://hdl.handle.net/1956/19524
On the convergence Rate of Operator splitting for weakly coupled Systems of Hamilton-Jacobi Equations.
Jakobsen, Espen R.; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-06-01T00:00:00ZOn the Convergence Rate of Operator splitting for Hamilton-Jacobi Equations with Source Terms
http://hdl.handle.net/1956/19523
On the Convergence Rate of Operator splitting for Hamilton-Jacobi Equations with Source Terms
Jakobsen, Espen R.; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-02-01T00:00:00ZOn strongly degenerate convection-diffusion Problems modeling Sedimentation-Consolidation Processes
http://hdl.handle.net/1956/19522
On strongly degenerate convection-diffusion Problems modeling Sedimentation-Consolidation Processes
Bürger, Raimund; Evje, Steinar; Karlsen, Kenneth Hvistendahl
Research report
1999-08-01T00:00:00ZOn Acoustic Streaming in Magnetohydrodynamics
http://hdl.handle.net/1956/19519
On Acoustic Streaming in Magnetohydrodynamics
Kildal, A.; Tjøtta, Sigve
Research report
1964-02-01T00:00:00ZOn stability in ideal compressible hydrodynamics
http://hdl.handle.net/1956/19521
On stability in ideal compressible hydrodynamics
Eckhoff, Knut S.
Research report
1975-05-01T00:00:00ZOn Energy Conversion in a Sigma coordinate Oceaan Model
http://hdl.handle.net/1956/19520
On Energy Conversion in a Sigma coordinate Oceaan Model
Eldevik, Tor
Research report
1999-08-01T00:00:00ZOn a stability problem in hydrodynamics. Part I
http://hdl.handle.net/1956/19518
On a stability problem in hydrodynamics. Part I
Engevik, Leif
Research report
1966-11-01T00:00:00ZOn a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of pressure filtration.
http://hdl.handle.net/1956/19516
On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of pressure filtration.
Bürger, Raimund; Frid, H.; Karlsen, Kenneth Hvistendahl
Research report
2002-01-01T00:00:00ZOn a Model for Continous Sedimentation in Vessels with Discontinously varying Cross-Sectional Area.
http://hdl.handle.net/1956/19517
On a Model for Continous Sedimentation in Vessels with Discontinously varying Cross-Sectional Area.
Bürger, Raimund; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik; Towers, J. D.
Research report
2002-04-01T00:00:00ZNumerical studies of the current response at Ormen Lange to a travelling storm.
http://hdl.handle.net/1956/19515
Numerical studies of the current response at Ormen Lange to a travelling storm.
Vikebø, Frode; Berntsen, Jarle; Furnes, Gunnar
Research report
2001-12-01T00:00:00ZNumerical Solution of the Polymer System by Front Tracking
http://hdl.handle.net/1956/19514
Numerical Solution of the Polymer System by Front Tracking
Haugse, V.; Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas; Natvig, Jostein Roald
Research report
1999-05-01T00:00:00ZNumerical Solution of the Buckley-Leverett Equation with a General Fractional Flow Function
http://hdl.handle.net/1956/19513
Numerical Solution of the Buckley-Leverett Equation with a General Fractional Flow Function
Pettersen, Øystein
Research report
1984-11-01T00:00:00ZNumerical Solution of Reservoir Flow Models based on large Time Step Operator splitting Algorithms
http://hdl.handle.net/1956/19512
Numerical Solution of Reservoir Flow Models based on large Time Step Operator splitting Algorithms
Espedal, Magne S.; Karlsen, Kenneth Hvistendahl
Research report
1999-01-01T00:00:00ZNumerical Methods for Flow in a Porous Media with Internal Boundaries
http://hdl.handle.net/1956/19510
Numerical Methods for Flow in a Porous Media with Internal Boundaries
Ersland, Brit Gunn; Espedal, Magne S.
Research report
1997-02-01T00:00:00ZNumerical methods for the simulation of continous sedimentation in ideal clarifier-thickener units.
http://hdl.handle.net/1956/19511
Numerical methods for the simulation of continous sedimentation in ideal clarifier-thickener units.
Bürger, Raimund; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik; Towers, J. D.
Research report
2001-10-01T00:00:00ZNonlinear propagation and interaction of collinear sound beams
http://hdl.handle.net/1956/19509
Nonlinear propagation and interaction of collinear sound beams
Vefring, Erlend H.
Research report
1989-10-01T00:00:00ZNonlinear equations of acoustics in inhomogeneous, thermoviscous fluids
http://hdl.handle.net/1956/19508
Nonlinear equations of acoustics in inhomogeneous, thermoviscous fluids
Reiso, Edel
Research report
1991-03-01T00:00:00ZNonlinear Effects During Transient Fluid Flow in Reservoirs as Encountered in Well-Test Analysis
http://hdl.handle.net/1956/19507
Nonlinear Effects During Transient Fluid Flow in Reservoirs as Encountered in Well-Test Analysis
Aanonsen, Sigurd Ivar
Research report
1985-03-01T00:00:00ZMass transport induced by wave motion in a rotating fluid
http://hdl.handle.net/1956/19505
Mass transport induced by wave motion in a rotating fluid
Mæland, Einar
Research report
1974-10-01T00:00:00ZMonotone Difference Approximation of BV Solutions to Degenerate Convection-Diffusion Equations
http://hdl.handle.net/1956/19506
Monotone Difference Approximation of BV Solutions to Degenerate Convection-Diffusion Equations
Evje, Steinar; Karlsen, Kenneth Hvistendahl
Research report
1998-03-01T00:00:00ZLinear waves and stability in ideal magnetohydrodynamics
http://hdl.handle.net/1956/19504
Linear waves and stability in ideal magnetohydrodynamics
Eckhoff, Knut S.
Research report
1987-05-01T00:00:00ZLinear and weakly nonlinear propagation of a pulsed sound beam
http://hdl.handle.net/1956/19503
Linear and weakly nonlinear propagation of a pulsed sound beam
Frøysa, Kjell-Eivind
Research report
1991-06-01T00:00:00ZKinetic Theory for Evolution of a Plasma in External Electromagnetic Fields toward a State characterized by Balance of Forces Transverse to the Magnetic Field
http://hdl.handle.net/1956/19502
Kinetic Theory for Evolution of a Plasma in External Electromagnetic Fields toward a State characterized by Balance of Forces Transverse to the Magnetic Field
Øien, Alf H.
Research report
1973-04-01T00:00:00ZIdentification of a nonlinear parameter in a parabolic equation by a linear approach
http://hdl.handle.net/1956/19500
Identification of a nonlinear parameter in a parabolic equation by a linear approach
Tai, Xue-Cheng; Kärkkäinen, Tommi
Research report
1993-03-01T00:00:00ZKinetic Equation for an Electron Gas (Non-Neutral Plasma) in Strong Fields and Inhomogenities
http://hdl.handle.net/1956/19501
Kinetic Equation for an Electron Gas (Non-Neutral Plasma) in Strong Fields and Inhomogenities
Øien, Alf H.
Research report
1978-06-01T00:00:00ZGlobal Convergence of Subspace Correction Methods for Convex Optimization Problems
http://hdl.handle.net/1956/19498
Global Convergence of Subspace Correction Methods for Convex Optimization Problems
Tai, Xue-Cheng; Xu, Jinchao
Research report
1998-03-01T00:00:00ZGravity currents at Ormen Lange
http://hdl.handle.net/1956/19499
Gravity currents at Ormen Lange
Vikebø, Frode; Berntsen, Jarle; Furnes, Gunnar
Research report
2001-12-01T00:00:00ZFresh Water driven primary Production in a Fjord
http://hdl.handle.net/1956/19497
Fresh Water driven primary Production in a Fjord
Berntsen, Jarle; Aksnes, Dag L.; Foldvik, Arne
Research report
1997-03-01T00:00:00ZELLAM-based Operator Splitting for Nonlinear Advection-Diffusion Equations
http://hdl.handle.net/1956/19496
ELLAM-based Operator Splitting for Nonlinear Advection-Diffusion Equations
Dahle, Helge K.
Research report
1995-06-01T00:00:00ZDiscrete Approximations of BV solutions to Doubly Nonlinear Degenerate Parabolic Equations
http://hdl.handle.net/1956/19494
Discrete Approximations of BV solutions to Doubly Nonlinear Degenerate Parabolic Equations
Evje, Steinar; Karlsen, Kenneth Hvistendahl
Research report
1998-05-01T00:00:00ZEffects of bottom drag and shelf slope steepnes on the currents along the shelf slope at Ormen Lange
http://hdl.handle.net/1956/19495
Effects of bottom drag and shelf slope steepnes on the currents along the shelf slope at Ormen Lange
Sørflaten, Agathe; Berntsen, Jarle; Alendal, Guttorm
Research report
2001-12-01T00:00:00ZCorrections to Classical Kinetic and Transport Theory for a Two-Temperature, Fully lonized Plasma in Electromagnetic Fields
http://hdl.handle.net/1956/19492
Corrections to Classical Kinetic and Transport Theory for a Two-Temperature, Fully lonized Plasma in Electromagnetic Fields
Øien, Alf H.
Research report
1977-06-01T00:00:00ZDegenerate Convection-Diffusion Equations and Implicit Monotone Difference Schemes
http://hdl.handle.net/1956/19493
Degenerate Convection-Diffusion Equations and Implicit Monotone Difference Schemes
Evje, Steinar; Karlsen, Kenneth Hvistendahl
Research report
1998-03-01T00:00:00ZConvergence Rate Analysis of an Asynchronous Space Decomposition Method for Convex Minimization
http://hdl.handle.net/1956/19489
Convergence Rate Analysis of an Asynchronous Space Decomposition Method for Convex Minimization
Tai, Xue-Cheng; Tseng, Paul
Research report
1998-08-01T00:00:00ZConvergence Rate Analysis of Domain Decomposition Methods for Obstacle Problems
http://hdl.handle.net/1956/19490
Convergence Rate Analysis of Domain Decomposition Methods for Obstacle Problems
Tai, Xue-Cheng
Research report
1999-01-01T00:00:00ZCorrected Operator Splitting for Nonlinear Parabollic Equations
http://hdl.handle.net/1956/19491
Corrected Operator Splitting for Nonlinear Parabollic Equations
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
1997-04-01T00:00:00ZConvergence of finite differences schemes for viscous and inviscid conservation laws with rough coefficients.
http://hdl.handle.net/1956/19488
Convergence of finite differences schemes for viscous and inviscid conservation laws with rough coefficients.
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-09-01T00:00:00ZContinuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations.
http://hdl.handle.net/1956/19487
Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations.
Jakobsen, Espen R.; Karlsen, Kenneth Hvistendahl
Research report
2001-08-01T00:00:00ZCombined effects of thermohaline and atmosphere forcing on the currents at Ormen Lange
http://hdl.handle.net/1956/19486
Combined effects of thermohaline and atmosphere forcing on the currents at Ormen Lange
Thiem, Øyvind; Eldevik, Tor; Berntsen, Jarle
Research report
2001-12-01T00:00:00ZCarl Ludvig Schreiner Godske (1906 - 1970). Forelesning holdt ved Matematisk institutt, Universitetet i Bergen 3.12.1997
http://hdl.handle.net/1956/19485
Carl Ludvig Schreiner Godske (1906 - 1970). Forelesning holdt ved Matematisk institutt, Universitetet i Bergen 3.12.1997
Utaaker, Kåre
Research report
1998-04-01T00:00:00ZAnalysis and Numerics of strongly degenerate Convection-diffusion Problems Modelling sedimentation-consolidation Processes
http://hdl.handle.net/1956/19483
Analysis and Numerics of strongly degenerate Convection-diffusion Problems Modelling sedimentation-consolidation Processes
Bürger, Raimund; Karlsen, Kenneth Hvistendahl
Research report
1999-09-01T00:00:00ZApplication of the Galerkin’s method on the problem of cellular convection induced by surface tension gradients.
http://hdl.handle.net/1956/19484
Application of the Galerkin’s method on the problem of cellular convection induced by surface tension gradients.
Mæland, Einar
Research report
1971-11-01T00:00:00ZAdaptive Characteristic Operator Splitting Techniques for Convection-Dominated Diffusion Problems in One and Two Space Dimensions
http://hdl.handle.net/1956/19480
Adaptive Characteristic Operator Splitting Techniques for Convection-Dominated Diffusion Problems in One and Two Space Dimensions
Dahle, Helge K.
Research report
1988-12-01T00:00:00ZAn Operator Splitting Method for Convection-Diffusion Equations
http://hdl.handle.net/1956/19482
An Operator Splitting Method for Convection-Diffusion Equations
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
1996-01-01T00:00:00ZAn algorithm for internal morging of two subsets with small extra storage requirements.
http://hdl.handle.net/1956/19481
An algorithm for internal morging of two subsets with small extra storage requirements.
