DSpace Community:http://hdl.handle.net/2381/4452017-06-29T02:10:39Z2017-06-29T02:10:39ZVirtual Element MethodsSutton, Oliver Jameshttp://hdl.handle.net/2381/399552017-06-27T02:27:11Z2017-06-26T13:44:08ZTitle: Virtual Element Methods
Authors: Sutton, Oliver James
Abstract: In this thesis we study the Virtual Element Method, a recent generalisation of the standard conforming Finite Element Method offering high order approximation spaces on computational meshes consisting of general polygonal or polyhedral elements. Our particular focus is on developing the tools required to use the method as the foundation of mesh adaptive algorithms which are able to take advantage of the flexibility offered by such general meshes.
We design virtual element discretisations of certain classes of linear second order elliptic and parabolic partial differential equations, and present a detailed exposition of their implementation aspects. An outcome of this is a concise and usable 50-line MATLAB implementation of a virtual element method for solving a model elliptic problem on general polygonal meshes, the code for which is included as an appendix. Optimal order convergence rates in the H1 and L2 norms are proven for the discretisation of elliptic problems. Alongside these, we derive fully computable residual-type a posteriori estimates of the error measured in the H1 and L2 norms for the methods we develop for elliptic problems, and in the L2(0; T;H1) and L∞(0; T;L2) norms for parabolic problems. In deriving the L∞(0; T;L2) error estimate, we introduce a new technique (which translates naturally back into the setting of conventional finite element methods) to produce estimates with effectivities which become constant for long time simulations. Mesh adaptive algorithms, designed around these methods and computable error estimates, are proposed and numerically assessed in a variety of challenging stationary and time-dependent scenarios.
We further propose a virtual element discretisation and computable coarsening/refinement indicator for a system of semilinear parabolic partial differential equations which we apply to a Lotka-Volterra type model of interacting species. These components form the basis of an adaptive method which we use to reveal a variety of new pattern-forming mechanisms in the cyclic competition model.2017-06-26T13:44:08ZDynamics of a Two Subpopulations System Including ImmigrationZincenko, A.Petrovskii, Sergeihttp://hdl.handle.net/2381/399422017-06-23T02:20:27Z2017-06-22T14:22:55ZTitle: Dynamics of a Two Subpopulations System Including Immigration
Authors: Zincenko, A.; Petrovskii, Sergei
Abstract: The phenomenon of replacement migration into declining population prompts development of multicomponent models in population dynamics. We propose a simple model of population including resident and migrant components with migration flow as an external input. The main assumption is that offspring that are born to migrants will have the same vital rates as the resident population. The proposed model is based on partial differential equation to take into account the age structure of the population. The formulae for exact solutions are derived. We focus on the case when native population declines in the absence of migration. Assuming a sufficiently large constant growth rate of migration we obtain asymptotic solutions as t → â. Using the asymptotic solutions, we have calculated "critical" value of growth rate of migrant inflow that is the value that provides, as time tends to infinity, the equal number of residents and migrants in the population. We provide numerical illustrations using demographic data for Germany in 2010.2017-06-22T14:22:55ZWhich Random Walk is Faster? Methods to Compare Different Step Length Distributions in Individual Animal MovementChoules, J. D.Petrovskii, Sergeihttp://hdl.handle.net/2381/399402017-06-23T02:20:22Z2017-06-22T14:04:28ZTitle: Which Random Walk is Faster? Methods to Compare Different Step Length Distributions in Individual Animal Movement
Authors: Choules, J. D.; Petrovskii, Sergei
Abstract: Good understanding of individual animal movement is needed in the context of epidemiology in order to predict the rate of spread of infectious diseases. It is also required for problems arising in nature conservation, biological invasion, pest monitoring, etc. A question that often appears in the centre of the movement studies is which movement pattern is 'faster' or more efficient. For instance, it is widely believed that the pattern quantified by a power law distribution of movement steps is faster than the Brownian motion. Here we show that the answer to this question may be not so straightforward and depends on the way how different step length distributions are compared.2017-06-22T14:04:28ZSpectral Sequences for Hochschild cohomology and graded centers of derived categoriesNeumann, FrankSzymik, Markushttp://hdl.handle.net/2381/399082017-06-15T02:22:10Z2017-06-14T15:08:22ZTitle: Spectral Sequences for Hochschild cohomology and graded centers of derived categories
Authors: Neumann, Frank; Szymik, Markus
Abstract: The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general. To illustrate this, we discuss modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-06-14T15:08:22ZAdaptive radial basis function interpolation using an error indicatorZhang, QiZhao, YangzhangLevesley, Jeremyhttp://hdl.handle.net/2381/398102017-05-18T02:21:52Z2017-05-17T15:34:19ZTitle: Adaptive radial basis function interpolation using an error indicator
Authors: Zhang, Qi; Zhao, Yangzhang; Levesley, Jeremy
Abstract: In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.2017-05-17T15:34:19ZA Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equationBanjai, LehelGeorgoulis, Emmanuil H.Lijoka, Oluwaseunhttp://hdl.handle.net/2381/397852017-05-16T02:22:27Z2017-05-15T13:54:03ZTitle: A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation
Authors: Banjai, Lehel; Georgoulis, Emmanuil H.; Lijoka, Oluwaseun
Abstract: A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339-363; Comput. Methods Appl. Mech. Engrg., 84 (1990), pp. 327-348]. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behavior of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions d = 1 and d = 2. Numerical experiments highlight the effectivness of the Trefftz method in problems with energy at high frequencies.2017-05-15T13:54:03ZOn the non-parallel instability of the rotating-sphere boundary layerSegalini, AntonioGarrett, Stephen J.http://hdl.handle.net/2381/397782017-05-16T02:22:21Z2017-05-15T10:38:27ZTitle: On the non-parallel instability of the rotating-sphere boundary layer
Authors: Segalini, Antonio; Garrett, Stephen J.
