<p>Vi har i denne masteroppgaven sett nærmere på Gröbner-baser og signaturskjemaet Unbalanced Oil and Vinegar. Vi har sett nærmere på Gröbner-basenes definisjoner og sett hvordan Gröbner-baser kan genereres ved Buchbergers algoritme. Videre har vi sett hvordan Gröbner-baser hjelper for å gi resten i divisjonsalgoritmen i den multivariable polynomringen unikhet og indirekte dermed løse idealmedlemskapsproblemet. Videre undersøkte vi mulighetene for å bruke Gröbner-baser til å lage et offentlig nøkkel-kryptosystem. Foreløpig er det ingen som har greid å lage et slikt kryptosystem som har stått imot visse angrep. Den neste delen av oppgaven tok for seg signaturskjemaet Unbalanced Oil and Vinegar. Vi har presentert teorien bak UOV og sett nærmere på et enkelt eksempel. I tillegg har vi undersøkt tre angrep på UOV, der det ene var basert på Gröbner-baser. Det viser seg at UOV virker resistent mot disse angrepene gitt at vi tar visse forholdsregler med parametrene. For angrepet basert på Gröbner-baser må systemet være av en viss størrelse for at sikkerheten skal ivaretas. Avslutningsvis har vi sett på forbedrede algoritmer for å beregne Gröbner-baser. Den første algoritmen vi så på var $F_4$ -algoritmen som tok i bruk lineæralgebra, mens den andre, $F_5$-algoritmen, tok utgangspunkt i å kutte ut unødvendige beregninger. Det viser seg at både $F_4$- og $F_5$-algoritmen kraftig forbedrer beregningstiden for å finne en Gröbner-basis.</p>

<p>I denne masteroppgaven ser vi nærmere på Gröbner-baser og kryptosystemet Hidden Field Equations. Vi begynner med å se på flere algebraiske begreper knyttet til Gröbner-baser, og hvordan Gröbner-baser kan beregnes vha. Buchbergers algoritme. I den siste delen av første kapittel ser vi hvordan Gröbner-baser kan brukes til å løse polynomiske ligningssystemer. Videre ser vi på kryptosystemet Hidden Field Equations. Vi begynner med den matematiske teorien som ligger bak HFE-systemet, for deretter å gi en beskrivelse av selve krypteringen. For å øke forståelsen av kryptosystemet, ser vi på et enkelt eksempel. Flere angrep på HFE har blitt forsøkt, og disse kan grovt sett deles inn i to klasser. Den ene typen angrep utnytter bestemte egenskaper i det konkrete kryptosystemet, mens den andre typen består av generelle algoritmer for å løse multivariate ligningssystemer. I denne oppgaven fokuserer vi på den siste typen, nærmere bestemt algoritmer som beregner en Gröbner-basis for et gitt ligningssystem. I kapittel 3 gjennomgår vi et Gröbner-basisangrep på HFE. I den siste delen av oppgaven ser vi på koblingen mellom Gröbner-baser og lineær algebra. Vi ser deretter på forbedringer av Buchbergers originale algoritme. Vi studerer F4-algoritmen som tar i bruk lineær algebra, og en videreutvikling av F4, kalt F5. F5 tar utgangspunkt i å kutte ut unødvendige beregninger ved å bruke det såkalte F5-kriteriet. Et tilsvarende kriterium, formulert av Gebauer og Möller, blir også gjennomgått.</p>

<p>PI by FFT har blitt implementert og sammenlignet med tre andre numeriske løsere på to 2D-modeller av partikkelakselleratorer. Løsningen med differenseskjema krever en transformasjon til den tilhørende Fokker-Planck-ligningen, som er vist i seksjon 3. Det blir også vist hvordan multiplikativ støy kan transformeres til additiv støy, og modeller i finans med mer generelle støykilder blir brukt til opsjonsprising. Kildekoden krever installasjon av det C++-baserte ROOT-rammeverket fra CERN.</p>

<p>We give an short introduction to the Camassa-Holm equation and its travelling wave solutions. Many well-known equations in mathematical physics describe geodesic flows on appropriate Lie groups. The choice of group and metric defines the Euler equation. We show that by choosing the group of diffeomorphisms on the circle and the Sobolev H^1-metric one gets the Camassa-Holm equation. The equation is shown to have a bi-Hamiltonian structure, and thus infinitely many conserved quantities. We introduce a new class of methods that can be applied to the Euler equation. We solve the Camassa-Holm equation by freezing some of the coefficients in the Euler equation and applying a Lie group integrator. In some situations the method is found to outperform existing schemes. The available numerical methods is reviewed and modified. We compare long term structure preservation for both smooth and non-smooth initial conditions for each method. Of special interest is the ability to handle wave collisions.</p>

<p>This master thesis has looked at numerical simulation of interacting bodies with delays, especially the situation involving interacting airguns in seismic source arrays. The equations describing the airguns have been derived and the interaction between the airguns has been studied. The resulting delay differential equations have been solved with methods that handle step sizes larger than the delays. The accuracy and efficiency of these methods have been investigated, and compared with Matlab solvers.</p>