Espelid, Terje 0.
Research report
1974-09-01T00:00:00ZA two-way nesting procedure for an ocean model with application to the Norwegian Sea.
http://hdl.handle.net/1956/19479
A two-way nesting procedure for an ocean model with application to the Norwegian Sea.
Heggelund, Yngve; Berntsen, Jarle
Research report
2000-11-01T00:00:00ZA total pressure-saturation formulation of two-phase flow incorporating dynamic effects in the capillary-pressure-saturation relationship.
http://hdl.handle.net/1956/19478
A total pressure-saturation formulation of two-phase flow incorporating dynamic effects in the capillary-pressure-saturation relationship.
Dahle, Helge K.; Celia, Michael A.; Hassanizadeh, S. Majid; Karlsen, Kenneth Hvistendahl
Research report
2002-01-01T00:00:00ZA Study of the Verigin Problem with Application to Analysis of Water Injection Tests
http://hdl.handle.net/1956/19477
A Study of the Verigin Problem with Application to Analysis of Water Injection Tests
Barkve, Tor
Research report
1985-03-01T00:00:00ZA Study of the Modelling Error in two Operator Splitting Algorithms for porous Media Flow
http://hdl.handle.net/1956/19476
A Study of the Modelling Error in two Operator Splitting Algorithms for porous Media Flow
Brusdal, Kari; Dahle, Helge K.; Karlsen, Kenneth Hvistendahl; Mannseth, Trond
Research report
1997-10-01T00:00:00ZA strongly degenerate convectio-diffusion Problem modeling centrifugation of flocculated Suspensions.
http://hdl.handle.net/1956/19475
A strongly degenerate convectio-diffusion Problem modeling centrifugation of flocculated Suspensions.
Bürger, Raimund; Karlsen, Kenneth Hvistendahl
Research report
2000-06-01T00:00:00ZA robust Nonconforming H²-Element
http://hdl.handle.net/1956/19472
A robust Nonconforming H²-Element
Nilssen, Trygve K.; Tai, Xue-Cheng; Winther, Ragnar
Research report
1999-03-01T00:00:00ZA Space Decomposition Method for Parabolic Equations
http://hdl.handle.net/1956/19473
A Space Decomposition Method for Parabolic Equations
Tai, Xue-Cheng
Research report
1997-02-01T00:00:00ZA streamline front tracking method for two- and three-phase flow including capillary forces.
http://hdl.handle.net/1956/19474
A streamline front tracking method for two- and three-phase flow including capillary forces.
Berre, Inga; Dahle, Helge K.; Karlsen, Kenneth Hvistendahl; Nordhaug, Hans F.
Research report
2001-10-01T00:00:00ZA robust finite element method for Darcy-Stokes flow.
http://hdl.handle.net/1956/19471
A robust finite element method for Darcy-Stokes flow.
Mardal, Kent André; Tai, Xue-Cheng; Winther, Ragnar
Research report
2001-01-01T00:00:00ZA random exchange model with constant decrements.
http://hdl.handle.net/1956/19470
A random exchange model with constant decrements.
Helland, Inge S.
Research report
1974-12-01T00:00:00ZA Note on the Wave Field in the Surface Zone
http://hdl.handle.net/1956/19469
A Note on the Wave Field in the Surface Zone
Engevik, Leif Eirik
Research report
1991-07-01T00:00:00ZA Note on the Stability of Steady Inviscid Helical Gas Flows
http://hdl.handle.net/1956/19468
A Note on the Stability of Steady Inviscid Helical Gas Flows
Storesletten, Leiv
Research report
1976-01-01T00:00:00ZA Note on Portfolio Management under non-Gaussian Logreturns
http://hdl.handle.net/1956/19466
A Note on Portfolio Management under non-Gaussian Logreturns
Benth, Fred Espen; Karlsen, Kenneth Hvistendahl; Reikvam, Kristin
Research report
2000-02-01T00:00:00ZA note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws.
http://hdl.handle.net/1956/19465
A note on Front tracking and the Equivalence between Viscosity Solutions of Hamilton-Jacobi Equations And Entropy Solutions of scalar Conservation Laws.
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-01-01T00:00:00ZA Note on Portfolio oprimization in a levy Market with local Substitution and habit formation
http://hdl.handle.net/1956/19467
A Note on Portfolio oprimization in a levy Market with local Substitution and habit formation
Benth, Fred Espen; Karlsen, Kenneth Hvistendahl; Reikvam, Kristin
Research report
2000-05-01T00:00:00ZA mathematical model for batch and continous thickening of flocculent suspensions in vessels with varying section.
http://hdl.handle.net/1956/19464
A mathematical model for batch and continous thickening of flocculent suspensions in vessels with varying section.
Bürger, Raimund; Damasceno, J. J. R; Karlsen, Kenneth Hvistendahl
Research report
2001-10-01T00:00:00ZA Hybrid Domain Decomposition Method for Advection-Diffusion Problems
http://hdl.handle.net/1956/19463
A Hybrid Domain Decomposition Method for Advection-Diffusion Problems
Espedal, Magne S.; Tai, Xue-Cheng
Research report
1996-09-01T00:00:00ZA Front Tracking Approach to a Two-Phase fluid Flow Model with Capillary Forces
http://hdl.handle.net/1956/19462
A Front Tracking Approach to a Two-Phase fluid Flow Model with Capillary Forces
Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas; Risebro, Nils Henrik; Frøyen, Johnny
Research report
1997-04-01T00:00:00ZA fast Level Set Method for Reservoir Simulation
http://hdl.handle.net/1956/19461
A fast Level Set Method for Reservoir Simulation
Karlsen, Kenneth Hvistendahl; Lie, Knut-Andreas; Risebro, Nils Henrik
Research report
1999-05-01T00:00:00ZA description of daphnicle dynamics based on kinetic theory: Attempts at analogue-modelling of swarming and behaviour of Daphnia
http://hdl.handle.net/1956/19460
A description of daphnicle dynamics based on kinetic theory: Attempts at analogue-modelling of swarming and behaviour of Daphnia
Øien, Alf H.
Research report
2001-06-01T00:00:00ZA note on the Wick product
http://hdl.handle.net/1956/19457
A note on the Wick product
Gjessing, Håkon K.
Research report
1993-03-01T00:00:00ZModeling In Vivo Interstitial Hydration-Pressure Relationships in Skin and Skeletal Muscle
http://hdl.handle.net/1956/19348
Modeling In Vivo Interstitial Hydration-Pressure Relationships in Skin and Skeletal Muscle
Øien, Alf Hartvig; Wiig, Helge
Journal article
A theoretical understanding of hydrostatic pressure-fluid volume relationships, or equations of state, of interstitial fluid in skin and skeletal muscle through mathematical/physical modeling is lacking. Here, we investigate at the microscopic level forces that seem to underlie and determine the movements of fluid and solid tissue elements on the microscopic as well as on the macroscopic level. Effects that occur during variation of hydration due to interaction between expanding glycosaminoglycans (GAGs) and the collagen interstitial matrix of tissue seem to be of major importance. We focus on these interactions that let effects from spherical GAGs expand and contract relative to collagen on the microscopic level as hydration changes and thereby generate a hydration-dependent electrostatic pressure on the extracellular matrix on the microscopic level. This pressure spreads to macroscopic levels and become a key factor for setting up equations of state for skin and skeletal muscle interstitia. The modeling for a combined skeletal muscle and skin tissue is one dimensional, i.e., a flat box that may mimic central transverse parts of tissue with more complex geometry. Incorporating values of GAG and collagen densities and fluid contents of skin and muscle tissues that are of an order of magnitude found in literature into the model gives interstitial hydrostatic pressure- fluid volume relationships for these tissues that agree well with experimental results.
2018-09-01T00:00:00ZWave breaking in the BBM-equation
http://hdl.handle.net/1956/19309
Wave breaking in the BBM-equation
Norevik, Anders Marifjæren
Master thesis
2019-04-02T00:00:00ZModelling memory decay after injuries using household survey data from Khartoum State, Sudan
http://hdl.handle.net/1956/19305
Modelling memory decay after injuries using household survey data from Khartoum State, Sudan
Heuch, Ivar; Abdalla, Safa; El Tayeb, Sally
Journal article
Background
Injuries represent an important cause of morbidity and mortality worldwide. In retrospective epidemiological studies, estimated rates of reported injuries often decline considerably when information is included from periods more than a few months before the data collection. Such low rates are usually regarded as a consequence of memory decay. It is largely unknown whether the extent of memory decay depends on external factors otherwise affecting injury rates.
Methods
A statistical model was introduced to separate the influence of external factors on true injury rates from effects on memory decay. The relationship between apparent rates and time elapsed between injury occurrence and data collection was described by a parametric regression model. Relationships between memory decay and external factors were modelled by effect modification of the relationship with time. The procedure was applied to data collected in a retrospective household survey, carried out in Khartoum State in 2010, which elicited information about injuries that had occurred during the last year. The survey included 5661 individuals in 973 households, reporting a total of 481 non-fatal injuries.
Results
In the data from Khartoum State, differences in memory recall were observed between socioeconomic groups, with considerably faster memory decay in the lower socioeconomic tertile. In this tertile the estimated probability that an injury which occurred 6 months ago was reported was only 18%, compared to probabilities of about 35% in the remainder of the population. In the lower socioeconomic tertile, in contrast to other groups, a simple exponential model was not sufficient for describing memory decay. Memory decay did not depend on sex, age, urban/rural status or education. Road traffic injuries were subject to less memory decay than injuries due to falls, mechanical causes and burns. Memory decay seriously affected crude overall injury rates and also to some degree estimated relative rates.
Conclusion
In the statistical analysis of retrospective injury data it is important to take into account the effects of memory decay.
2018-06-20T00:00:00ZWell-posedness Issues for Nonlinear Partial Differential Equations Appearing in the Modeling of Long Water Waves
http://hdl.handle.net/1956/19299
Well-posedness Issues for Nonlinear Partial Differential Equations Appearing in the Modeling of Long Water Waves
Teyekpiti, Vincent
Doctoral thesis
<p>Modelling of wave motion in a fluid is usually based on classical systems which are obtained by the hypotheses that the flow is irrotational and the bottom is even. In such a context, the influence of vorticity is entirely disregarded in the formulation of the governing equations. Although this consideration is justified in many circumstances, there are also a fair number of observed cases in near-shore hydrodynamics and open channel flow where this approach is unsuitable. In this thesis, the influence of constant background vorticity on the properties of shock waves in a shallow water system is considered and the governing equations are derived. An analysis of the shock-wave solutions of the system detailed in the body of this paper shows that stationary jumps can be described in terms of two non-dimensional parameters, one being the Froude number and the other incorporating the background vorticity. It is shown that these two parameters completely determine the strength of the jump. Moreover, in many practical situations, the assumption of a flat bottom is too restrictive. If this theory is to describe the physics of an underlying problem adequately, then it is important to introduced uneven bed in the formulation of the governing equations. This is done in this thesis where it is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally lead to singular solutions featuring Dirac delta distributions. These singular solutions feature a Rankine-Hugoniot deficit and the method of complex-valued weak asymptotic is used to provide a firm link between the Rankine-Hugoniot deficit and the singular parts of the weak solutions.</p>
<p>Furthermore, it is shown that a shallow water system for interfacial waves in the case of a neutrally buoyant two-layer fluid setup ceases to be strictly hyperbolic and the standard theory of hyperbolic conservation laws cannot be used to solve the Riemann problem. Nevertheless, it is shown that the Riemann problem can still be solved uniquely using singular shocks which contain Dirac delta distributions travelling with the shock. The solution is characterized in terms of the complex-valued weak asymptotic method and it is established that the two solution concepts coincide.</p>
<p>The thesis also made a significant contribution to the Brio system which is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set v = 0.</p>
<p>In the present contribution, the focus is on such an example, a hyperbolic conservation law appearing in ideal magnetohydrodynamics. For this conservation law, solutions cannot be found using the classical techniques of conservation laws. Consequently, certain Riemann problems have no weak solutions in the traditional Lax admissible sense. It was argued by some authors that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Although solutions of this type were exhibited, uniqueness was not obtained. In this thesis, a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws is introduced. In addition, a criterion which leads to an admissibility condition for singular solutions of the original system is developed and it is shown that such admissible solutions are unique in the framework developed in this thesis.</p>
2018-12-18T00:00:00ZExistence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics
http://hdl.handle.net/1956/19298
Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics
Kalisch, Henrik; Mitrovic, Darko; Teyekpiti, Vincent
Journal article
2018-10-30T00:00:00ZCubical and cosimplicial descent
http://hdl.handle.net/1956/19292
Cubical and cosimplicial descent
Dundas, Bjørn Ian; Rognes, John
Journal article
We prove that algebraic K ‐theory, topological Hochschild homology and topological cyclic homology satisfy cubical and cosimplicial descent at connective structured ring spectra along 1‐connected maps of such ring spectra.