Abstract: We present a new solution for the steady boundary-layer flow over the rotating sphere that also accounts for the eruption of the boundary layer at the equator and other higher-order viscous effects. Non-parallel corrections to the local type I and type II convective instability modes of this flow are also computed as a function of spin rate. Our instability results are associated with the previously observed spiral vortices and remarkable agreement between our predictions of the number of vortices and experimental observations is found. Vortices travelling at 70 %–80 % of the local surface speed are found to be the most amplified for sufficient spin rates, also consistent with prior experimental observations.2017-05-15T10:38:27ZApplications of Quaternionic Holomorphic Geometry to minimal surfacesLeschke, K.Moriya, K.http://hdl.handle.net/2381/397752017-05-13T03:13:53Z2017-05-12T15:29:05ZTitle: Applications of Quaternionic Holomorphic Geometry to minimal surfaces
Authors: Leschke, K.; Moriya, K.
Abstract: In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface.2017-05-12T15:29:05ZInteraction of human migration and wealth distributionVolpert, V.Petrovskii, SergeiZincenko, A.http://hdl.handle.net/2381/397602017-05-11T02:22:46Z2017-05-10T15:05:00ZTitle: Interaction of human migration and wealth distribution
Authors: Volpert, V.; Petrovskii, Sergei; Zincenko, A.
Abstract: Dynamics of human populations depends on various economical and social factors. Their migration is partially determined by the economical conditions and it can also influence these conditions. This work is devoted to the analysis of the interaction of human migration and wealth distribution. The model consists of a system of equations for the population density and for the wealth distribution with conventional diffusion terms and with cross diffusion terms describing human migration determined by the wealth gradient and wealth flux determined by human migration. Wealth production and consumption depend on the population density while the natality and mortality rates depend on the level of wealth. In the absence of cross diffusion terms, dynamics of solutions is described by travelling wave solutions of the corresponding reaction-diffusion systems of equations. We show persistence of such solutions for sufficiently small cross diffusion coefficients. This result is based on the perturbation methods and on the spectral properties of the linearized operators.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above. Following the embargo period the above license will apply.2017-05-10T15:05:00ZFortune favours the brave: Movement responses shape demographic dynamics in strongly competing populationsPotts, Jonathan R.Petrovskii, Sergei V.http://hdl.handle.net/2381/397592017-05-11T02:22:45Z2017-05-10T14:56:57ZTitle: Fortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations
Authors: Potts, Jonathan R.; Petrovskii, Sergei V.
Abstract: Animal movement is a key mechanism for shaping population dynamics. The effect of interactions between competing animals on a population's survival has been studied for many decades. However, interactions also affect an animal's subsequent movement decisions. Despite this, the indirect effect of these decisions on animal survival is much less well-understood. Here, we incorporate movement responses to foreign animals into a model of two competing populations, where inter-specific competition is greater than intra-specific competition. When movement is diffusive, the travelling wave moves from the stronger population to the weaker. However, by incorporating behaviourally induced directed movement towards the stronger population, the weaker one can slow the travelling wave down, even reversing its direction. Hence movement responses can switch the predictions of traditional mechanistic models. Furthermore, when environmental heterogeneity is combined with aggressive movement strategies, it is possible for spatially segregated co-existence to emerge. In this situation, the spatial patterns of the competing populations have the unusual feature that they are slightly out-of-phase with the environmental patterns. Finally, incorporating dynamic movement responses can also enable stable co-existence in a homogeneous environment, giving a new mechanism for spatially segregated co-existence.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above. Following the embargo period the above license applies.2017-05-10T14:56:57ZBabuška-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshesGeorgoulis, EmmanuilMakridakis, CharalambosPryer, Tristanhttp://hdl.handle.net/2381/397082017-04-28T02:21:13Z2017-04-27T14:22:23ZTitle: Babuška-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshes
Authors: Georgoulis, Emmanuil; Makridakis, Charalambos; Pryer, Tristan
Abstract: We prove the inf-sup stability of the interior penalty class of discontinuous Galerkin schemes in unbalanced mesh-dependent norms, under a mesh condition allowing for a general class of meshes, which includes many examples of geometrically graded element neighbourhoods. The inf-sup condition results in the stability of the interior penalty Ritz projection in $L^2$ as well as, for the first time, quasi-best approximations in the $L^2$-norm which in turn imply a priori error estimates that do not depend on the global maximum meshsize in that norm. Some numerical experiments are also given.2017-04-27T14:22:23ZSmall Cocycles, Fine Torus Fibrations, and a Z^2 Subshift with NeitherClark, AlexSadun, Lorenzohttp://hdl.handle.net/2381/396732017-04-20T02:18:01Z2017-04-19T15:45:19ZTitle: Small Cocycles, Fine Torus Fibrations, and a Z^2 Subshift with Neither
Authors: Clark, Alex; Sadun, Lorenzo
Abstract: Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free ZdZd actions on Cantor sets admit “small cocycles.” These represent classes in H1H1 that are mapped to small vectors in RdRd by the Ruelle–Sullivan (RS) map. We show that there exist Z2Z2 actions where no such small cocycles exist, and where the image of H1H1 under RS is Z2Z2 . Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of “virtual eigenvalues,” i.e., elements of RdRd that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.2017-04-19T15:45:19ZCalculating Exceedance Probabilities Using a Distributionally Robust MethodFaridafshin, FarzadGrechuk, BogdanNaess, Arvidhttp://hdl.handle.net/2381/396692017-06-14T13:18:30Z2017-04-19T14:49:39ZTitle: Calculating Exceedance Probabilities Using a Distributionally Robust Method
Authors: Faridafshin, Farzad; Grechuk, Bogdan; Naess, Arvid
Abstract: Calculation of exceedance probabilities or the inverse problem of finding the level corresponding to a given exceedance probability occurs in many practical applications. For instance, it is often of interest in offshore engineering to evaluate the wind, wave, current, and sea ice properties with annual exceedance probabilities of, e.g., 10−1, 10−2, and 10−3, or so-called 10-year, 100-year, and 1000-year values. A methodology is provided in this article to calculate a tight upper bound of the exceedance probability, given any probability distribution from a wide range of commonly used distributions. The approach is based on a generalization of the Chebyshev inequality for the class of distributions with a logarithmically concave cumulative distribution function, and has the potential to relieve the often-debated exercise of determining an appropriate probability distribution function based on limited data, particularly in terms of tail behavior. Two numerical examples are provided for illustration.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-04-19T14:49:39ZRegimes of electrostatic collapse of a highly charged polyelectrolyte in a poor solventTom, Anvy MolyVemparala, SatyavaniRajesh, R.Brilliantov, Nikolai V.http://hdl.handle.net/2381/396512017-04-12T02:22:08Z2017-04-11T11:12:38ZTitle: Regimes of electrostatic collapse of a highly charged polyelectrolyte in a poor solvent
Authors: Tom, Anvy Moly; Vemparala, Satyavani; Rajesh, R.; Brilliantov, Nikolai V.
