Matematisk institutt
http://hdl.handle.net/10852/5
Fri, 30 Jul 2021 07:20:19 GMT2021-07-30T07:20:19ZNonlinear model for waves on water covered by ice - Resonant interaction equations and analytical solution of zeroth harmonic “group line” measured in experiments
http://hdl.handle.net/10852/86577
Nonlinear model for waves on water covered by ice - Resonant interaction equations and analytical solution of zeroth harmonic “group line” measured in experiments
Salman, Faten
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/865772021-01-01T00:00:00ZOn subrecursive representation of irrational numbers: Contractors and Baire sequences
http://hdl.handle.net/10852/86560
On subrecursive representation of irrational numbers: Contractors and Baire sequences
Kristiansen, Lars
We study the computational complexity of three representations of irrational numbers: standard Baire sequences, dual Baire sequences and contractors. Our main results: Irrationals whose standard Baire sequences are of low computational complexity might have dual Baire sequences of arbitrarily high computational complexity, and vice versa, irrationals whose dual Baire sequences are of low complexity might have standard Baire sequences of arbitrarily high complexity. Further- more, for any subrecursive class S closed under primitive recursive oper- ations, the class of irrationals that have a contractor in S is exactly the class of irrationals that have both a standard and a dual Baire sequence in S. Our results implies that a subrecursive class closed under primitive recursive operations contains the continued fraction of an irrational number α if and only if there is a contractor for α in the class.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/865602021-01-01T00:00:00ZFinite-volume methods with dense neural networks
http://hdl.handle.net/10852/86536
Finite-volume methods with dense neural networks
Kvitvang, Åsmund Danielsen
Hyperbolic conservation laws are an important part in classical physics to be able to mathematically describe the actions of nature. To obtain approximate solutions of such problems, several numerical methods have been developed, most of which with both advantages and disadvantages in terms of accuracy, efficiency and implementation simplicity. Inspired by modern computer science we will in this thesis propose numerical methods based on flux approximations obtained by using dense neural networks (DNNs). We will investigate the accuracy and efficiency by performing experiments with Burgers' equation. The main result of this thesis is a proposed numerical method for approximating solutions of two-dimensional nonlinear conservation laws. As there does not exist exact solution formulas of such two-dimensional problems, a possible approach is to use fine-resolution solvers in order to properly approximate the solutions. These solvers are extremely time consuming, and the hope is that the use of pre-trained DNN models will lead to a precise and efficient numerical method. We will also explore the possibility of using a physics-informed loss-function for approximating solutions of one-dimensional conservation laws, and further discuss how this may be applied to the two-dimensional methods. The DNN based numerical methods tested in this thesis yielded promising results with respect to both accuracy and efficiency. Due to time limitations of this study we have restricted ourselves to only studying Burgers' equation with a narrow sample of parameters. Thus, some uncertainty follows with the results, and thereby uncertainty in the conclusions. However, there are strong indications that the proposed models are valuable, given the right set of parameters.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/865362021-01-01T00:00:00ZStochastic Modelling in Energy Markets - From the Spot Price to Derivative Contracts
http://hdl.handle.net/10852/86437
Stochastic Modelling in Energy Markets - From the Spot Price to Derivative Contracts
Lavagnini, Silvia
The production of renewable energy is growing world-wide, and -- as a result -- power production is becoming increasingly dependent on weather factors such as temperature, wind and precipitation. All of these factors are hard to predict, and this causes power prices to change rapidly and unpredictably, and makes the modelling of financial risk in energy markets particularly challenging. This thesis develops new models and tools to be used in this direction.
Energy markets can be divided into three main sectors: there is (1) a spot market for short-term delivery contracts, (2) a forward market for delivery in a future time at a price set today, and (3) an option market where the contracts traded allow, but not oblige, the buyer to buy or sell the asset in a future time at a price set today. Buying and selling electricity in these markets, while managing the financial risk, requires accurate mathematical models.