2018-01-01T00:00:00ZDeformations of rational surface singularities and reflexive modules with an application to flops
http://hdl.handle.net/1956/19254
Deformations of rational surface singularities and reflexive modules with an application to flops
Gustavsen, Trond Stølen; Ile, Runar
Journal article
Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops.
2018-01-01T00:00:00ZEcological and evolutionary dynamics of interconnectedness and modularity
http://hdl.handle.net/1956/19252
Ecological and evolutionary dynamics of interconnectedness and modularity
Nordbotten, Jan Martin; Levin, Simon A.; Szathmáry, Eörs; Stenseth, Nils Christian
Journal article
In this contribution, we develop a theoretical framework for linking microprocesses (i.e., population dynamics and evolution through natural selection) with macrophenomena (such as interconnectedness and modularity within an ecological system). This is achieved by developing a measure of interconnectedness for population distributions defined on a trait space (generalizing the notion of modularity on graphs), in combination with an evolution equation for the population distribution. With this contribution, we provide a platform for understanding under what environmental, ecological, and evolutionary conditions ecosystems evolve toward being more or less modular. A major contribution of this work is that we are able to decompose the overall driver of changes at the macro level (such as interconnectedness) into three components: (i) ecologically driven change, (ii) evolutionarily driven change, and (iii) environmentally driven change.
2018-01-23T00:00:00ZA Simulation Study of Goodness-of-Fit Tests for Binary Regression with Applications to Norwegian Intensive Care Registry Data
http://hdl.handle.net/1956/19047
A Simulation Study of Goodness-of-Fit Tests for Binary Regression with Applications to Norwegian Intensive Care Registry Data
Nygaard, Ellisif
Master thesis
When using statistical methods to fit a model, the consensus is that it is possible to represent
a complex reality in the form of a simpler model. It is helpful to systematically measure a
model’s ability to capture the underlying system which controls the data generation in the
population being examined.
One of the possible tools we can apply to evaluate model adequacy is goodness-offit (GOF) tests. Summary GOF statistics are computed for a specific fitted model, then
attributed an asymptotic distribution, and finally the null hypothesis that the model fits the
data adequately is tested. A great challenge, when the model is a binary regression model
and it has one or several continuous covariates, is to verify which asymptotic distributions
the GOF statistics in fact have (Hosmer et al., 1997).
In this thesis, we will evaluate the validity of the distributions of some established GOF
test statistics mentioned in the literature. We have chosen so-called global GOF tests, where
user input is not necessary. Tests demanding user input, such as the Hosmer-Lemeshow test,
have been shown to have some considerable disadvantages. Hosmer et al. (1997) states that
number of groups (which are determined by user discretion) can influence whether the GOF
test rejects the model fit or not.
Binary regression models present a specific set of challenges with regards to GOF
measures, especially in situations where at least one covariate is continuous. There appears to
be no broad general agreement on which GOF statistics are reliable options when fitting such
models. This thesis aims to extend the current knowledge in this area. A modified version of
one of the statistics is introduced. The GOF tests studied are later applied in a data analysis
on real data set from the Norwegian Intensive Care Registry (NIR).
An exploration was performed in the attempt to suggest a suitable tool to evaluate the
discrepancies between the estimated logistic probabilities and the outcome variable, and how
different GOF tests will behave for different categories of discrepancies.
2019-01-31T00:00:00Z"Eg tenkte jo det hadde nåke med n å gjer, eg skjønte jo det". Bruk av Five Practices i arbeid med generalisering av figurmønster
http://hdl.handle.net/1956/18976
"Eg tenkte jo det hadde nåke med n å gjer, eg skjønte jo det". Bruk av Five Practices i arbeid med generalisering av figurmønster
Haugland, Lena Sjursen
Master thesis
I denne oppgåva har eg brukt ei alternativ undervisningsmetode kalla Five Practices i arbeidet med generalisering av figurmønster. Ved å gjennomføre to identiske testar, ein før og ein etter prosjektperioden, har eg sett på kva type løysingsstrategiar elevar brukar når dei løyser oppgåver med figurmønster. Eg har også sett på om det kjem fram endringar i bruk av løysingsstrategiar frå den fyrste til den andre testen. Vidare har eg prøvd å finne ut om undervisningsmetoden har hatt noko å seie for elevane sine val av løysingsstrategi. Totalt atten elevar frå to ulike klassar i programfaget R2 deltok på testane. På bakgrunn av resultata på testane, plukka eg ut ni elevar til intervju. Ved hjelp av konvensjonell og teoridreven analyse der datamaterialet vart koda og kategorisert, har eg delt løysingsstrategiane som kom fram i studien inn i to hovudkategoriar, rekursiv og eksplisitt. Under rekursiv kategori finn ein strategiane finne differansen, dobling med justering og addisjonsmetoden, der addisjonsmetoden er henta frå Mason (1996). Finne differansen handlar om at eleven finn ein differanse mellom kvar figur i mønsteret. Dobling med justering går ut på å doble talet på perlar i ein figur og justere ved å trekke frå dei perlene ein får for mykje i forhold til neste figur. Addisjonsmetoden går ut på å bruke føregåande ledd for å finne eit uttrykk for det neste leddet i mønsteret. Under eksplisitt kategori har eg plassert strategiane kvadratmetoden, rektangelmetoden og fullt kvadrat med justering. Alle desse tre strategiane handlar om å sjå ein struktur i mønsteret basert på geometriske figurer som dei kjenner frå før. Studien min viser at generalisering av figurmønster er vanskeleg for fleire elevar. Halvparten av elevane som er med i studien klarte ikkje å generere eit formelt uttrykk for mønsteret i oppgåva på den fyrste testen, som vart gjennomført før prosjektperioden. Etter prosjektperioden klarte tre av desse elevane å finne eit eksplisitt uttrykk for eit vilkårleg ledd i mønsteret. Blant dei elevane som genererte eksplisitte løysingsstrategiar på begge testane, kunne ein registrere mindre endringar. Vidare kjem det fram at fleire elevar set ord på at dei har lært noko i prosjektperioden. Det er likevel vanskeleg å knytte desse funna opp mot kva rolle arbeidsmetoden har hatt å seie for utviklinga dei har hatt.
2018-01-01T00:00:00ZÅ omsetje mellom mellom ulike representasjonar av funksjonar. Ei kvalitativ undersøking av elevar sine løysingsstrategiar i matematikkfaget 1P.
http://hdl.handle.net/1956/18975
Å omsetje mellom mellom ulike representasjonar av funksjonar. Ei kvalitativ undersøking av elevar sine løysingsstrategiar i matematikkfaget 1P.
Floen, Kine Renate
Master thesis
I denne studien har løysingsstrategiar for å omsetje mellom ulike representasjonar av lineære funksjonar vore i fokus. Følgande forskingsspørsmål har vore grunnlag for arbeidet: 1. Kva løysingsstrategiar brukar elevar i matematikkfaget 1P når dei skal omsetje mellom ulike representasjonar av lineære funksjonar. 2. Kva kjenneteiknar ulike strategiar på ulike nivå? Det har i denne studien blitt gjennomført eit semistrukturert intervju med ni elevar. Bakgrunn for utvalet er gjort ut i frå ein diagnostisk test som undersøkjer kva strategiar elevar bruker for å omsetje mellom ulike representasjonar av lineære funksjonar. Under arbeidet med analysen blei det funnet ei rekke strategiar der kjenneteikn på desse strategiane er brukt for å utvikle tre nivå. Nivå A (imitativ) finn ein kjenneteikn som bruk av omvegar, prøve- og feilemetode og imitative tilnærmingar. For nivå B (hybrid) er også eit kjenneteikn bruk av omvegar, men med færre steg enn på nivå A. Andre kjenneteikn vil vere mønstersniffing og ei hybrid tilnærming. For nivå C (kreativ) finn ein kjenneteikn som å omsetje direkte, tolke og kreative tilnærmingar. Studien viser at på nivå A vil ein finne strategiar som tar i bruk ei imitativ tilnærming der ein leiter etter kjente prosedyrar og formlar. Dette kjem fram ved bruk av omvegar der ein . Ved mangel på kjente prosedyrar vil strategiar på nivå A ta i bruk ein prøve- og feilemetode der ein gjerne bruker aritmetikk og ikkje algebra for å finne svaret. Strategiar på nivå B vil ta i bruk ei hybrid tilnærming med kjenneteikn som både imitativ, men også kreativ tilnærming. Imitative tilnærmingar kjem fram ved bruk av omvegar, men med færre steg enn nivå A. Samtidig vil strategiane vere meir kreative ved at ein brukar strategiar der ein leiter etter mønster for å løyse ei oppgåve. Strategiar på nivå C vil bruke ei kreativ tilnærming ved å bruke lærte prosedyrar til å vurdere, noko som saman med å tolke kan gjere strategiane meir effektive. Ei slik kreativitet gjer at elevar kan omsetje direkte. Kjennskap til ulike strategiar vil vere nyttig for lærarar å bruke i undervisninga. Ved å ha eit slikt verktøy kan ein lettare kjenne igjen dei ulike kjenneteikn for strategiar på ulike nivå og dermed hjelpe elevar til å utvikle sine strategiar.
2018-01-01T00:00:00ZWater renewal in a threshold fjord: A simulation study of submerged fresh water discharge
http://hdl.handle.net/1956/18846
Water renewal in a threshold fjord: A simulation study of submerged fresh water discharge
Sand, Ragnhild Evebø
Master thesis
Threshold fjords are characterized by low concentrations of oxygen in the depths of the basin, caused by weak and/or infrequent exchange of the basin water. A three-dimensional ocean model, The Bergen Ocean Model (BOM), has been used to estimate effects of a submerged fresh water discharge in a threshold fjord. The principle of discharging fresh water at a depth and thus creating artificial upwelling is known to give promising results for the primary production. The aim of this study was to examine if a discharge at the bottom of the basin could increase the ventilation and/or circulation of the basin water. Three preliminary experiments and four main experiments were performed and the results indicated an effect of such discharges. Both the circulation and ventilation of the basin water was increased by the intervention of a submerged discharge. The main effect was a reduction in the density of the basin water, as the dense bottom water was mixed with the discharge and elevated due to buoyancy. This lead to high density intrusion of coastal water along the sill. In the four main experiments, the area and the flux of the discharge were adjusted. These adjustments indicated that a greater discharge area or a higher flux lead to more entrainment of the dense bottom water.
2018-12-15T00:00:00ZDifferent aspects of the hodograph transform and Riemann invariants for the shallow water equations
http://hdl.handle.net/1956/18845
Different aspects of the hodograph transform and Riemann invariants for the shallow water equations
Acharya, Dipti
Master thesis
The aim of this work is to visualize irrotational long wave on a sloping beach
by following the approach of Carrier and Greenspan [5]. We first derive
the non-linear shallow-water equations for sloping beach and then find the
Riemann invariants. The Riemann invariants are then used to implement a
proper hodograph transformation in order to transform the equations into
linear form. By using separation of variables the exact solutions of the linear
equations are found and the results are plotted for different values of runup and run-down time. Furthermore, in this study we obtain shallow-water
equations for shear flow which are also called Benney equations [10]. These
equations are written in a vector form [1] to find the characteristic form and
the Riemann invariants of the shallow-water equations for shear flow over a
flat bed.
2018-12-15T00:00:00ZAnalysis of the probability of default in peer-to-peer lending. Application of different classification techniques.
http://hdl.handle.net/1956/18844
Analysis of the probability of default in peer-to-peer lending. Application of different classification techniques.
Evjen, Endre Kvåle
Master thesis
In this thesis, peer-to-peer lending is explored and analyzed with the objective of fitting a model to accurately predict if borrowers default on their loans or not. The foundation for the thesis is a dataset from LendingClub, a peer-to-peer lending platform based in San Francisco, USA. Detailed information of borrowers’ financial history, personal characteristics and the specifics of each loan is used to predict the probability of default for the various loans in the portfolio. Methods used include elastic net regularization of logistic regression, boosting of decision trees, and bagging with random forests. The results are compared using accuracy metrics and a profitability measure, before a final model selection is carried out.
2018-12-22T00:00:00ZVideregåendeelevers strategier i brøkoppgaver. Hvilke deler av brøkbegrepet er synlige i strategiene?
http://hdl.handle.net/1956/18807
Videregåendeelevers strategier i brøkoppgaver. Hvilke deler av brøkbegrepet er synlige i strategiene?