Abstract: We perform extensive molecular dynamics simulations of a highly charged, collapsed, flexible polyelectrolyte chain in a poor solvent for the case when the electrostatic interactions, characterized by the reduced Bjerrum length ℲB, are strong. We find the existence of several sub-regimes in the dependence of the gyration radius of the chain Rg on ℲB characterized by Rg ∼ Ⅎ−γB. In contrast to a good solvent, the exponent γ for a poor solvent crucially depends on the size and valency of the counterions. To explain the different sub-regimes, we generalize the existing counterion fluctuation theory by including a more complete account of all possible volume interactions in the free energy of the polyelectrolyte chain. We also show that the presence of condensed counterions modifies the effective attraction among the chain monomers and modulates the sign of the second virial coefficient under poor solvent conditions.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-04-11T11:12:38ZBeyond Navier–Stokes equations: capillarity of ideal gasGorban, Alexander N.Karlin, I. V.http://hdl.handle.net/2381/396502017-04-12T02:22:08Z2017-04-11T11:06:09ZTitle: Beyond Navier–Stokes equations: capillarity of ideal gas
Authors: Gorban, Alexander N.; Karlin, I. V.
Abstract: The system of Navier–Stokes–Fourier equations is one of the most celebrated systems of equations in modern science. It describes dynamics of fluids in the limit when gradients of density, velocity and temperature are sufficiently small, and loses its applicability when the flux becomes so non-equilibrium that the changes of velocity, density or temperature on the length compatible with the mean free path are non-negligible. The question is: how to model such fluxes? This problem is still open. (Despite the fact that the first ‘final equations of motion’ modified for analysis of thermal creep in rarefied gas were proposed by Maxwell in 1879.) There are, at least, three possible answers: (i) use molecular dynamics with individual particles, (ii) use kinetic equations, like Boltzmann’s equation or (iii) find a new system of equations for description of fluid dynamics with better accounting of non-equilibrium effects. These three approaches work at different scales. We explore the third possibility using the recent findings of capillarity of internal layers in ideal gases and of saturation effect in dissipation (there is a limiting attenuation rate for very short waves in ideal gas and it cannot increase infinitely). One candidate equation is discussed in more detail, the Korteweg system proposed in 1901. The main ideas and approaches are illustrated by a kinetic system for which the problem of reduction of kinetics to fluid dynamics is analytically solvable.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-04-11T11:06:09ZModern Mathematical Methods for Actuarial SciencesKaya, Ahmethttp://hdl.handle.net/2381/396132017-04-04T02:20:54Z2017-04-03T08:09:18ZTitle: Modern Mathematical Methods for Actuarial Sciences
Authors: Kaya, Ahmet
Abstract: In the ruin theory, premium income and outgoing claims play an important role. We introduce several ruin type mathematical models and apply various mathematical methods to find optimal premium price for the insurance companies. Quantum theory is one of the significant novel approaches to compute the finite time non-ruin probability. More exactly, we apply the discrete space Quantum mechanics formalism (see main thesis for formalism) and continuous space Quantum mechanics formalism (see main thesis for formalism) with the appropriately chosen Hamiltonians.
Several particular examples are treated via the traditional basis and quantum mechanics formalism with the different eigenvector basis. The numerical results are also obtained using the path calculation method and compared with the stochastic modeling results.
In addition, we also construct various models with interest rate. For these models, optimal premium prices are stochastically calculated for independent and dependent claims with different dependence levels by using the Frank copula method.2017-04-03T08:09:18ZAdaptive large-scale mantle convection simulationsCox, Samuel Peterhttp://hdl.handle.net/2381/395712017-03-28T02:17:11Z2017-03-27T10:20:14ZTitle: Adaptive large-scale mantle convection simulations
Authors: Cox, Samuel Peter
Abstract: The long-term motion of the Earth's mantle is of considerable interest to geologists and geodynamists in explaining the evolution of the planet and its internal and surface history. The inaccessible nature of the mantle necessitates the use of computer simulations to further our understanding of the processes underlying the motion of tectonic plates.
Numerical methods employed to solve the equations describing this motion lead to linear systems of a size which stretch the current capabilities of supercomputers to their limits. Progress towards the satisfactory simulation of this process is dependent upon the use of new mathematical and computational ideas in order to bring the largest problems within the reach of current computer architectures.
In this thesis we present an implementation of the discontinuous Galerkin method, coupled to a more traditional finite element method, for the simulation of this system. We also present an a posteriori error estimate for the convection-diffusion equation without reaction, using an exponential fitting technique and artificial reaction to relax the restrictions upon the derivative of the convection field that are usually imposed within the existing literature. This error bound is used as the basis of an h-adaptive mesh refinement strategy. We present an implementation of the calculation of this bound alongside the simulation and the indicator, in a parallelised C++ code, suitable for use in a distributed computing setting.
Finally, we present an implementation of the discontinuous Galerkin method into the community code ASPECT, along with an adaptivity indicator based upon the proven a posteriori error bound. We furnish both implementations with numerical examples to explore the applicability of these methods to a number of circumstances, with the aim of reducing the computational cost of large mantle convection simulations.2017-03-27T10:20:14ZA dissipative force between colliding viscoelastic bodies: Rigorous approachBrilliantov, Nikolay V.Pimenova, Anastasiya V.Goldobin, Denis S.http://hdl.handle.net/2381/395442017-03-23T03:21:13Z2017-03-22T13:48:38ZTitle: A dissipative force between colliding viscoelastic bodies: Rigorous approach
Authors: Brilliantov, Nikolay V.; Pimenova, Anastasiya V.; Goldobin, Denis S.