This thesis is concerned with the modelling of these markets. It both develops concrete models and more abstract mathematical tools which can be used for this challenging task. In particular, it focuses on spot price modelling, by taking into account the dependence between spot price behaviour and weather variables such as wind speed. Moreover, it focuses on forward price modelling and pricing of options written on forward contracts with delivery period, which are typical in the energy markets. Two central challenges which are addressed in this thesis are model accuracy and computational complexity, both of which are improved upon by using deep learning.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/864372021-01-01T00:00:00ZAccurate discretization of poroelasticity without Darcy stability -- Stokes–Biot stability revisited
http://hdl.handle.net/10852/86364
Accurate discretization of poroelasticity without Darcy stability -- Stokes–Biot stability revisited
Mardal, Kent-Andre; Rognes, Marie E.; Thompson, Travis
Abstract In this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/863642021-01-01T00:00:00ZHeisenberg modules and Balian–Low theorems - Applications of operator algebras to Gabor analysis
http://hdl.handle.net/10852/86266
Heisenberg modules and Balian–Low theorems - Applications of operator algebras to Gabor analysis
Enstad, Ulrik Bo Rufus
The main way to study a periodic signal is to decompose it into a sum of simple signals, namely sine waves. However, when a signal changes substantially over time, such as a piece of music, different methods are needed. One method is to use Gabor frames. A Gabor frame represents a given signal in a way that emphasizes the signal’s frequency content at each point in time. For instance, a Gabor frame will represent an audio signal analogously to how sheet music is written.
Constructing good Gabor frames is not an easy task, and this problem has connections to many other areas in mathematics. In my dissertation, I have connected this problem to an area called operator algebras. A basic theorem about Gabor frames is the Balian-Low theorem, which is rooted in the uncertainty principle from quantum mechanics. I have shown that this theorem has a conceptual interpretation in operator algebras. Moreover, one can generally talk about Gabor frames in an abstract setting, namely in the context of an abelian topological group. I have completely classified the groups to which the Balian-Low theorem extends. One of the groups to which it extends is the rational adele group from number theory.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/862662020-01-01T00:00:00ZClustering and automatic labelling within time series of cate- gorical observations - with an application to marine log messages
http://hdl.handle.net/10852/85991
Clustering and automatic labelling within time series of cate- gorical observations - with an application to marine log messages
Gramuglia, Emanuele; Storvik, Geir Olve; Stakkeland, Morten
System logs or log files containing textual messages with associated time stamps are generated by many technologies and systems. The clustering technique proposed in this paper provides a tool to discover and identify patterns or macrolevel events in this data. The motivating application is logs generated by frequency converters in the propulsion system on a ship, while the general setting is fault identification and classification in complex industrial systems. The paper introduces an offline approach for dividing a time series of log messages into a series of discrete segments of random lengths. These segments are clustered into a limited set of states. A state is assumed to correspond to a specific operation or condition of the system, and can be a fault mode or a normal operation. Each of the states can be associated with a specific, limited set of messages, where messages appear in a random or semi‐structured order within the segments. These structures are in general not defined a priori. We propose a Bayesian hierarchical model where the states are characterised both by the temporal frequency and the type of messages within each segment. An algorithm for inference based on reversible jump MCMC is proposed. The performance of the method is assessed by both simulations and operational data.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/859912021-01-01T00:00:00ZA mean counting function for Dirichlet series and compact composition operators
http://hdl.handle.net/10852/85967
A mean counting function for Dirichlet series and compact composition operators
Brevig, Ole Fredrik; Perfekt, Karl-Mikael
We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/859672021-01-01T00:00:00ZOn a rough perturbation of the Navier–Stokes system and its vorticity formulation
http://hdl.handle.net/10852/85863
On a rough perturbation of the Navier–Stokes system and its vorticity formulation
Nilssen, Torstein; Hofmanova, Martina; Leahy, James-Michael
We introduce a rough perturbation of the Navier–Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of strong solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong–Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/858632021-01-01T00:00:00ZStatistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry
http://hdl.handle.net/10852/85860
Statistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry
Lawrence, Christopher; Trulsen, Karsten; Gramstad, Odin