Bringeland, Helen
Master thesis
Behr, Lesh, Post & Silver (1983) mener at rasjonale tall er blant de mest komplekse og viktigste matematiske idéene barn i grunnskolen skal lære og jeg ønsket derfor å se nærmere på brøkkunnskapene til elever etter endt grunnskole. I denne oppgaven har jeg undersøkt hvilke oppgavespesifikke strategier elever har i et sett med brøkoppgaver og hvilke deler av brøkbegrepet som er synlige i dem. Jeg har også fått muligheten til å sammenligne 1P- og 1T-elevers strategier og de ulike delene av brøk som synliggjøres i dem. Oppgaven bygger på Behr et al. (1983) sin modell med de fem delene av brøkbegrepet: Del-hele, måling, operator, kvotient og forhold. 42 elever har gjennomført en kartleggingsprøve med brøkoppgaver fra nasjonale prøver. Elevene hadde akkurat startet på videregående skole, etter å ha gått på ulike ungdomsskoler spredt over hele landet. En slik kartleggingsprøve er i utgangspunktet kvantitativ og gir mulighet til å sammenligne elevenes besvarelser. Etter gjennomlesing av elevbesvarelsene ble strategiene sortert, navngitt og beskrevet. Hver oppgave ga meg 3-5 ulike strategier som jeg deretter kunne analysere kvalitativt og se hvilke deler av brøkbegrepet som er synlige i elevenes tenkning. Undersøkelsen min viser at elever benytter ulike oppgavespesifikke strategier i møte med brøkoppgaver. Likevel er det ofte en eller to strategier som majoriteten av elevene velger å bruke. Dette er blitt presentert i en profil med oversikt over majoritetens strategier og brøkdelene i dem. I mange strategier blir flere deler av brøkbegrepet synlig, noe som viser at det ikke nødvendigvis går an å skille de fem brøkdelene fra hverandre i strategiene. Alle deler av brøkbegrepet er representert i elevstrategiene, noen oftere enn andre. Det er store likheter blant 1T- og 1P-elevers strategier og brøkdelene i dem, men med noen antydninger til at 1P-elever oftere benytter såkalte backup-strategier.
Revised version: some spelling errors corrected
2018-12-19T00:00:00ZLagrangian dynamics for solid multi-body systems with the Moving Frame formalism
http://hdl.handle.net/1956/18756
Lagrangian dynamics for solid multi-body systems with the Moving Frame formalism
Sætran, Nicolai Mikal
Master thesis
Classical mechanics is the study of the motion of particles, solid bodies or of systems of bodies, and the effects of forces and moments on them. The dynamical behavior may either be studied by the application of Newtonian mechanics, Lagrangian mechanics or Hamiltonian Mechanics. Either approach produces mathematical equations describing the time evolution of the multi-body system. The resulting equations of motion for many bodies are typically non-linear and of large scale. The Lagrangian approach is more abstract than the Newtonian approach in the sense that it employs more advanced theory from mathematics, and has therefore traditionally mostly been a topic studied in mathematics, physics or on master level engineering. The Moving Frame Method by H. Murakami et. Impelluso is a formalism which makes advanced results from classical mechanics and group theory available to bachelor level engineers. The method allows engineers without deep understanding of these fields to describe the dynamic behavior of systems that would otherwise be too complex to approach. The method is therefore a framework on which engineers may rely without having a background in group theory or calculus of variations. In this thesis the focus will be on further generalizing the multi-body Moving Frame Method. We will restrict ourselves to multi-body systems with generalized coordinates which are all free rotations. These systems may all be idealized as N-body three-dimensional pendulums (These are sometimes named open kinematic chains). The three main contributions of this thesis is a coordinate free formulation of the Euler-Lagrange equations under the Moving Frame Formalism. An algorithmic approach to generating the equations of motion. The application of quaternions as representation of rotations in numerical simulation of multi-body systems rotating in three dimensions.
2018-12-11T00:00:00ZOn the optimization of iterative schemes for solving non-linear and/or coupled PDEs
http://hdl.handle.net/1956/18755
On the optimization of iterative schemes for solving non-linear and/or coupled PDEs
Storvik, Erlend
Master thesis
In this thesis we study the optimization of iterative schemes as both linearization methods, and as splitting methods for solving non-linear and coupled partial differential equations (PDEs). We consider two equations that are describing processes in porous media; Richards’ equation, a possibly degenerate, non-linear and elliptic/parabolic equation that models flow of water in saturated/unsaturated porous media, and Biot’s equations, a coupled system of equations that models flow in deformable porous media. For Richards’ equation we compare the numerical properties of several linearization schemes, including the Newton-Raphson method, the modified Picard method and the L-scheme. Additionally, we prove convergence of the linearly and globally convergent L-scheme and discuss theoretically and practically how to choose its stabilization parameter optimally in the sense that convergence is obtained in the least amount of iterations. The second aim of the thesis is to effectively solve the quasi-static, linear Biot model. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot’s equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization parameter. We propose a new approach to optimize this parameter, and show theoretically that it does not only depend on the mechanical properties and the coupling coefficient, but also on the fluid’s flow properties. The type of analysis presented in this thesis is not restricted to a particular spatial discretization, but we require it to be inf-sup stable. The convergence proof also applies to low-compressible or incompressible fluids, and low-permeable porous media. We perform illustrative numerical examples, including a well-known benchmark problem, Mandel’s problem. The results largely agree with the theoretical findings. Furthermore, we show numerically that for conditionally inf-sup stable discretizations, the performance of the fixed-stress splitting scheme behaves in a manner which contradicts the theory provided for inf-sup stable discretizations.
2018-12-11T00:00:00ZAn Exploratory Analysis of Multi-Class Uncertainty Approximation in Bayesian Convolutional Neural Networks
http://hdl.handle.net/1956/18735
An Exploratory Analysis of Multi-Class Uncertainty Approximation in Bayesian Convolutional Neural Networks
Murray, Sean Meling
Master thesis
Neural networks are an important and powerful family of models, but they have lacked practical ways of estimating predictive uncertainty. Recently, researchers from the Bayesian machine learning community developed a technique called Monte Carlo (MC) dropout which provides a theoretically grounded approach to estimating predictive uncertainty in dropout neural networks. Some researchers have developed ad hoc approximations of these uncertainty estimates for use in convolutional neural networks. We extend their research to a multi-class setting, and find that ad hoc approximations of predictive uncertainty in some cases provides useful information about a model’s confidence in its predictions. Furthermore, we develop a novel approximation of uncertainty that in some respects performs better than those currently being used. Finally, we test these approximations in practice and compare them to other methods suggested in the literature. In our setting we find that the ad hoc approximations perform adequately, but not as well as those already suggested by experts.
2018-11-22T00:00:00ZLikelihood Estimation of Jump-Diffusions. Extensions from Diffusions to Jump-Diffusions, Implementation with Automatic Differentiation, and Applications
http://hdl.handle.net/1956/18720
Likelihood Estimation of Jump-Diffusions. Extensions from Diffusions to Jump-Diffusions, Implementation with Automatic Differentiation, and Applications
Lunde, Berent Ånund Strømnes
Master thesis
This thesis considers the problem of likelihood- based parameter estimation for time-homogeneous jump-diffusion processes. The problem is that there often is no analytic solution to the stochastic differential equations driving the process. Thus, the transition density of the process is unknown. In this thesis we build on the solution presented in Preston and Wood (2012), where the transition density of a time- homogeneous diffusion process is approximated by a saddlepoint approximation based on the approximated solution following from discretization schemes, which in turn stems from an Itô-Taylor expansion of the stochastic differential equation. The mathematical tools for understanding the method in Preston and Wood (2012) and the extended methods to jump- diffusions are developed. We reproduce the results found here, and extend the analysis with maximum likelihood estimation for benchmark processes such as the geometric Brownian motion, the Ornstein-Uhlenbeck process, the Cox- Ingersoll-Ross process, and the Merton model. We also investigate the use of the renormalized saddlepoint approximation in the context of maximum likelihood estimation. The implementation of the methods is carried out with the newly released parallel programming package, Template Model Builder, which uses automatic differentiation among other things. We therefore give an introduction to the basics of automatic differentiation in the context of our computational problems, and also extend the Template Model Builder package to e.g. allow for complex numbers. Thereafter we apply the methods developed in previous chapters to the analysis of stock prices modelled as nonlinear stochastic differential equations, with and without jumps. Finally we briefly analyse some models for stochastic volatility.
2016-06-02T00:00:00ZA continous Dependence Result for nonlinear degenerate parabolic Equations with spatially dependent flux Function.
http://hdl.handle.net/1956/18683
A continous Dependence Result for nonlinear degenerate parabolic Equations with spatially dependent flux Function.
Evje, Steinar; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik
Research report
2000-06-01T00:00:00ZIs there an Association between vitamin D status and risk of chronic low back pain? A nested case-Control analysis in the Nord-Trøndelag Health Study
http://hdl.handle.net/1956/18617
Is there an Association between vitamin D status and risk of chronic low back pain? A nested case-Control analysis in the Nord-Trøndelag Health Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Mai, Xiao-Mei; Langhammer, Arnulf; Zwart, John-Anker
Journal article
<p>Objectives: To explore potential associations between vitamin D status and risk of chronic low back pain (LBP) in a Norwegian cohort, and to investigate whether relationships depend on the season of blood sample collection.</p>
<p>Design: A nested case–control study in a prospective data set.</p>
<p>Setting: The Norwegian community-based Nord-Trøndelag Health Study (HUNT). Data were collected in the HUNT2 (1995–1997) and HUNT3 (2006–2008) surveys.</p>
<p>Main outcome measure: Chronic LBP, defined as LBP persisting at least 3 months continuously during the past year.</p>
<p>Participants: Among individuals aged 19–55 years without LBP in HUNT2, a data set was generated including 1685 cases with LBP in HUNT3 and 3137 controls without LBP.</p>
<p>Methods: Blood samples from the participants collected in HUNT2 were analysed for serum 25-hydroxyvitamin D (25(OH)D) level. Associations with LBP in HUNT3 were evaluated by unconditional logistic regression analysis with adjustment for age, sex, work status, physical activity at work and in leisure time, education, smoking, and body mass index.</p>
<p>Results: No association between vitamin D status and risk of chronic LBP was found in the total data set (OR per 10 nmol/L 25(OH)D=1.01, 95% CI 0.97 to 1.06) or in individuals with blood samples collected in summer/autumn (OR per 10 nmol/L 25(OH)D=0.99, 95% CI 0.93 to 1.06). For blood samples drawn in winter/spring, associations differed significantly between women and men (p=0.004). Among women a positive association was seen (OR per 10 nmol/L 25(OH)D=1.11, 95% CI 1.02 to 1.20), but among men no significant association was observed (OR per 10 nmol/L 25(OH)D=0.90, 95% CI 0.81 to 1.01).</p>
<p>Conclusions: Overall, no association between vitamin D status and risk of LBP was demonstrated. The association suggested in women for the winter/spring season cannot be regarded as established.</p>
2017-11-25T00:00:00ZPhysical activity level at work and risk of chronic low back pain: A follow-up in the Nord-Trøndelag Health Study
http://hdl.handle.net/1956/18531
Physical activity level at work and risk of chronic low back pain: A follow-up in the Nord-Trøndelag Health Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Zwart, John-Anker
Journal article
<p>Background: Physical activity in leisure time seems to reduce the risk of low back pain, but it is not known whether occupational activity, as recorded in a representative working population, produces a higher or lower risk.</p>
<p>Objective: To study associations between physical activity level at work and risk of chronic low back pain.</p>
<p>Methods: Associations were examined in a Norwegian prospective study using data from the HUNT2 and HUNT3 surveys carried out in the whole county of Nord-Trøndelag. Participants were 7580 women and 7335 men who supplied information about physical activity level at work. Levels considered were sedentary work, work involving walking but no heavy lifting, work involving walking and heavy lifting, and particularly strenuous physical work. Nobody in the cohort was affected by chronic low back pain at baseline. After 11 years, participants reported whether they suffered from chronic low back pain. Generalized linear modelling with adjustment for potential confounders was applied to assess associations with risk factors.</p>
<p>Results: In age-adjusted analyses both women and men showed statistically significant associations between physical activity at work and risk of chronic low back pain, suggesting positive relationships. For particularly strenuous physical work the relative risk of chronic low back pain was 1.30 (95% CI: 1.00–1.71) in women and 1.36 (95% CI 1.17–1.59) in men, compared to sedentary work. Women still showed a general association with activity level after adjustment for education, leisure time physical activity, BMI, smoking and occupational category. In men, the higher risk was only maintained for particularly strenuous work.</p>
<p>Conclusion: In this cohort, women had a higher risk of chronic low back pain with work involving walking and heavy lifting or particularly strenuous work, compared to sedentary work. Men participating in particularly strenuous work also experienced a higher risk of chronic low back pain.</p>
2017-04-10T00:00:00ZFish louse and treatment options: A mathematical approach
http://hdl.handle.net/1956/18458
Fish louse and treatment options: A mathematical approach
Utåker, Therese Sjøstrøm
Master thesis
2018-09-04T00:00:00ZAnalytical solutions for aquifer thermal energy storage
http://hdl.handle.net/1956/18419
Analytical solutions for aquifer thermal energy storage
Nordbotten, Jan Martin
Journal article
The concept of aquifer thermal energy storage involves injection of water at elevated temperature, and possibly nonambient salinity, into a host aquifer. We consider axisymmetric injection, wherein both the composition and temperature of the injected fluid differ from the fluid in the target aquifer. In this setting, we derive the governing equations within a vertically integrated framework, and show their self‐similar structure. We subsequently derive explicit approximate solutions to the self‐similar equations for parameter ranges of relevance to thermal energy storage (small density and viscosity differences). The analysis is supported by numerical validation, covering the relevant parameter regime. The resulting comparisons demonstrate the mathematical qualities of the analytical approximations. A study based on field data from analogue sites justifies the assertions regarding the magnitude of the dimensionless parameters used in the analysis.