Abstract: A collision of viscoelastic bodies is analysed within a mathematically rigorous approach. We develop a perturbation scheme to solve continuum mechanics equation, which deals simultaneously with strain and strain rate in the bulk of the bodies' material. We derive dissipative force that acts between particles and express it in terms of particles' deformation, deformation rate and material parameters. It differs noticeably from the currently used dissipative force, found within the quasi-static approximation and does not suffer from inconsistencies of this approximation. The proposed approach may be used for other continuum mechanics problems where the bulk dissipation is addressed.2017-03-22T13:48:38ZA Generalist Predator Regulating Spread of a Wildlife Disease: Exploring Two Infection Transmission ScenariosSen, M.Banerjee, M.Morozov, A.http://hdl.handle.net/2381/395432017-03-23T03:21:13Z2017-03-22T13:39:47ZTitle: A Generalist Predator Regulating Spread of a Wildlife Disease: Exploring Two Infection Transmission Scenarios
Authors: Sen, M.; Banerjee, M.; Morozov, A.
Abstract: Ecoepidemiology is a well-developed branch of theoretical ecology, which explores interplay between the trophic interactions and the disease spread. In most ecoepidemiological models, however, the authors assume the predator to be a specialist, which consumes only a single prey species. In few existing papers, in which the predator was suggested to be a generalist, the alternative food supply was always considered to be constant. This is obviously a simplification of reality, since predators can often choose between a number of different prey. Consumption of these alternative prey can dramatically change their densities and strongly influence the model predictions. In this paper, we try to bridge the gap and explore a generic eco-epidemiological system with a generalist predator, where the densities of all prey are dynamical variables. The model consists of two prey species, one of which is subject to an infectious disease, and a predator, which consumes both prey species. We investigate two main scenarios of infection transmission mode: (i) the disease transmission rate is predator independent and (ii) the transmission rate is a function of predator density. For both scenarios we fulfil an extensive bifurcation analysis. We show that including a second dynamical prey in the system can drastically change the dynamics of the single prey case. In particular, the presence of a second prey impedes disease spread by decreasing the basic reproduction number and can result in a substantial drop of the disease prevalence. We demonstrate that with efficient consumption of the second prey species by the predator, the predator-dependent disease transmission can not destabilize interactions, as in the case with a specialist predator. Interestingly, even if the population of the second prey eventually vanishes and only one prey species finally remains, the system with two prey species may exhibit different properties to those of the single prey system.2017-03-22T13:39:47ZModelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special IssueMorozov, A.http://hdl.handle.net/2381/395422017-03-23T03:21:13Z2017-03-22T10:03:33ZTitle: Modelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue
Authors: Morozov, A.2017-03-22T10:03:33ZNoise-Produced Patterns in Images Constructed from Magnetic Flux Leakage DataGoldobin, D. S.Pimenova, A. V.Levesley, J.Elkington, P.Bacciarelli, M.http://hdl.handle.net/2381/395412017-03-23T03:21:12Z2017-03-22T09:58:52ZTitle: Noise-Produced Patterns in Images Constructed from Magnetic Flux Leakage Data
Authors: Goldobin, D. S.; Pimenova, A. V.; Levesley, J.; Elkington, P.; Bacciarelli, M.
Abstract: Magnetic flux leakage measurements help identify the position, size and shape of corrosion-related defects in steel casings used to protect boreholes drilled into oil and gas reservoirs. Images constructed from magnetic flux leakage data contain patterns related to noise inherent in the method. We investigate the patterns and their scaling properties for the case of delta-correlated input noise, and consider the implications for the method’s ability to resolve defects. The analytical evaluation of the noise-produced patterns is made possible by model reduction facilitated by large-scale approximation. With appropriate modification, the approach can be employed to analyze noise-produced patterns in other situations where the data of interest are not measured directly, but are related to the measured data by a complex linear transform involving integrations with respect to spatial coordinates.
Description: Mathematics Subject Classification: 78A30 / 78M34 / 60G602017-03-22T09:58:52ZMagnetic Flux Leakage Method: Large-Scale ApproximationPimenova, A. V.Goldobin, D. S.Levesley, J.Ivantsov, A. O.Elkington, P.Bacciarelli, M.http://hdl.handle.net/2381/395402017-03-23T03:21:12Z2017-03-22T09:55:42ZTitle: Magnetic Flux Leakage Method: Large-Scale Approximation
Authors: Pimenova, A. V.; Goldobin, D. S.; Levesley, J.; Ivantsov, A. O.; Elkington, P.; Bacciarelli, M.
Abstract: We consider the application of the magnetic flux leakage (MFL) method to the detection of defects in ferromagnetic (steel) tubulars. The problem setup corresponds to the cases where the distance from the casing and the point where the magnetic field is measured is small compared to the curvature radius of the undamaged casing and the scale of inhomogeneity of the magnetic field in the defect-free case. Mathematically this corresponds to the planar ferromagnetic layer in a uniform magnetic field oriented along this layer. Defects in the layer surface result in a strong deformation of the magnetic field, which provides opportunities for the reconstruction of the surface profile from measurements of the magnetic field. We deal with large-scale defects whose depth is small compared to their longitudinal sizes—these being typical of corrosive damage. Within the framework of large-scale approximation, analytical relations between the casing thickness profile and the measured magnetic field can be derived.
Description: Mathematics Subject Classification: 78A30 / 78M34 / 78A552017-03-22T09:55:42ZThree Waves of Chemical DynamicsGorban, A. N.Yablonsky, G. S.http://hdl.handle.net/2381/395392017-03-23T03:21:10Z2017-03-22T09:46:50ZTitle: Three Waves of Chemical Dynamics
Authors: Gorban, A. N.; Yablonsky, G. S.