Experimental and numerical evidence have shown that nonuniform bathymetry may alter significantly the statistical properties of surface elevation in irregular wave fields. The probability of “rogue” waves is increased near the edge of the upslope as long-crested irregular waves propagate into shallower water. The present paper studies the statistics of wave kinematics in long-crested irregular waves propagating over a shoal with a Monte Carlo approach. High order spectral method is employed as wave propagation model, and variational Boussinesq model is employed to calculate wave kinematics. The statistics of horizontal fluid velocity can be different from statistics in surface elevation as the waves propagate over uneven bathymetry. We notice strongly non-Gaussian statistics when the depth changes abruptly in sufficiently shallow water. We find an increase in kurtosis in the horizontal velocity around the downslope area. Furthermore, the effects of the bottom slope with different incoming waves are discussed in terms of kurtosis and skewness. Finally, we investigate the evolution of kurtosis and skewness of the horizontal velocity over a sloping bottom in a deeper regime. The vertical variation of these statistical quantities is also presented.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/858602021-01-01T00:00:00ZIndependent increment processes: a multilinearity preserving property
http://hdl.handle.net/10852/85799
Independent increment processes: a multilinearity preserving property
Benth, Fred Espen; Detering, Nils; Krühner, Paul
We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space B. It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of B being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/857992020-01-01T00:00:00ZAnalysis and approximation of mixed-dimensional pdes on 3d-1d domains coupled with lagrange multipliers
http://hdl.handle.net/10852/85779
Analysis and approximation of mixed-dimensional pdes on 3d-1d domains coupled with lagrange multipliers
Kuchta, Miroslav; Laurino, Federica; Mardal, Kent-Andre; Zunino, Paolo
Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled problem. Such a problem poses several challenges from the standpoint of existence of solutions and numerical approximation. For the coupling conditions across dimensions, we consider the combination of essential and natural conditions, which are basically the combination of Dirichlet and Neumann conditions. To ensure a meaningful formulation of such conditions, we use the Lagrange multiplier method suitably adapted to the mixed-dimensional case. The well-posedness of the resulting saddle-point problem is analyzed. Then, we address the numerical approximation of the problem in the framework of the finite element method. The discretization of the Lagrange multiplier space is the main challenge. Several options are proposed, analyzed, and compared, with the purpose of determining a good balance between the mathematical properties of the discrete problem and flexibility of implementation of the numerical scheme. The results are supported by evidence based on numerical experiments.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/857792021-01-01T00:00:00ZThe polynomial endomorphisms of graph algebras
http://hdl.handle.net/10852/85778
The polynomial endomorphisms of graph algebras
Johansen, Rune; Sørensen, Adam Peder Wie; Szymanski, Wojciech
-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/857782020-01-01T00:00:00ZAsymptotic, Convergent, and Exact Truncating Series Solutions of the Linear Shallow Water Equations for Channels with Power Law Geometry
http://hdl.handle.net/10852/85685
Asymptotic, Convergent, and Exact Truncating Series Solutions of the Linear Shallow Water Equations for Channels with Power Law Geometry
Pedersen, Geir Kleivstul
The present study was originally motivated by some intriguing exact solutions for waves propagating in nonuniform media. In particular, for special depth profiles reflected waves did not appear and ray theory became exact. Herein, geometrical optics is employed to obtain asymptotic series for waves of general shapes in nonuniform narrow channels, within the framework of linear shallow water theory. While being kept simple, the series incorporate higher order contributions that may describe the evolution of waves with high accuracy. The higher orders also contain a secondary wave system. For selected classes of geometries and wave shapes explicit solutions are calculated and compared to numerical solutions. Apart from the vicinity of shorelines, say, higher order expansions generally may provide very accurate approximations to the full solutions. The asymptotic series are analyzed for different wave shapes and are found to be convergent for cases where the basic wave profiles have compact support (finite length). A number of new, closed form, exact solutions are also found. The asymptotic expansion is put into a context by employing it for the transmission of waves from a uniform channel section into a nonuniform one. Additional results and side topics are presented in a supplement.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/856852021-01-01T00:00:00ZVoronoi cells of varieties
http://hdl.handle.net/10852/85682
Voronoi cells of varieties
Cifuentes, Diego; Ranestad, Kristian; Sturmfels, Bernd; Weinstein, Madeleine
Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/856822020-01-01T00:00:00ZA financial market with singular drift and no arbitrage
http://hdl.handle.net/10852/85620
A financial market with singular drift and no arbitrage
Agram, Nacira; Øksendal, Bernt
We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay θ>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as θ>0. This implies that there is no arbitrage in the market in that case. However, when θ goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/856202020-01-01T00:00:00ZInfluence of Submarine Landslide Failure and Flow on Tsunami Genesis
http://hdl.handle.net/10852/85503
Influence of Submarine Landslide Failure and Flow on Tsunami Genesis
Zengaffinen-Morris, Thomas