2017-02-01T00:00:00ZFinite volume methods for elasticity with weak symmetry
http://hdl.handle.net/1956/18401
Finite volume methods for elasticity with weak symmetry
Keilegavlen, Eirik; Nordbotten, Jan Martin
Journal article
We introduce a new cell‐centered finite volume discretization for elasticity with weakly enforced symmetry of the stress tensor. The method is motivated by the need for robust discretization methods for deformation and flow in porous media and falls in the category of multi‐point stress approximations (MPSAs). By enforcing symmetry weakly, the resulting method has flexibility beyond previous MPSA methods. This allows for a construction of a method that is applicable to simplexes, quadrilaterals, and most planar‐faced polyhedral grids in both 2D and 3D, and in particular, the method amends a convergence failure in previous MPSA methods for certain simplex grids. We prove convergence of the new method for a wide range of problems, with conditions that can be verified at the time of discretization. We present the first set of comprehensive numerical tests for the MPSA methods in three dimensions, covering Cartesian and simplex grids, with both heterogeneous and nearly incompressible media. The tests show that the new method consistently is second order convergent in displacement, despite being the lowest order, with a rate that mostly is between 1 and 2 for stresses. The results further show that the new method is more robust and computationally cheaper than previous MPSA methods.
2017-11-01T00:00:00ZDerived Logarithmic Hochschild Homology
http://hdl.handle.net/1956/18266
Derived Logarithmic Hochschild Homology
Lundemo, Tommy
Master thesis
2018-06-27T00:00:00ZModel Spaces in Riemannian and Sub-Riemannian Geometries
http://hdl.handle.net/1956/18265
Model Spaces in Riemannian and Sub-Riemannian Geometries
Berge, Eirik
Master thesis
The development of Riemannian geometry has been highly influenced by certain spaces with maximal symmetry called model spaces. Following in the footsteps of Klein's Erlangen program, model spaces fit with the approach of investigating the symmetries of a geometric object to understand the object itself. This thesis examines model spaces in the sub-Riemannian setting. The scientific contribution of the thesis is the classification of all sub-Riemannian model spaces with step and rank three.
2018-06-27T00:00:00ZLie Groups and the Principle of Virtual Work Applied to Systems of Linked Rigid Bodies. A generic algorithm for the equations of motion using the moving frame method in engineering
http://hdl.handle.net/1956/18264
Lie Groups and the Principle of Virtual Work Applied to Systems of Linked Rigid Bodies. A generic algorithm for the equations of motion using the moving frame method in engineering
Rykkje, Thorstein Ravneberg
Master thesis
This thesis explores the possibility of a generic algorithm for systems of linked rigid bodies using the moving frame method (MFM) in engineering developed by H. Murakami and T.Impelluso. The project entails the construction of a generic algorithm for the equations of motion and the validation of the equations generated by said algorithm. The validation is done by comparing the equations of motion generated by the algorithm to equations evaluated manually. Furthermore, the resulting behavior from integration is compared to those from the Hamilton canonical equations of motion. Finally, a real-life model is built to see if the theory holds in reality. Naturally, when taking the step to bring in reality, friction and air resistance will have to be taken into account. Therefore, the equations of motion were supplemented with friction models constructed to match the movement of the real-life model.
2018-06-27T00:00:00ZGeogebra og funksjonar. Elevane si tilnærming til representasjonsskifte i arbeid med funksjonar, med og utan bruk av Geogebra, i matematikk på vidaregåande skule.
http://hdl.handle.net/1956/18263
Geogebra og funksjonar. Elevane si tilnærming til representasjonsskifte i arbeid med funksjonar, med og utan bruk av Geogebra, i matematikk på vidaregåande skule.
Westermoen, Torstein Taranger
Master thesis
Digitale hjelpemiddel har for lengst fått sin plass i matematikkundervisninga i norsk skule. I den vidaregåande skulen har dataprogram med grafteiknar også langt på veg erstatta kalkulatorar, som digitalt hjelpemiddel i arbeid med funksjonar. Frå og med våren 2015 vart det og eit krav, å nytte grafteiknar til å svare på einskilde oppgåver på eksamen. I den samanheng vart det interessant å sjå nærmare på korleis elevane arbeider med funksjonar med grafteiknar (i denne oppgåva Geogebra) samanlikna med det å bruke tradisjonelle metodar (papir, blyant og eventuelt kalkulator). Målet for denne oppgåva byggjer på denne problemstillinga: Elevane si tilnærming til representasjonsskifte i arbeid med funksjonar, med og utan bruk av Geogebra, i matematikk på første trinn på vidaregåande skule. Problemstillinga gav også følgjande forskingsspørsmål: Kva utfordringar har elevane i 1P og 1T når dei løyser oppgåver i funksjonar i dei to kontekstane, med og utan Geogebra? For å svare på dette har eg undersøkt korleis elevane arbeider med representasjonsskifte i funksjonar og om det er noko skilnad på korleis dei lukkast i dei to kontekstane. Det er åttiseks elevar frå 1P og 1T som har vore med i undersøkinga. Først gjennomførte eg ein skriftleg prøve i to delar. Den fyrste delen inneheld oppgåver som skulle løysast utan bruk av Geogebra, medan del to består av oppgåver som skulle løysast med bruk av Geogebra. Alle elevane svara på prøven, og i etterkant intervjua eg tre av desse, blant anna på bakgrunn av interessante svar frå den skriftlege prøven. Resultatet frå prøven og intervjua vart koda og analysert for å best mogleg svare på problemstillinga og forskingsspørsmål(et). Undersøkinga viser at det var signifikante forskjellar på kva elevane fekk til av oppgåver, ved å samanlikne svara deira i dei to kontekstane, med og utan bruk av Geogebra. Både elevane i 1P og i 1T hadde betraktelig lågare score på den delen som var med Geogebra. Dette heng nok saman med eit anna funn, nemleg alle dei «manglande svara», særleg i den sistnemnte konteksten. I undersøkinga kjem det fram at mange elevar har vanskar med å gå i frå ein representasjon i funksjonar til ein annan. Enkelte elevar teiknar grafen til ein lineærfunksjon som eit punkt, andre blandar stigningstal og konstantledd. Vi ser også teikn til at ein del elevar vegrar seg mot å lese av x- eller y-verdiar frå ein graf, men vil heller rekne desse ut ved hjelp av likninga til grafen. Nokre av desse vanskane kjem berre fram i dei oppgåvene der ein ikkje skulle bruke Geogebra, medan andre dukkar opp også her. Interessant er det å sjå at enkelte «drar med seg» løysingsstrategiar (algoritmar) frå den eine konteksten og over i den andre, med blanda hell. Andre utfordringar med Geogebra går på tekniske innstillingar, slik som avrunding av desimalar og skalering av aksar.
2018-06-20T00:00:00ZForståelse av matematiske tekstoppgaver. Hva bør man som lærer ha tenkt gjennom når man ønsker at elevene skal jobbe med tekstoppgaver i matematikk?
http://hdl.handle.net/1956/18234
Forståelse av matematiske tekstoppgaver. Hva bør man som lærer ha tenkt gjennom når man ønsker at elevene skal jobbe med tekstoppgaver i matematikk?
Stordrange, Kjetil Kjellesvik
Master thesis
Denne studien tar for seg matematiske tekstoppgavers oppbygning og ser på ulike typer
utfordringer elever kommer over i møte med tekstoppgaver. Utfordringene handler i stor grad
om det å oversette teksten i tekstoppgaver til matematikk som man kan regne med.
Studien baserer seg på elevers besvarelser på noen utvalgte matematiske tekstoppgaver.
Oppgavene ble gitt til en spesifikk skole og var forsøkt tilpasset skolens sosiomatematiske
normer.
I tillegg blir det i studien utført to intervjuer med personer med erfaring i å konstruere
matematiske tekstoppgaver.
Resultatene vitner om mange misforståelser i elevenes besvarelser, noe som viser seg i
elevenes løsningsforslag. Dermed dukker det opp flere forhåndsregler man som lærer eller
oppgavekonstruktør må ta hensyn til når det kommer til matematiske tekstoppgaver. Viktige
begrep i denne sammenheng er presisjon, realisme og sosiomatematiske normer.
2018-06-21T00:00:00ZNumerical solution of drift-diffusion and electrostatic potential equations for solar cells.
http://hdl.handle.net/1956/18233
Numerical solution of drift-diffusion and electrostatic potential equations for solar cells.
Engelsen, Daniel
Master thesis
This thesis gives an introduction to how solar panels work and some of the problems for solar cells that need to be solved. Further, it gives an introduction to some physical equations that describe the electrical behavior inside the panels. We try to solve these equations numerically for different parameter values and with different numerical techniques. An analysis of some of the convergence and stability of each numerical system is given, as well as numerical results with numerical values and plots.
2018-06-22T00:00:00ZNumerical solution of the the time-dependent Schrödinger equation.
http://hdl.handle.net/1956/18232
Numerical solution of the the time-dependent Schrödinger equation.
Nedrelid, Kjell
Master thesis
On the atomic scale the time-dependent Schrödinger equation is used but for systems with three or
more particles like the helium atom it is not possible to solve the Schrödinger equation exactly.
The motivation here is to calculate a numerical solution for the helium atom in the ground state. To
this end a numerical method must be used and the choice is the summations by part method with
simultaneous approximation terms at any boundaries. We analysed and ran numerical verification
tests after coding in Matlab on the following PDE's. The transport equation, the heat equation and
the Schrödinger equation for particle in a box. The results for the transport equation and the heat
equation seems to verify the code but for the particle in a box we seem to very quickly hit the roundoff
limit and this made the verification difficult for higher order.
We also tested a little on the hydrogen atom before calculating the ground state of helium.
2018-06-22T00:00:00ZNumerical studies of rotationally controlled lock exchange flow over a sill
http://hdl.handle.net/1956/18231
Numerical studies of rotationally controlled lock exchange flow over a sill
Asplin, Magdeli Holmøy
Master thesis
Numerical experiments of lab scale on dense water inflow in a channel initially filled with freshwater, where a sill is present. The focus has been on the dynamics at the top of the sill, and especially on the behavior of the flow when rotation is added.
2018-06-21T00:00:00ZAdvection Upscaling for Heat Transport in Fractured Geothermal Reservoir
http://hdl.handle.net/1956/18230
Advection Upscaling for Heat Transport in Fractured Geothermal Reservoir
Ninic, Marija
Master thesis
The main objective in this thesis is to give numerical solution for advection term of heat transport equation in fractured geothermal reservoir. To get better insight in the fracture influence on the processes we will present all numerical results for domain with and without fracture field. Fracture field is modeled explicitly using discrete fracture model. Advection term of heat transport equation is discretized in space using upwind scheme for time discretization we use implicit method: Euler's backward scheme. For advection term upscaling we use the known flux values that we get from numerical solution of pressure equation. To get comutationally more efficient transport solver we will use upscaling and upgridding. We will use flow-based indicators for upgridding fine scale grid. In the case of fractured domain beside standard flow-based indicators: permeability, velocity and time of flight we will use distance to the nearest fracture and the combination of distance and time of flight as indicators. At the end we will compare results on different coarse grids with result on fine scale grid.