Abstract: Three epochs in development of chemical dynamics are presented. We try to understand the modern research programs in the light of classical works.2017-03-22T09:46:50ZGeneration of mechanical force by grafted polyelectrolytes in an electric fieldBrilliantov, N. V.Budkov, Yu. A.Seidel, C.http://hdl.handle.net/2381/395302017-03-22T03:22:09Z2017-03-21T10:25:12ZTitle: Generation of mechanical force by grafted polyelectrolytes in an electric field
Authors: Brilliantov, N. V.; Budkov, Yu. A.; Seidel, C.
Abstract: We study theoretically and by means of molecular dynamics (MD) simulations the generation of mechanical force by grafted polyelectrolytes in an external electric field, which favors its adsorption on the grafting plane. The force arises in deformable bodies linked to the free end of the chain. Varying the field, one controls the length of the nonadsorbed part of the chain and hence the deformation of the target body, i.e., the arising force too. We consider target bodies with a linear force-deformation relation and with a Hertzian one. While the first relation models a coiled Gaussian chain, the second one describes the force response of a squeezed colloidal particle. The theoretical dependences of generated force and compression of the target body on an applied field agree very well with the results of MD simulations. The analyzed phenomenon may play an important role in future nanomachinery, e.g., it may be used to design nanovices to fix nanosized objects.2017-03-21T10:25:12ZForward-Invariant Peeling in Chemical Dynamics: a Simple Case StudyGorban, A. N.http://hdl.handle.net/2381/395282017-03-22T03:22:08Z2017-03-21T10:12:37ZTitle: Forward-Invariant Peeling in Chemical Dynamics: a Simple Case Study
Authors: Gorban, A. N.
Abstract: Forward-invariant peeling aims to produce forward-invariant subset from a given set in phase space. The structure of chemical kinetic equations allows us to describe the general operations of the forward-invariant peeling. For example, we study a simple reaction network with three components A1,A2,A3 and reactions A1 → A2 → A3 → A1, 2A1 ⇌ 3A2 (without any stoichiometric conservation law). We assume that kinetics obey the classical mass action law and reaction rate constants are positive intervals 0 <ki min ≤ ki ≤ ki max< ∞. Kinetics of this system is described by a system of differential inclusions. We produce forward-invariant sets for these kinetic inclusions from the sets { c | ci ≥ 0, ∑ ci ≥ ε } by the forward-invariant peeling (for sufficiently small ε> 0). In particular, this construction proves persistence of this kinetic system (a positive solution cannot approach the origin even asymptotically, as t → ∞).
Description: Mathematics Subject Classification: 37C10, 34D20, 93D052017-03-21T10:12:37ZGeneralized Mass Action Law and Thermodynamics of Nonlinear Markov ProcessesGorban, A. N.Kolokoltsov, V. N.http://hdl.handle.net/2381/395272017-03-22T03:22:08Z2017-03-21T09:59:51ZTitle: Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes
Authors: Gorban, A. N.; Kolokoltsov, V. N.
Abstract: The nonlinear Markov processes are measure-valued dynamical systems which preserve positivity. They can be represented as the law of large numbers limits of general Markov models of interacting particles. In physics, the kinetic equations allow Lyapunov functionals (entropy, free energy, etc.). This may be considered as a sort of inheritance of the Lyapunov functionals from the microscopic master equations. We study nonlinear Markov processes that inherit thermodynamic properties from the microscopic linear Markov processes. We develop the thermodynamics of nonlinear Markov processes and analyze the asymptotic assumption, which are sufficient for this inheritance.
Description: Mathematics Subject Classification: 80A30 / 60J25 / 60J60 / 60J75 / 82B402017-03-21T09:59:51ZThe Impact of Fragmented Habitat's Size and Shape on Populations with Allee EffectAlharbi, W. G.Petrovskii, S. V.http://hdl.handle.net/2381/395242017-03-22T03:22:05Z2017-03-21T09:34:56ZTitle: The Impact of Fragmented Habitat's Size and Shape on Populations with Allee Effect
Authors: Alharbi, W. G.; Petrovskii, S. V.
Abstract: This study aims to explore the ways in which population dynamics are affected by the shape and size of fragmented habitats. Habitat fragmentation has become a key concern in ecology over the past 20 years as it is thought to increase the threat of extinction for a number of plant and animal species; particularly those close to the fragment edge. In this study, we consider this issue using mathematical modelling and computer simulations in several domains of various shape and with different strength of the Allee effect. A two-dimensional reaction-diffusion equation (taking the Allee effect into account) is used as a model. Extensive simulations are performed in order to determine how the boundaries impact the population persistence. Our results indicate the following: (i) for domains of simple shape (e.g. rectangle), the effect of the critical patch size (amplified by the Allee effect) is similar to what is observed in 1D space, in particular, the likelihood of population survival is determined by the interplay between the domain size and thee strength of the Allee effect; (ii) in domains of complicated shape, for the population to survive, the domain area needs to be larger than the area of the corresponding rectangle. Hence, it can be concluded that domain size and shape both have crucial effect on population survival.
Description: Mathematics Subject Classification: 92D40 / 35B36 / 35Q92 / 37N252017-03-21T09:34:56ZModelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special IssueMorozov, A.Petrovskii, S.http://hdl.handle.net/2381/395232017-03-21T03:21:58Z2017-03-20T17:05:07ZTitle: Modelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue
Authors: Morozov, A.; Petrovskii, S.2017-03-20T17:05:07ZFast Sampling of Evolving Systems with Periodic TrajectoriesTyukin, I. Yu.Gorban, A. N.Tyukina, T. A.Al-Ameri, J. M.Korablev, Yu. A.http://hdl.handle.net/2381/395222017-03-21T03:21:57Z2017-03-20T16:55:54ZTitle: Fast Sampling of Evolving Systems with Periodic Trajectories
Authors: Tyukin, I. Yu.; Gorban, A. N.; Tyukina, T. A.; Al-Ameri, J. M.; Korablev, Yu. A.