<p>We present a general approach for Monte Carlo computation of conditional expectations of the form E[(T)|S = s] given a sufficient statistic S. The idea of the method was first introduced by Lillegård and Engen [4], and has been further developed by Lindqvist and Taraldsen [7, 8, 9]. If a certain pivotal structure is satised in our model, the simulation could be done by direct sampling from the conditional distribution, by a simple parameter adjustment of the original statistical model. In general it is shown by Lindqvist and Taraldsen [7, 8] that a weighted sampling scheme needs to be used. The method is in particular applied to the nonhomogeneous Poisson process, in order to develop exact goodness-of-fit tests for the null hypothesis that a set of observed failure times follow the NHPP of a specic parametric form. In addition exact confidence intervals for unknown parameters in the NHPP model are considered [6]. Different test statistics W=W(T) designed in order to reveal departure from the null model are presented [1, 10, 11]. By the method given in the following, the conditional expectation of these test statistics could be simulated in the absence of the pivotal structure mentioned above. This extends results given in [10, 11], and answers a question stated in [1]. We present a power comparison of 5 of the test statistics considered under the nullhypothesis that a set of observed failure times are from a NHPP with log linear intensity, under the alternative hypothesis of power law intensity. Finally a convergence comparison of the method presented here and an alternative approach of Gibbs sampling is given.</p>

<p>In this thesis we present an approximate recursive algorithm for calculations of discrete Markov random fields defined on graphs. We write the probability distribution of a Markov random field as a function of interaction parameters, a representation well suited for approximations. The algorithm we establish is a forward-backward algorithm, where the forward part recursively decomposes the probability distribution into a product of conditional distributions. Next we establish two different backward parts to our algorithm. In the first one we are able to simulate from the probability distribution, using the decomposed system. The second one enables us to calculate the marginal distributions for all the nodes in the Markov random field. All the approximations in our algorithm are controlled by a positive parameter, and when this parameter is equal to 0, our algorithm is by definition an exact algorithm. We investigate the performance of our algorithm by the CPU time, and by evaluating the quality of the approximations in various ways. As an example of the usage of our algorithm, we estimate an unknown picture from a degenerated version, using the marginal posterior mode estimate. This is a classical Bayesian problem.</p>

<p>Bayesian closed-skew Gaussian inversion is defined as a generalization of traditional Bayesian Gaussian inversion. Bayesian inversion is often used in seismic inversion, and the closed-skew model is able to capture the skewness in the variable of interest. Different stationary prior models are presented, but the generalization comes at a cost, simulation from high-dimensional pdfs and parameter inference from data is more complicated. An efficient algorithm to generate realizations from the high-dimensional closed-skew Gaussian distribution is presented. A full-likelihood is used for parameter estimation of stationary prior models under exponential dependence structure. The simulation algorithms and estimators are evaluated on synthetic examples. Also a closed-skew T-distribution is presented to include heavy tails in the pdf and the model is presented with some examples. In the last part the simulation algorithm, the different prior models and parameter estimators are demonstrated on real data from a well in the Sleipner Øst field. The full-likelihood estimator seems to be the best estimator for data with exponential dependence structure</p>

<p>Spectral discretization in space and time of the weak formulation of a partial differential equations (PDE) is studied. The exact solution to the PDE, with either Dirichlet or Neumann boundary conditions imposed, is approximated using high order polynomials. This is known as a spectral Galerkin method. The main focus of this work is the solution algorithm for the arising algebraic system of equations. A direct fast tensor-product solver is presented for the Poisson problem in a rectangular domain. We also explore the possibility of using a similar method in deformed domains, where the geometry of the domain is approximated using high order polynomials. Furthermore, time-dependent PDE's are studied. For the linear convection-diffusion equation in $mathbb{R}$ we present a tensor-product solver allowing for parallel implementation, solving $mathcal{O}(N)$ independent systems of equations. Lastly, an iterative tensor-product solver is considered for a nonlinear time-dependent PDE. For most algorithms implemented, the computational cost is $mathcal O (N^{p+1})$ floating point operations and a memory required of $mathcal O (N^{p})$ floating point numbers for $mathcal O (N^{p})$ unknowns. In this work we only consider $p=2$, but the theory is easily extended to apply in higher dimensions. Numerical results verify the expected convergence for both the iterative method and the spectral discretization. Exponential convergence is obtained when the solution and domain geometry are infinitely smooth.</p>

<p>Heterogeneous and parallel computing systems are increasingly appealing to high-performance computing. Among heterogeneous systems, the GPUs have become an attractive device for compute-intensive problems. Their many-core architecture, primarily customized for graphics processing, is now widely available through programming architectures that exploit parallelism in GPUs. We follow this new trend and attempt an implementation of a classical mathematical model describing incompressible single-phase fluid flow through a porous medium. The porous medium is an oil reservoir represented by means of corner-point grids. Important geological and mathematical properties of corner-point grids will be discussed. The model will also incorporate pressure- and rate-controlled wells to be used for some realistic simulations. Among the test models are the 10th SPE Comparative Solution Project Model 2. After deriving the underlying mathematical model, it will be discretised using the numerical technique of Mimetic Finite Difference methods. The heterogeneous system utilised is a desktop computer with an NVIDIA GPU, and the programming architecture to be used is CUDA, which will be described. Two different versions of the final discretised system have been implemented; a traditional way of using an assembled global stiffness sparse matrix, and a Matrix-free version, in which only the element stiffness matrices are used. The former version evaluates two GPU libraries; CUSP and THRUST. These libraries will be briefly decribed. The linear system is solved using the iterative Jacobi-preconditioned conjugate gradient method. Numerical tests on realistic and complex reservoir models shows significant performance benefits compared to corresponding CPU implementations.</p>