2018-06-20T00:00:00Z«Oppfatter ikke matematikk som nyttig for min del, siden filmregissører har ikke særlig bruk for faget». En kvantitativ studie av Vg1-elever sine oppfatninger om matematikkfagets nytteverdi
http://hdl.handle.net/1956/18229
«Oppfatter ikke matematikk som nyttig for min del, siden filmregissører har ikke særlig bruk for faget». En kvantitativ studie av Vg1-elever sine oppfatninger om matematikkfagets nytteverdi
Østreim, Rune
Master thesis
Elever sine oppfatninger om matematikk er med på å påvirke læringen i faget. I undervisningstimene kommer ofte spørsmålet om lærestoffet sin relevans opp, og hvilken nytteverdi eleven har av det som skal læres. Samtidig knyttes viktigheten av relevans opp mot gjennomføringsgrad i videregående skole, og statlig-initierte prosjekter som NyGiv og FYR er med på å støtte opp under dette. I denne masteroppgaven stilles spørsmålet om hva som kjennetegner elevenes oppfatninger om matematikkfagets nytteverdi, og tematikken belyses fra flere innfallsvinkler. Er det noen forskjeller på grad av oppfattet nytteverdi basert på kjønn, karakter i faget eller hva slags utdanningsprogram eleven går på? Har graden av oppfattet nytteverdi endret seg sammenlignet med en studie fra 2005? Finnes det noen sammenhenger mellom elevenes oppfatning om nytteverdi og mestring, og hvilken påvirkning har hjemmet? Til slutt, hvordan argumenterer elevene når de selv skal sette ord på hvorfor matematikk oppfattes som nyttig, og hva forteller disse argumentene oss? Problemstillingen belyses med utgangspunkt i en kvantitativ studie som tar for seg elever på en videregående skole, på både studieforberedende- og yrkesfaglig utdanningsprogram. Studien finner at elevene i 2018 oppfatter matematikkfagets nytteverdi i enda sterke grad enn i 2005. Det er ikke signifikante forskjeller i oppfatning om nytteverdi basert på kjønn og utdanningsprogram, mens det motsatte er tilfelle når det gjelder mestring, og de spesielt målt i form av karakter. Ved å skille ut ulike deler av utvalget blir bildet ytterligere nyansert. Det viser seg også at elever har ulike argumentstyper på hvorfor matematikk er nyttig, et funn som kan sees i lys av allerede eksisterende modeller for strukturering av oppfatninger. Det blir også tydelig at det er gunstig for læringen at elevene faktisk kan sette ord på hvorfor matematikk oppfattes som nyttig.
2018-06-20T00:00:00ZModelling memory decay after injuries using household survey data from Khartoum State, Sudan
http://hdl.handle.net/1956/18221
Modelling memory decay after injuries using household survey data from Khartoum State, Sudan
Heuch, Ivar; Abdalla, Safa; El Tayeb, Sally
Journal article
<p>Background: Injuries represent an important cause of morbidity and mortality worldwide. In retrospective epidemiological studies, estimated rates of reported injuries often decline considerably when information is included from periods more than a few months before the data collection. Such low rates are usually regarded as a consequence of memory decay. It is largely unknown whether the extent of memory decay depends on external factors otherwise affecting injury rates.</p>
<p>Methods: A statistical model was introduced to separate the influence of external factors on true injury rates from effects on memory decay. The relationship between apparent rates and time elapsed between injury occurrence and data collection was described by a parametric regression model. Relationships between memory decay and external factors were modelled by effect modification of the relationship with time. The procedure was applied to data collected in a retrospective household survey, carried out in Khartoum State in 2010, which elicited information about injuries that had occurred during the last year. The survey included 5661 individuals in 973 households, reporting a total of 481 non-fatal injuries.</p>
<p>Results: In the data from Khartoum State, differences in memory recall were observed between socioeconomic groups, with considerably faster memory decay in the lower socioeconomic tertile. In this tertile the estimated probability that an injury which occurred 6 months ago was reported was only 18%, compared to probabilities of about 35% in the remainder of the population. In the lower socioeconomic tertile, in contrast to other groups, a simple exponential model was not sufficient for describing memory decay. Memory decay did not depend on sex, age, urban/rural status or education. Road traffic injuries were subject to less memory decay than injuries due to falls, mechanical causes and burns. Memory decay seriously affected crude overall injury rates and also to some degree estimated relative rates.</p>
<p>Conclusion: In the statistical analysis of retrospective injury data it is important to take into account the effects of memory decay.</p>
2018-06-20T00:00:00ZHigh-accuracy phase-field models for brittle fracture based on a new family of degradation functions
http://hdl.handle.net/1956/18182
High-accuracy phase-field models for brittle fracture based on a new family of degradation functions
Sargado, Juan Michael; Keilegavlen, Eirik; Berre, Inga; Nordbotten, Jan Martin
Journal article
Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients. These in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations that enforce stress equilibrium and govern phase-field evolution. These equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature.
2018-02-01T00:00:00ZConforming Discretizations of Mixed-Dimensional Partial Differential Equations
http://hdl.handle.net/1956/18159
Conforming Discretizations of Mixed-Dimensional Partial Differential Equations
Boon, Wietse Marijn
Doctoral thesis
<p>Mixed-dimensional partial differential equations (PDEs) are coupled equations defined on connected manifolds of different dimensionalities. Two main examples of mixed-dimensional PDEs are considered in this dissertation, namely flow in fractured porous media and mechanics of composite materials. We focus on the discretization of these examples using hierarchical finite elements defined on coupled manifolds of codimension one, successively. By uncovering their underlying structure, we use the corresponding tools to define, analyze and discretize mixed-dimensional partial differential equations.</p>
<p>Our first example concerning mixed-dimensional PDEs arises in the context of fracture flow. Here, the planar fractures, intersection lines, as well as intersection points are represented as lower-dimensional manifolds. In turn, the entire embedded fracture network forms a mixed-dimensional geometry. We continue by defining the conservation and constitutive laws on the mixed-dimensional geometry, leading to a hierarchically coupled system of partial differential equations. Next, we extend these concepts from flow to derive the governing equations concerning mechanics of materials with thin inclusions in an analogous manner. Together, the embedded features and their surroundings form the mixed-dimensional geometry and the behavior of the system can be captured by prescribing significantly different material parameters. The analysis of these systems introduces several new concepts including mixed-dimensional function spaces and semi-discrete differential operators.</p>
<p>With the aim of discretization, we use finite element exterior calculus to construct mixed finite element schemes on the mixed-dimensional geometry. We focus on two families of mixed-dimensional finite elements, hierarchically ordered by dimensionality. We refer to these families as the first and second kind and show that both are of interest in the context of fracture flow, with different behavior in terms of convergence and computational cost. On the other hand, the mixed formulation of the mechanics equations requires the family of elements of the second kind.</p>
<p>For fracture flow, stability and optimal convergence of the discretization method are shown with the use of weighted, mixed-dimensional Sobolev spaces. A novel way of incorporating the fracture aperture leads to a scheme capable of handling arbitrarily small and spatially varying apertures. In case of fractures pinching out, the degeneration of the equations eliminates the possibility for flow resulting in a natural termination of fractures.</p>
<p>In a benchmark study concerning flow through fractured porous media, the proposed scheme is compared to various other numerical methods. Four two-dimensional test cases of varying complexity are considered, specifically designed to highlight the typical difficulties with modeling flow through fracture networks.With eight participating numerical schemes, a clear view is given of the performance and limitations of the state-of-the-art numerical schemes.</p>
<p>Finally, we consider the evolution of the water table and identify the water table itself as a lower-dimensional manifold. Its location is governed by a partial differential equation which is coupled to the underlying saturated region. To solve this problem, a numerical scheme is proposed which maps the problem to a stationary reference domain. We analyze the properties of this scheme and successfully apply it to a real world problem concerning ground flow patterns surrounding meandering streams.</p>
2018-06-01T00:00:00ZEfficient water table evolution discretization using domain transformation
http://hdl.handle.net/1956/18158
Efficient water table evolution discretization using domain transformation
Boon, Wietse; Balbarini, Nicola; Binning, Philip John; Nordbotten, Jan Martin
Journal article
Domain transformation methods are useful techniques for solving problems on non-stationary domains. In this work, we consider the evolution of the water table in an unconfined aquifer. This nonlinear, time-dependent problem is greatly simplified by using a mapping from the physical domain to a reference domain and is then further reduced to a single, (nonlinear) partial differential equation. We show well-posedness of the approach and propose a stable and convergent discretization scheme. Numerical results are presented supporting the theory.
2017-02-01T00:00:00ZRobust Discretization of Flow in Fractured Porous Media
http://hdl.handle.net/1956/18157
Robust Discretization of Flow in Fractured Porous Media
Boon, Wietse; Nordbotten, Jan Martin; Yotov, Ivan
Journal article
Flow in fractured porous media represents a challenge for discretization methods due to the disparate scales and complex geometry. Herein we propose a new discretization, based on the mixed finite element method and mortar methods. Our formulation is novel in that it employs the normal fluxes as the mortar variable within the mixed finite element framework, resulting in a formulation that couples the flow in the fractures with the surrounding domain with a strong notion of mass conservation. The proposed discretization handles complex, nonmatching grids and allows for fracture intersections and termination in a natural way, as well as spatially varying apertures. The discretization is applicable to both two and three spatial dimensions. A priori analysis shows the method to be optimally convergent with respect to the chosen mixed finite element spaces, which is supported by numerical examples.
2018-01-01T00:00:00ZFractal structures in freezing brine
http://hdl.handle.net/1956/18082
Fractal structures in freezing brine
Alyaev, Sergey; Keilegavlen, Eirik; Nordbotten, Jan Martin; Pop, Iuliu Sorin
Journal article
The process of initial ice formation in brine is a highly complex problem. In this paper, we propose a mathematical model that captures the dynamics of nucleation and development of ice inclusions in brine. The primary emphasis is on the interaction between ice growth and salt diffusion, subject to external forcing provided by temperature. Within this setting two freezing regimes are identified, depending on the rate of change of the temperature: a slow freezing regime where a continuous ice domain is formed; and a fast freezing regime where recurrent nucleation appears within the fluid domain. The second regime is of primary interest, as it leads to fractal-like ice structures. We analyse the critical threshold between the slow and fast regimes by identifying the explicit rates of external temperature control that lead to self-similar salt-concentration profiles in the fluid domains. Subsequent heuristic analysis provides estimates of the characteristic length scales of the fluid domains depending on the time-variation of the temperature. The analysis is confirmed by numerical simulations.
2017-09-01T00:00:00ZCost efficient environmental survey paths for detecting continuous tracer discharges
http://hdl.handle.net/1956/18044
Cost efficient environmental survey paths for detecting continuous tracer discharges
Alendal, Guttorm
Journal article
Designing monitoring programs for detecting potential tracer discharges from unknown locations is challenging. The high variability of the environment may camouflage the anticipated anisotropic signal from a discharge, and there are a number of discharge scenarios. Monitoring operations may also be costly, constraining the number of measurements taken. By assuming that a discharge is active, and a prior belief on the most likely seep location, a method that uses Bayes' theorem combined with discharge footprint predictions is used to update the probability map. Measurement locations with highest reduction in the overall probability of a discharge to be active can be identified. The relative cost between reallocating and measurements can be taken into account. Three different strategies are suggested to enable cost efficient paths for autonomous vessels.
2017-07-01T00:00:00ZParameter Estimation of Multivariate Factor Stochastic Volatility Models
http://hdl.handle.net/1956/18033
Parameter Estimation of Multivariate Factor Stochastic Volatility Models
Wahl, Jens Christian
Master thesis
Volatility is a crucial aspect of risk management and important to accurately quantify. A broad range of models and methods tackle this problem, but there is no consensus to exactly which method or model that solves this problem best. We use maximum likelihood and Hamiltonian Monte Carlo to estimate parameters in multivariate factor stochastic volatility models and compare the two alternative methods with the new interweaving strategy proposed in Kastner et al. (2017). Through simulation studies, we show that convergence of the likelihood is unstable and very data dependent. We investigate possible restrictions on our parameters by calculating the characteristic function of our model. We find that restricting the loading matrix (in two dimensions) makes convergence more stable. Furthermore, we introduce the “Nested Laplace Approximation” (where we integrate over the latent variables in a sequential way) and compare to the classical Laplace approximation on two state space models. We also compare the methods on exchange data from 2005-2015. All methods give similar results, but Hamiltonian Monte Carlo is sensitive to the choice of priors.