Abstract: We propose a novel method for fast and scalable evaluation of periodic solutions of systems of ordinary differential equations for a given set of parameter values and initial conditions. The equations governing the system dynamics are supposed to be of a special class, albeit admitting nonlinear parametrization and nonlinearities. The method enables to represent a given periodic solution as sums of computable integrals and functions that are explicitly dependent on parameters of interest and initial conditions. This allows invoking parallel computational streams in order to increase speed of calculations. Performance and practical implications of the method are illustrated with examples including classical predator-prey system and models of neuronal cells.
Description: Mathematics Subject Classification: 93B30 / 34A05 / 92B99 / 93B152017-03-20T16:55:54ZTri-trophic Plankton Models Revised: Importance of Space, Food Web Structure and Functional Response ParametrisationEgilmez, H. I.Morozov, A. Yu.http://hdl.handle.net/2381/395192017-03-21T03:21:56Z2017-03-20T16:19:32ZTitle: Tri-trophic Plankton Models Revised: Importance of Space, Food Web Structure and Functional Response Parametrisation
Authors: Egilmez, H. I.; Morozov, A. Yu.
Abstract: Revealing mechanisms of efficient top-down control in eutrophic ecosystems remains a long term challenge in theoretical ecology. In this paper, we revisit the role of environmental heterogeneity, food web structure and shape of the predator functional response in persistence and stabilization of a planktonic system with high nutrient supply. We consider a 1D vertically resolved tri-trophic planktonic food web composed of a primary producer, an intermediate predator and a highly mobile top predator, such as a system of phytoplankton, microzooplankton and copepods. We explore the realistic scenario in which the top predator is omnivorous, i.e. when copepods can feed both on phytoplankton and microzooplankton. We show that the interplay between heterogeneity of the environment due to for instance, a light gradient in the water column, and trophic interaction between species can result in an efficient top-down control which would otherwise be impossible in the corresponding well-mixed system. We also find that allowing the top predator to feed on the primary producer may dramatically impede the coexistence of the three trophic levels, with only two levels generally surviving. The coexistence of all three trophic levels within a wide range of parameters becomes possible only when the top predator exhibits active food source switching behaviour. We also show the phenomenon of bistability in the considered tri-trophic food web: a small initial amount of the top predator should lead to its extinction whereas introduction of a supercritical initial amount will eventually result in establishment of the population. The demonstrated mechanism of top-down control seems to be rather generic and might be a good candidate to explain stability in some other non-planktonic tri-trophic ecosystems.
Description: Mathematics Subject Classification: 47A75 / 45K05 / 92D402017-03-20T16:19:32ZPreface. Bifurcations and Pattern Formation in Biological ApplicationsMorozov, A.Ptashnyk, M.Volpert, V.http://hdl.handle.net/2381/395172017-03-21T03:21:55Z2017-03-20T16:05:40ZTitle: Preface. Bifurcations and Pattern Formation in Biological Applications
Authors: Morozov, A.; Ptashnyk, M.; Volpert, V.
Abstract: In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena.2017-03-20T16:05:40ZA random walk description of individual animal movement accounting for periods of restTilles, Paulo F. C.Petrovskii, Sergei V.Natti, Paulo L.http://hdl.handle.net/2381/395162017-03-21T03:21:54Z2017-03-20T14:54:02ZTitle: A random walk description of individual animal movement accounting for periods of rest
Authors: Tilles, Paulo F. C.; Petrovskii, Sergei V.; Natti, Paulo L.
Abstract: Animals do not move all the time but alternate the period of actual movement (foraging) with periods of rest (e.g. eating or sleeping). Although the existence of rest times is widely acknowledged in the literature and has even become a focus of increased attention recently, the theoretical approaches to describe animal movement by calculating the dispersal kernel and/or the mean squared displacement (MSD) rarely take rests into account. In this study, we aim to bridge this gap. We consider a composite stochastic process where the periods of active dispersal or ‘bouts’ (described by a certain baseline probability density function (pdf) of animal dispersal) alternate with periods of immobility. For this process, we derive a general equation that determines the pdf of this composite movement. The equation is analysed in detail in two special but important cases such as the standard Brownian motion described by a Gaussian kernel and the Levy flight described by a Cauchy distribution. For the Brownian motion, we show that in the large-time asymptotics the effect of rests results in a rescaling of the diffusion coefficient. The movement occurs as a subdiffusive transition between the two diffusive asymptotics. Interestingly, the Levy flight case shows similar properties, which indicates a certain universality of our findings.2017-03-20T14:54:02ZCatching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization.Petrovskaya, NataliaPetrovskii, Sergeihttp://hdl.handle.net/2381/395152017-03-21T03:21:54Z2017-03-20T14:47:07ZTitle: Catching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization.
Authors: Petrovskaya, Natalia; Petrovskii, Sergei
Abstract: Synchronization of population dynamics in different habitats is a frequently observed phenomenon. A common mathematical tool to reveal synchronization is the (cross)correlation coefficient between time courses of values of the population size of a given species where the population size is evaluated from spatial sampling data. The corresponding sampling net or grid is often coarse, i.e. it does not resolve all details of the spatial configuration, and the evaluation error-i.e. the difference between the true value of the population size and its estimated value-can be considerable. We show that this estimation error can make the value of the correlation coefficient very inaccurate or even irrelevant. We consider several population models to show that the value of the correlation coefficient calculated on a coarse sampling grid rarely exceeds 0.5, even if the true value is close to 1, so that the synchronization is effectively lost. We also observe 'ghost synchronization' when the correlation coefficient calculated on a coarse sampling grid is close to 1 but in reality the dynamics are not correlated. Finally, we suggest a simple test to check the sampling grid coarseness and hence to distinguish between the true and artifactual values of the correlation coefficient.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2017-03-20T14:47:07Z$hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshesCangiani, AndreaDong, ZhaonanGeorgoulis, Emmanuil H.http://hdl.handle.net/2381/394732017-03-16T03:21:58Z2017-03-15T16:01:58ZTitle: $hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes
Authors: Cangiani, Andrea; Dong, Zhaonan; Georgoulis, Emmanuil H.
Abstract: We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of \emph{total} degree, say $p$, defined in the physical coordinate system, as opposed to standard dG-time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete $hp$-dG scheme using less degrees of freedom for each time step, compared to standard dG time-stepping schemes employing tensorized space-time, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with \emph{arbitrary} number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments.