Submarine mass movements are important sources for tsunamis with potential destructive consequences at coastlines. A famous example from historical records is, for instance, the 1929 Grand Banks tsunami in Canada. This tsunami of several meters' height hit, among other coasts, the south coast of Newfoundland. The event resulted in fatalities and destroyed homes. This doctoral thesis aims to relate the properties of submarine mass movements to tsunami genesis. Thereby, we apply a numerical landslide model that treats the mass movement as a deformable fluid. The main research finding is that the tsunami genesis is sensitive to, among other parameters, the initial yield strength of the mass. The lower the initial yield strength, the larger the velocity and acceleration, which induces a larger maximum tsunami height. The work of this thesis is important, because a better understanding of the physical processes that drive the tsunami genesis can bring prognostic tsunami modelling a step forward.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10852/855032021-01-01T00:00:00ZPartially linear monotone methods with automatic variable selection and monotonicity direction discovery
http://hdl.handle.net/10852/85488
Partially linear monotone methods with automatic variable selection and monotonicity direction discovery
Engebretsen, Solveig; Glad, Ingrid Kristine
In many statistical regression and prediction problems, it is reasonable to assume monotone relationships between certain predictor variables and the outcome. Genomic effects on phenotypes are, for instance, often assumed to be monotone. However, in some settings, it may be reasonable to assume a partially linear model, where some of the covariates can be assumed to have a linear effect. One example is a prediction model using both high‐dimensional gene expression data, and low‐dimensional clinical data, or when combining continuous and categorical covariates. We study methods for fitting the partially linear monotone model, where some covariates are assumed to have a linear effect on the response, and some are assumed to have a monotone (potentially nonlinear) effect. Most existing methods in the literature for fitting such models are subject to the limitation that they have to be provided the monotonicity directions a priori for the different monotone effects. We here present methods for fitting partially linear monotone models which perform both automatic variable selection, and monotonicity direction discovery. The proposed methods perform comparably to, or better than, existing methods, in terms of estimation, prediction, and variable selection performance, in simulation experiments in both classical and high‐dimensional data settings.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/854882020-01-01T00:00:00ZCoupling time-indexed and big-M formulations for real-time train scheduling during metro service disruptions
http://hdl.handle.net/10852/85483
Coupling time-indexed and big-M formulations for real-time train scheduling during metro service disruptions
Huang, Yeran; Mannino, Carlo; Yang, Lixing; Tang, Tao
Track disruptions in metro systems may lead to severe train delays with many passengers stranded at platforms, unable to board on overloaded trains. Dispatchers may put in place different recovery actions, such as alternating train directions and allowing short turns. The objective is to alleviate the inconvenience for passengers and to regain the nominal train regularity. To characterize this process, this paper develops nonlinear mixed integer programming (NMIP) models with two different recovery strategies to reschedule trains during the disruption. For solving models in real time, the hybrid formulation, which couples big-M and time-indexed formulations, is proposed to linearize the proposed model as the mixed integer linear programming (MILP) model. Then, a two-stage approach is designed for handling the real-time detected information (like dynamic arriving passengers and end time of the disruption), including offline task (to select the best recovery strategy) and online task (to implement the best strategy and update timetable). Finally, the numerical experiments from Beijing metro Line 2 are implemented to verify the performance and effectiveness of the proposed hybrid formulation and two-stage approach.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/854832020-01-01T00:00:00ZThe mechanisms behind perivascular fluid flow
http://hdl.handle.net/10852/85482
The mechanisms behind perivascular fluid flow
Daversin-Catty, Cecile; Vinje, Vegard; Mardal, Kent-Andre; Rognes, Marie E.
Flow of cerebrospinal fluid (CSF) in perivascular spaces (PVS) is one of the key concepts involved in theories concerning clearance from the brain. Experimental studies have demonstrated both net and oscillatory movement of microspheres in PVS (Mestre et al. (2018), Bedussi et al. (2018)). The oscillatory particle movement has a clear cardiac component, while the mechanisms involved in net movement remain disputed. Using computational fluid dynamics, we computed the CSF velocity and pressure in a PVS surrounding a cerebral artery subject to different forces, representing arterial wall expansion, systemic CSF pressure changes and rigid motions of the artery. The arterial wall expansion generated velocity amplitudes of 60–260 μ m/s, which is in the upper range of previously observed values. In the absence of a static pressure gradient, predicted net flow velocities were small (<0.5 μ m/s), though reaching up to 7 μ m/s for non-physiological PVS lengths. In realistic geometries, a static systemic pressure increase of physiologically plausible magnitude was sufficient to induce net flow velocities of 20–30 μ m/s. Moreover, rigid motions of the artery added to the complexity of flow patterns in the PVS. Our study demonstrates that the combination of arterial wall expansion, rigid motions and a static CSF pressure gradient generates net and oscillatory PVS flow, quantitatively comparable with experimental findings. The static CSF pressure gradient required for net flow is small, suggesting that its origin is yet to be determined.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10852/854822020-01-01T00:00:00Z