2018-06-22T00:00:00ZImplementation of an MPFA/MPSA-FV Solver for the Unsaturated Flow in Deformable Porous Media
http://hdl.handle.net/1956/17905
Implementation of an MPFA/MPSA-FV Solver for the Unsaturated Flow in Deformable Porous Media
Varela, Jhabriel
Master thesis
The Unsaturated Flow In Deformable Porous Media (UFIDPM) plays a crucial role in several academic and industrial applications such as; cracks induced by desiccation, collapsing soils, ground movement involving expansive soils, lateral earth surfaces, stability of vertical excavations, natural slopes subjected to environmental changes, construction and operation of a dam, etc. Typically, studying these systems with experimental techniques is impractical. On the other hand, numerical simulations allow us to investigate several scenarios in a short time and with reduced costs. At the moment, only Finite Element Method (FEM) based codes were available. FEM often shows conservative issues and when is applied to UFIDPM instabilities in the limit of incompressibility have been reported. Due to this issues, we were motivated to develop the first Cell-Centered Finite Volume Method (CCFVM) based code for solving UFIDPM. The governing equations are derived following the extended Biot’s theory of three-dimensional consolidation which results in a coupled hydro-mechanical system. For the flow problem, we use the Richards’ equation whereas the mechanical problem is modeled using the linear elasticity equations. For the spatial discretization, we use multi-points approximations schemes. Specifically, Multi Point Flux Approximation (MPFA) for the flow problem and Multi Point Stress Approximation (MPSA) for the mechanical problem. Moreover, the time discretization of the equations was obtained using Backward Euler (BE). The code was implemented in MATLAB R2017b were two core toolboxes (MRST and FV-BIOT) were used. The resulting non-linear set of equations was solved using the Newton method together with Automatic Differentiation (AD). To test the capability of the code several sub-problems were validated. Furthermore, we present a numerical application where we focus on the desiccation process of a clayey soil in a Petri-dish. In this experiment, the water content reduction is caused by instantaneous water evaporation controlled by atmospheric conditions. Finally, by carefully post-processing the resulting stress field, the zones of tensile stress concentration, which corresponds to the areas where cracks are more likely to initiate, are identified.
2018-06-22T00:00:00ZHorisontkunnskap i aksjon. Om lærernes syn på relevansen av avansert matematisk kunnskap i undervisningsarbeidet
http://hdl.handle.net/1956/17836
Horisontkunnskap i aksjon. Om lærernes syn på relevansen av avansert matematisk kunnskap i undervisningsarbeidet
Lauw, Sabine
Master thesis
In this master’s thesis I have examined in which ways Norwegian secondary teachers in mathematics think they need advanced mathematical knowledge in their teaching practice, and I have discussed this against the concept of Horizon Content Knowledge. I have also looked at what kinds of knowledge might be useful in their teaching practice. I have found nine categories for ways of usage of advanced mathematical knowledge, and I have sorted them into three main areas: (1) deep content knowledge and professional weight, (2) flexibility and (3) different ways of working with students. The first area includes professional confidence and use of content knowledge as a resource in their teaching practice. The second area includes independence of textbooks and contingency. The third area includes differentiation in teaching, motivation of students, teaching with the student’s future in mind, guidance of students and understanding what it is like to learn mathematics from the student’s perspective. My study shows that the teachers use their advanced mathematical knowledge to a large extent. This knowledge gives them professional confidence and is a resource that can been draw upon directly in their teaching too. But confidence and mathematical insight also gives them a high degree of flexibility according to the use of textbooks and the selection of subject matters, and also flexibility to act in the moment. This flexibility enables the teacher to differentiate better, motivate, guide, position students for their mathematical future and to better understand what it is like to learn mathematics. Different forms of teacher’s advanced mathematical knowledge might be useful in their teaching practice. They use their mathematical insight which they have achieved during different tertiary courses, but also knowledge of mathematical connections and metamathematical knowledge. Both teaching experience and pedagogical content knowledge and also their capacity to reflect are necessary for the transformation of content knowledge to useful teaching practice.; I denne oppgaven har jeg undersøkt på hvilken måte lærere på ungdomstrinn og videregående trinn mener at de har bruk for avansert matematisk kunnskap i sin undervisningspraksis og diskutert dette opp mot begrepet horisontkunnskap. Jeg har også sett på hva slags former for avansert kunnskap som kan være nyttig for undervisningsarbeidet. Jeg har intervjuet fem lærere fra ungdomstrinnet og fem lærere fra videregående trinn som alle hadde mer enn 60 studiepoeng i ren matematikk. Ved hjelp av datastyrt koding og kategorisering av datamaterialet har jeg på bakgrunn av lærernes beskrivelser utviklet ni kategorier for nyttige anvendelser av den avanserte matematiske kunnskapen i undervisningen. Disse kategoriene kan sorteres i de tre hovedområdene (1) faglig dybde og tyngde, (2) fleksibilitet og (3) måter å jobbe med elevene på. Faglig dybde og tyngde omfatter kategoriene faglig trygghet og faglig ressurs i undervisningen. Fleksibilitet omfatter kategorien å fristille seg fra læreboka og å kunne håndtere uforutsette situasjoner. Måter å jobbe med elevene på omfatter differensiering, motivering, veiledning, å ha et blikk fremover og å forstå hvordan det er å lære matematikk sett fra elevenes perspektiv. Studien min viser at lærerne i stor grad har bruk for avansert matematisk kunnskap i sitt undervisningsarbeid. Denne kunnskapen gir læreren faglig selvtillit og utgjør også en kunnskapsressurs som kan anvendes direkte i undervisningen. Men selvtillit og faglig innsikt gir i tillegg læreren en høy grad av fleksibilitet både i forhold til lærebok og utvelgelse av fagstoff, og også fleksibilitet til å håndtere de spontane situasjonene som oppstår underveis i undervisningen. Fleksibiliteten gjør læreren bedre i stand til å kunne differensiere, motivere, veilede, posisjonere elevene for sin matematiske fremtid og til å forstå hvordan det oppleves å lære matematikk. Ulike typer av lærerens avanserte matematiske kunnskap kan være nyttig i undervisningen. De bruker den matematikkfaglige innsikten som de har tilegnet seg i løpet av ulike kurs på universitets- og høyskolenivå, men også kunnskap om sammenhenger og kunnskap på metanivå, det vil si kunnskap om matematikkens karakter og egenart. I prosessen med å omdanne avansert fagkunnskap til nyttig undervisningspraksis, er både erfaring og didaktisk kunnskap nødvendig, i tillegg til lærernes egen refleksjonsevne.
2018-06-20T00:00:00ZMatematikk i kjemi. Hva er så vanskelig?
http://hdl.handle.net/1956/17826
Matematikk i kjemi. Hva er så vanskelig?
Fardal, Aina Elisabeth
Master thesis
I denne masteroppgaven har matematikk i kjemi hatt hovedfokus. Følgende forskningsspørsmål har vært grunnlag for arbeidet: • Med utgangspunkt i matematikkemner som kreves i kjemi, hvilke matematikkoppgaver viser flest elever som skal starte Kjemi 1 vansker med å løse? Hva kjennetegner disse ikke-korrekte løsningsmetodene? • Er det forskjell på elevenes prestasjon på relevante matematikkoppgaver med og uten kjemikontekst? • Hva avslører elevene, som scoret lavt på oppgaver med forhold og proporsjoner på kartleggingstest i matematikk, om sin proporsjonale resonnering? Femtifire elever har deltatt i undersøkelsen. Disse gjennomførte en kartleggingstest i matematikk høsten 2017. Denne testen og flere analoge kjemioppgaver har blitt analysert kvantitativt og kvalitativt. McNemar test er brukt for å sammenligne prestasjon på oppgaver med og uten kjemikontekst. Et semistrukturert intervju ble gjennomført med 4 elever for å få større innsikt i elevenes proporsjonale resonnering. Tilslutt ble en ettertest gjennomført på 29 elever for å følge opp funnene fra intervjuene. Studien viser blant annet at noen elever har vansker med områder i matematikk som forhold og intensive størrelser, skissering av grafer, sammenhengen mellom tall på standardform, brøk og desimaltall, forkorting av brøker og rasjonale uttrykk og logaritmer. Det er en signifikant forskjell i prestasjon på logaritmeoppgaver med og uten kjemikontekst, men dette kan skyldes forskjeller i tilgang på kalkulator. Elever med lave prestasjoner på oppgaver med forhold og intensive størrelser virker å løse slike oppgaver ulikt i matematikk og kjemi. I matematikk ble en bygge opp-strategi og halvering/dobling brukt. I kjemi ble beregninger med intensive størrelser brukt, men ofte feil. Kontekst påvirket flere av elevenes oppfatning av den multiplikative sammenhengen mellom to størrelser. Dette kunne trigge en additiv strategi og bruk av proporsjonale metoder i ikke-proporsjonale situasjoner. Enkelte elever virker å ha en svak oppfattelse av enheten. I blandingsforhold ble hele løsningen og del vann oppfattet likt, noe som påvirket tolkningen av del-del og del-hele og dermed forskjellen mellom ekstensive og intensives størrelser.
2018-06-20T00:00:00ZTime Series: Forecasting and Evaluation Methods With Concentration On Evaluation Methods for Density Forecasting
http://hdl.handle.net/1956/17766
Time Series: Forecasting and Evaluation Methods With Concentration On Evaluation Methods for Density Forecasting
Grindheim, Therese
Master thesis
The main focus of this thesis are density forecasts and the corresponding evaluation methods. A density forecast is an estimate of the probability density of predicted values. Density forecasts and the related evaluation methods have been little explored compared to point and interval forecasts, therefore we have chosen to focus on this topic. We go through a detailed description of three evaluation methods for density forecasts. To measure the performance of two of the density forecast evaluation methods we perform a Monte Carlo simulation. We simulate data sets with different data generating mechanisms to measure the size and power for the chosen evaluation methods. Based on our results from the Monte Carlo simulation, we continue with one evaluation method and apply it on empirical data, more specifically on economical, financial and insurance time series data.
2018-06-06T00:00:00ZOptimal design of photovoltaic power plants
http://hdl.handle.net/1956/17736
Optimal design of photovoltaic power plants
Eimhjellen, Einar
Master thesis
The configuration of the photovoltaic panels is important to maximize the energy output of a photovoltaic power plant. In this thesis, I have developed an algorithm calculating the optimal solution for the design of a solar farm with fixed panels. The design of the photovoltaic power plant is decided by the configuration of the following parameters: tilt and azimuth of the panels, height and length of the rows, distance between the rows and number of rows. The parameters are constructed to maximize the net present value of the solar farm in a given field. The calculations will be based on local measurements and estimations. We will analyse and optimize the design of the solar farms for three different locations.
2018-05-30T00:00:00ZTheta-Regularity and Log-Canonical Threshold
http://hdl.handle.net/1956/17735
Theta-Regularity and Log-Canonical Threshold
Øygarden, Morten
Master thesis
The log-canonical threshold is an invariant that is widely used in modern birational geometry. It contains information regarding the singularities of sheaves of ideals. The theta-regularity index is a regularity condition for coherent sheaves on principally polarized abelian varities, that in many ways is an analogue to the Castelnuovo-Mumford regularity index for projective spaces. Amongst other properties, theta-regularity contains information on when a coherent sheaf is generated by its global sections. The main result of this thesis is an inequality relating the log-canonical threshold and theta-regularity of non-trivial ideal sheaves on principally polarized abelian varieties.
2018-05-30T00:00:00ZKlassestørrelsens betydning for undervisningens innhold. En sammenligning av læreres undervisningspraksis i liten og stor klasse
http://hdl.handle.net/1956/17704
Klassestørrelsens betydning for undervisningens innhold. En sammenligning av læreres undervisningspraksis i liten og stor klasse
Mentzoni, Torsten
Master thesis
The importance of class size for the content of the teaching is a master's thesis that, using qualitative methods, tries to answer the question of how is the content of mathematics teaching in the upper secondary school affected by the class size? By starting from Schoenfeldt's (2014) framework for measuring the mathematical content of the teaching, I conducted a case study with two sets of a small and a large class observation with the same teacher as well as an interview of the teacher to look for answers to the question. The studies in this work show that reduced class size and increased quality of mathematics education coincide, and that smaller classes are associated with increased quality in the mathematical content of teaching. The findings are clear but can not rule out that students' varied motivation for different mathematics subjects is a contributing factor.; Klassestørrelsens betydning for undervisningens innhold er en masteroppgave som ved hjelp av kvalitative metoder prøver å besvare spørsmålet om hvordan innholdet i matematikk-undervisningen i den videregående skole påvirkes av klassestørrelsen. Ved å ta utgangspunkt i Schoenfeldts(2014) rammeverk for å måle undervisningens matematiske innhold fortok jeg et case-studie med to sett observasjoner av stor og liten klasse med samme lærer samt intervju av læreren for å lete etter svar på spørsmålet. Undersøkelsene i dette arbeidet viser at redusert klassestørrelse og økt kvalitet på matematikkundervisningen sammenfaller, og at mindre klasser henger sammen med økt kvalitet i det matematiske innholdet i undervisninger. Funnene er tydelige men kan ikke utelukke at elevers varierende motivasjon til ulike matematikkfag er en medvirkende årsak.