Description: AMS subject classifications. 65N30, 65M60, 65J102017-03-15T16:01:58ZThe non-conforming virtual element method for the Stokes equationsCangiani, AndreaGyrya, VitaliyManzini, Gianmarcohttp://hdl.handle.net/2381/394722017-03-16T03:21:58Z2017-03-15T15:51:49ZTitle: The non-conforming virtual element method for the Stokes equations
Authors: Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco
Abstract: We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.
Description: AMS subject classifications. 65N30, 65N12, 65G99, 76R992017-03-15T15:51:49ZMapping cones in the bounded derived category of a gentle algebraCanakci, IlkePauksztello, DavidSchroll, Sibyllehttp://hdl.handle.net/2381/394632017-03-15T03:21:29Z2017-03-14T16:07:26ZTitle: Mapping cones in the bounded derived category of a gentle algebra
Authors: Canakci, Ilke; Pauksztello, David; Schroll, Sibylle
Abstract: Gentle algebras are a class of algebras that are derived tame. They therefore provide a concrete setting to study the structure of the (bounded) derived category in detail. In this article we explicitly describe the triangulated structure of the bounded derived category of a gentle algebra by describing its triangles. In particular, we develop a graphical calculus which gives the indecomposable summands of the mapping cones of morphisms in a canonical basis of the Hom-space between any two indecomposable complexes.
Description: 34 pages, many figures2017-03-14T16:07:26ZAdaptive Discontinuous Galerkin Methods for Interface ProblemsSabawi, Younis Abidhttp://hdl.handle.net/2381/393862017-02-28T03:44:55Z2017-02-27T12:07:31ZTitle: Adaptive Discontinuous Galerkin Methods for Interface Problems
Authors: Sabawi, Younis Abid
Abstract: The aim of this thesis is to derive adaptive methods for discontinuous Galerkin approximations for both elliptic and parabolic interface problems. The derivation of adaptive method, is usually based on a posteriori error estimates. To this end, we present a residual-type a posteriori error estimator for interior penalty discontinuous Galerkin (dG) methods for an elliptic interface problem involving possibly curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. Respective upper and lower bounds of the error in the respective dG-energy norm with respect to the estimator are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. Moreover, a contraction property for a standard adaptive algorithm utilising these a posteriori bounds, with a bulk refinement criterion is also shown, thereby proving that the a posteriori bounds can lead to a convergent adaptive algorithm subject to some mesh restrictions. This work is also concerned with the derivation of a new L1∞(L2)-norm a posteriori error bound for the fully discrete adaptive approximation for non-linear interface parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the interior penalty discontinuous Galerkin finite element method.
The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto [48], enabling us to use the a posteriori error estimators derived for elliptic interface models and to obtain optimal order in both L1∞(L2) and L1∞(L2) + L2(H¹) norms. The effectiveness of all the error estimators and the proposed algorithms is confirmed through a series of numerical experiments.2017-02-27T12:07:31ZRepresentation theory of the Drinfeld doubles of a family of Hopf algebras II: corrections and new resultsErdmann, KarinGreen, Edward L.Snashall, NicoleTaillefer, Rachelhttp://hdl.handle.net/2381/393702017-02-25T04:03:47Z2017-02-24T14:45:11ZTitle: Representation theory of the Drinfeld doubles of a family of Hopf algebras II: corrections and new results
Authors: Erdmann, Karin; Green, Edward L.; Snashall, Nicole; Taillefer, Rachel
Abstract: We return to the fusion rules for the Drinfeld double of the duals of the generalised Taft algebras that we studied in [Erdmann et al., J. Pure Appl. Algebra 2006]. We first correct some proofs and statements in [Erdmann et al., J. Pure Appl. Algebra 2006] that were incorrect, using stable homomorphisms. We then complete this with new results on fusion rules for the modules we had not studied in [Erdmann et al., J. Pure Appl. Algebra 2006] and a classification of endotrivial and algebraic modules.2017-02-24T14:45:11ZGaussian process regression with functional covariates and multivariate responseWang, BoChen, TaoXu, Aipinghttp://hdl.handle.net/2381/393242017-03-06T16:41:08Z2017-02-03T15:36:06ZTitle: Gaussian process regression with functional covariates and multivariate response
Authors: Wang, Bo; Chen, Tao; Xu, Aiping
Abstract: Gaussian process regression (GPR) has been shown to be a powerful and effective nonparametric method for regression, classification and interpolation, due to many of its desirable properties. However, most GPR models consider univariate or multivariate covariates only. In this paper we extend the GPR models to cases where the covariates include both functional and multivariate variables and the response is multidimensional. The model naturally incorporates two different types of covariates: multivariate and functional, and the principal component analysis is used to de-correlate the multivariate response which avoids the widely recognised difficulty in the multi-output GPR models of formulating covariance functions which have to describe the correlations not only between data points but also between responses. The usefulness of the proposed method is demonstrated through a simulated example and two real data sets in chemometrics.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above. Following the embargo period the above license applies.2017-02-03T15:36:06ZOn the effects of changing mortality patterns on investment, labour and consumption under uncertaintyEwald, Christian-OliverZhang, Aihuahttp://hdl.handle.net/2381/393202017-04-12T14:09:25Z2017-02-02T09:56:40ZTitle: On the effects of changing mortality patterns on investment, labour and consumption under uncertainty
Authors: Ewald, Christian-Oliver; Zhang, Aihua
Abstract: In this paper we extend the consumption-investment life cycle model for an
uncertain-lived agent, proposed by Richard (1974), to allow for
flexible labor supply. We further study the consumption, labor supply and portfolio
decisions of an agent facing age-dependent mortality risk, as presented by
UK actuarial life tables spanning the time period from 1951-2060 (including
mortality forecasts). We find that historical changes in mortality produces
significant changes in portfolio investment (more risk taking), labour (de- crease of hours) and consumption level (shift to higher level) contributing
up to 5% to GDP growth during the period from 1980 until 2010.