2018-05-08T00:00:00ZModel development for efficient simulation of CO2 storage
http://hdl.handle.net/1956/17695
Model development for efficient simulation of CO2 storage
Bjørnarå, Tore Ingvald
Doctoral thesis
<p>Carbon capture and storage (CCS) is an important component of several initiatives to reduce global greenhouse gas emissions by injecting and storing CO2 in underground reservoirs. Simulation technology plays an important role in providing storage capacity estimates and analyzing long-term safety and risk factors of leakage to the surface. Two of several important questions that need to be answered before a storage project may be approved is how fast and how much CO2 can be injected without compromising the integrity of the sealing caprock, which stops it from migrating to the surface.</p>
<p>To evaluate the integrity we rely on mathematical models, but due to the large extent of the area that needs to be considered and the many processes that are involved the calculations can quickly become large and complicated and very time consuming to solve. For screening purposes of potential storage sites, or investigation of potential storage sites when little data is available, many model realizations are needed, thus fast and robust yet accurate numerical techniques are not only tractable but also essential.</p>
<p>To understand what happens to the CO2, formation water and rock during injection and storage, we have thoroughly reviewed the main processes that are relevant to the integrity of the reservoir and sealing formations. These main processes are fluid flow, stress change and temperature change and they are all coupled where for instance a change in pore pressure and temperature due to CO2 injection causes deformations and stress alterations that can affect the integrity of the injection reservoir and caprock.</p>
<p>Considering the low solubility of CO2 in formation water under typical storage con- ditions (depth, temperature, pressure and salinity) we have illustrated that it is a good approximation to treat the injected CO2 and formation water as two separate fluids. Miscibility is therefore not an important process to consider in relation to long-term mechanical integrity and this simplifies the mathematical description of fluid flow.</p>
<p>Whether the thermo-hydro-mechanical coupling, where the temperature change is also considered, is important to evaluate is less obvious. Through examples we show that the in situ temperature is important to consider when estimating material proper- ties, but the effect of the cold (CO2) injection, relative to the storage formation, is very local and mainly affects the near-field of the injector. The cooling effect reduces the spreading of the CO2, but has little effect on the pore pressure. In general, cold injec- tion (relative to formation temperature) lowers the fracture pressure of the rock and the limit for maximum sustainable injection rate, and therefore, ignoring non-isothermal effects can underestimate the risk of failure, and vice versa for hot injection. A risk analysis of reactivation of faults in the sealing formation in the CO2 storage project at In Salah, Algeria, revealed that the thermal effect can make the difference between safe and risky storage.</p>
<p>To achieve a procedure for faster numerical evaluation, the layering structure and high aspect ratio of typical storage reservoirs can be used to simplify the mathematical description of the internal physical processes using a method of dimensional reduction. This has previously been found particularly attractive in simulating the migration of CO2 in the context of CCS. Since hydro-mechanical coupling is particularly essential to consider when evaluating the integrity of the caprock, we have extended this concept to also include the geomechanical processes. The underlying assumptions of negligible vertical flow compared to horizontal flow (Vertical Equilibrium, VE, assumption) and linearly varying displacement across the thickness of the reservoir (Linear Vertical Deflection, LVD, assumption) has proved promising in providing significant savings in computational time and effort with up to more than ten times faster calculations compared to a full-dimensional model. It has also been demonstrated that such models can retain a high accuracy when applied to realistic field data, such as the conditions at the CO2 storage plant at In Salah, Algeria. Also, the range of applicability of the dimensionally reduced model is to a leading order the thickness of the reduced domain and accurate solutions in the order of 0.1 % and less difference in solution compared to a full-dimensional formulation for aquifers up to 100 meters thick has been achieved.</p>
2018-02-16T00:00:00ZVertically integrated models for coupled two‐phase flow and geomechanics in porous media
http://hdl.handle.net/1956/17694
Vertically integrated models for coupled two‐phase flow and geomechanics in porous media
Bjørnarå, Tore Ingvald; Nordbotten, Jan Martin; Park, Joonsang
Journal article
Models of reduced dimensionality have been found to be particularly attractive in simulating the fate of injected CO2 in supercritical state in the context of carbon capture and storage. This is motivated by the confluence of three aspects: the strong buoyant segregation of the lighter CO2 phase above water, the relatively long time scales associated with storage, and finally the large aspect ratios that characterize the geometry of typical storage aquifers. However, to date, these models have been confined to considering only the flow problem, as the coupling between reduced dimensionality models for flow and models for geomechanical response has previously not been developed. Herein, we develop a fully coupled, reduced dimension, model for multiphase flow and geomechanics. It is characterized by the aquifer(s) being of lower dimension(s), while the surrounding overburden and underburden being of full dimension. The model allows for general constitutive functions for fluid flow (relative permeability and capillary pressure) and uses the standard Biot coupling between the flow and mechanical equations. The coupled model retains all the simplicities of reduced‐dimensional models for flow, including less stiff nonlinear systems of equations (since the upscaled constitutive functions are closer to linear), longer time steps (since the high grid resolution in the vertical direction can be avoided), and less degrees of freedom. We illustrate the applicability of the new coupled model through both a validation study and a practical computational example.
2016-02-01T00:00:00ZData on gender contrasts in the risk of incident myocardial infarction by age. The Tromsø Study 1979-2012
http://hdl.handle.net/1956/17670
Data on gender contrasts in the risk of incident myocardial infarction by age. The Tromsø Study 1979-2012
Albrektsen, Grethe; Heuch, Ivar; Løchen, Maja-Lisa; Thelle, Dag Steinar; Wilsgaard, Tom; Njølstad, Inger; Bønaa, Kaare
Journal article
The data presented in this article relate to the research article entitled “Risk of incident myocardial infarction by gender: Interactions with serum lipids, blood pressure and smoking. The Tromsø Study 1979–2012” (Albrektsen et al., 2017) [1]. Data quantify the gender differences in the risk of myocardial infarction (MI) in terms of incidence rate ratios (IRR), in subgroups defined by serum lipids, blood pressure and smoking among persons aged 35–54 years, 55–74 years and 75–94 years, respectively. Data also describe the age- and gender-specific linear associations with the coronary heart disease (CHD) risk factors. IRRs for combined categories of age, gender and a CHD risk factor, with each category compared to the same reference group, are also shown. IRRs were calculated as estimates of relative risk in Poisson regression analyses of person-years at risk. Among 33,859 individuals at risk, a total of 622, 1308 and 816 were diagnosed with MI at ages 35–54, 55–74 and 75–94 years, respectively.
2017-08-01T00:00:00ZPostinjection Normal Closure of Fractures as a Mechanism for Induced Seismicity
http://hdl.handle.net/1956/17515
Postinjection Normal Closure of Fractures as a Mechanism for Induced Seismicity
Ucar, Eren; Berre, Inga; Keilegavlen, Eirik
Journal article
Understanding the controlling mechanisms underlying injection-induced seismicity is important for optimizing reservoir productivity and addressing seismicity-related concerns related to hydraulic stimulation in Enhanced Geothermal Systems. Hydraulic stimulation enhances permeability through elevated pressures, which cause normal deformations and the shear slip of preexisting fractures. Previous experiments indicate that fracture deformation in the normal direction reverses as the pressure decreases, e.g., at the end of stimulation. We hypothesize that this normal closure of fractures enhances pressure propagation away from the injection region and significantly increases the potential for postinjection seismicity. To test this hypothesis, hydraulic stimulation is modeled by numerically coupling flow in the fractures and matrix, fracture deformation, and matrix deformation for a synthetic reservoir in which the flow and mechanics are strongly affected by a complex three-dimensional fracture network. The role of the normal closure of fractures is verified by comparing simulations conducted with and without the normal closure effect.
2017-10-04T00:00:00ZNumerical modeling of shear stimulation in naturally fractured geothermal reservoirs
http://hdl.handle.net/1956/17514
Numerical modeling of shear stimulation in naturally fractured geothermal reservoirs
Ucar, Eren
Doctoral thesis
<p>Shear-dilation-based hydraulic stimulations are conducted to create enhanced geothermal systems
(EGS) from low permeable geothermal reservoirs, which are initially not amenable to energy
production. Reservoir stimulations are done by injecting low-pressurized fluid into the naturally
fractured formations. The injection aims to activate critically stressed fractures by decreasing
frictional strength and ultimately cause a shear failure. The shear failure leads to a permanent
permeability enhancement of the fractures, which contributes to the overall reservoir permeability,
owing to the damage in fracture surface characteristics during the shear failure. Shear stimulation
is considered a key for geothermal energy development; however, seismicity is a critical
by-product, which has to be controlled. Numerical modeling can provide a deeper understanding on
governing mechanisms, which is essential for reservoir assessments and the control of seismicity.
The primary goal of this thesis is to aid further development of EGS by contributing to the current
state-of-the-art for numerical modeling of shear-dilation-based hydraulic stimulations.</p>
<p>Numerical modeling of shear-dilation-based hydraulic stimulations requires mathematical modeling of
flow and mechanical deformation in fractured formations. The initial focus of the thesis is the
modeling of the mechanical deformation of naturally fractured rock. The deformation and stress
state of the rock are controlled by the deformation of pre-existing fractures, which is governed
by different equations than the deformation of the surrounding formation. A cell-centered
finite-volume approach is developed where the fractures are represented as co-dimension one
inclusions in the domain. The method is capable of modeling deformation considering open and
closed fractures with complex and nonlinear relationships governing the displacements and tractions
at the fracture surfaces. The method aims to provide benefits for studies including flow and
deformation couplings in a discontinuous rock.</p>
<p>Hydraulic stimulations are essentially coupled hydro-mechanical processes, where the deformation
of fractures has an impact on the permeability as well as on the stress state of the rock. We
develop a computational model, which has the capability to capture these interrelations in two- or
three-dimensional domains. Considering the significance of the pre-existing fractures, we model the
reservoir as a network of explicitly represented large-scale fractures immersed in a permeable rock
matrix. The model can forecast the permeability evolution of geothermal reservoirs with complex
fracture networks.</p>
<p>To be able to mitigate the seismic hazards, the contributing processes and the interaction between
them should be examined. The computational model developed here also has the capability to
investigate the induced seismicity. By using the developed model, a novel hypothesis regarding the
induced seismicity generated after the termination of injections has been tested. During the fluid
injections, the pressure builds up inside the fractures, which causes normal deformation and
increases the void place between the fracture surfaces. The termination of the injections reverses
the void increase; i.e., the fracture starts to close owing to the pressure decrease. We identify that the fracture closure is one of the mechanisms that are responsible for the
induced seismicity generated after the termination of injections.</p>
2018-03-22T00:00:00ZModelling of the Shear Dilation Based Hydraulic Stimulation in Enhanced
Geothermal Systems Considering Fractures in Different Scales
http://hdl.handle.net/1956/17513
Modelling of the Shear Dilation Based Hydraulic Stimulation in Enhanced
Geothermal Systems Considering Fractures in Different Scales
Ucar, Eren; Berre, Inga; Keilegavlen, Eirik
Conference object
<p>A numerical approach for modelling of shear dilation
of existing fractures in hydraulic stimulation of
geothermal reservoirs at low elevated pressures is
presented.</p>
<p>The fractured rock in the reservoir is modelled as a
combination of explicitly represented fractures and the
rock matrix surrounding these fractures. The efficient
modelling of slip-induced permeability enhancement
requires coupling of the fluid flow in fractured rock
with the mechanical deformation of the rock matrix
and the shear dilation of the fractures. For flow
simulations, conductive fractures are represented in
the domain as high-permeable discontinuities;
therefore they dominate the overall flow behaviour.
The rock matrix is represented by a low permeability,
capturing the effect of small-scale fractures. For the
mechanical deformation problem, the rock matrix is
assumed to be a linear elastic material, while the
fractures in the rock matrix are introduced as internal
boundaries. The shear dilation of the fractures is
calculated by a joint deformation model (JDM), which
connects the shear slip in the fracture surfaces and
additional permeability caused by shear displacement.
The flow simulations and the mechanical deformation
of the rock matrix are both obtained by finite volume
discretizations.</p>
<p>Several numerical experiments designed by
resembling realistic reservoir parameters are
conducted to provide better understanding of the shear
dilation mechanism. Moreover, fractures present in
different scales in a geothermal reservoir. Ignoring the
effect of small-scale fractures to the fluid flow in the
matrix may result in an overestimate of the
permeability enhancement. Hence, the influence of
rock matrix permeability on fracture aperture and the
overall flow behaviour of the reservoir are examined.</p>
2016-09-01T00:00:00Z