Description: JEL Subject Classi cation: G11; J11; J22; C61; 18 months embargo from publication; The file associated with this record is under embargo until 18 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2017-02-02T09:56:40ZDetailed balance in micro- and macrokinetics and micro-distinguishability of macro-processesGorban, A. N.http://hdl.handle.net/2381/392172017-01-19T03:16:37Z2017-01-18T16:43:31ZTitle: Detailed balance in micro- and macrokinetics and micro-distinguishability of macro-processes
Authors: Gorban, A. N.
Abstract: We develop a general framework for the discussion of detailed balance and analyse its microscopic background. We find that there should be two additions to the well-known T- or PT-invariance of the microscopic laws of motion:
1. Equilibrium should not spontaneously break the relevant T- or PT-symmetry.
2. The macroscopic processes should be microscopically distinguishable to guarantee persistence of detailed balance in the model reduction from micro- to macrokinetics.
We briefly discuss examples of the violation of these rules and the corresponding violation of detailed balance.2017-01-18T16:43:31ZMultilevel sparse grid kernels collocation with radial basis functions for elliptic and parabolic problemsZhao, Yangzhanghttp://hdl.handle.net/2381/391482017-01-17T03:23:11Z2017-01-16T11:28:58ZTitle: Multilevel sparse grid kernels collocation with radial basis functions for elliptic and parabolic problems
Authors: Zhao, Yangzhang
Abstract: Radial basis functions (RBFs) are well-known for the ease implementation as
they are the mesh-free method [31, 37, 71, 72]. In this thesis, we modify the
multilevel sparse grid kernel interpolation (MuSIK) algorithm proposed in [48]
for use in Kansa’s collocation method (referred to as MuSIK-C) to solve elliptic
and parabolic problems. The curse of dimensionality is a significant challenge
in high dimension approximation. A full grid collocation method requires O(Nd)
nodal points to construct an approximation; here N is the number of nodes in
one direction and d means the dimension. However, the sparse grid collocation
method in this thesis only demand O(N logd1(N)) nodes. We save much more
memory cost using sparse grids and obtain a good performance as using full grids.
Moreover, the combination technique [20, 54] allows the sparse grid collocation
method to be parallelised. When solving parabolic problems, we follow Myers
et al.’s suggestion in [90] to use the space-time method, considering time as
one spatial dimension. If we apply sparse grids in the spatial dimensions and
use time-stepping, we still need O(N2 logd1(N)) nodes. However, if we use the
space-time method, the total number of nodes is O(N logd(N)).
In this thesis, we always compare the performance of multiquadric (MQ) basis
function and the Gaussian basis function. In all experiments, we observe that
the collocation method using the Gaussian with scaling shape parameters does
not converge. Meanwhile, in Chapter 3, there is an experiment to show that the
space-time method with MQ has a similar convergence rate as a time-stepping
method using MQ in option pricing. From the numerical experiments in Chapter
4, MuSIK-C using MQ and the Gaussian always give more rapid convergence
and high accuracy especially in four dimensions (T R3) for PDEs with smooth
conditions. Compared to some recently proposed mesh-based methods, MuSIK-C
shows similar performance in low dimension situation and better approximation
in high dimension. In Chapter 5, we combine the Method of Lines (MOL) and our
MuSIK-C to obtain good convergence in pricing one asset European option and
the Margrabe option, that have non-smooth initial conditions.2017-01-16T11:28:58ZDynamic Cooperative InvestmentAlmualim, Anwar Hassan Alihttp://hdl.handle.net/2381/391462017-01-17T03:23:15Z2017-01-16T11:16:31ZTitle: Dynamic Cooperative Investment
Authors: Almualim, Anwar Hassan Ali
Abstract: In this thesis we develop dynamic cooperative investment schemes in discrete and
continuous time. Instead of investing individually, several agents may invest joint
capital into a commonly agreed trading strategy, and then split the uncertain
outcome of the investment according to the pre-agreed scheme, based on their
individual risk-reward preferences. As a result of cooperation, each investor is able
to get a share, which cannot be replicated with the available market instruments,
and because of this, cooperative investment is usually strictly profitable for all
participants, when compared with an optimal individual strategy. We describe
cooperative investment strategies which are Pareto optimal, and then propose a
method to choose the most ‘fair’ Pareto optimal strategy based on equilibrium
theory. In some cases, uniqueness and stability for the equilibrium are justified.
We study a cooperative investment problem, for investors with different risk preferences,
coming from expected utility theory, mean-variance theory, mean-deviation
theory, prospect theory, etc. The developed strategies are time-consistent; that
is the group of investors have no reasons to change their mind in the middle of
the investment process. This is ensured by either using a dynamic programming
approach, by applying the utility model based on the compound independence
axiom.
For numerical experiments, we use a scenario generation algorithm and stochastic
programming model for generating appropriate scenario tree components of the
S&P 100 index. The algorithm uses historical data simulation as well as a GARCH
model.2017-01-16T11:16:31ZGaussian Process and Functional Data Methods for Mortality ModellingWu, Ruhaohttp://hdl.handle.net/2381/391432017-01-17T03:23:10Z2017-01-16T10:46:09ZTitle: Gaussian Process and Functional Data Methods for Mortality Modelling
Authors: Wu, Ruhao
Abstract: Modelling the demographic mortality trends is of great importance due to its considerable impact on welfare policy, resource allocation and government planning. In this thesis, we propose to use various statistical methods, including Gaussian process (GP), principal curve, multilevel functional principal component analysis (MFPCA) for forecasting and clustering of human mortality data. This thesis is actually composed of three main topics regarding mortality modelling. In the first topic, we propose a new Gaussian process regression method and apply it to the modelling and forecasting of age-specific human mortality rates for a single population. The proposed method incorporates a weighted mean function and the spectral mixture covariance function, hence provides better performance in forecasting long term mortality rates, compared with the conventional GPR methods. The performance of the proposed method is also compared with Lee-Miller model and the functional data model by Hyndman and Ullah (2007) in the context of forecasting the French total mortality rates. Then, in the second topic, we extend mortality modelling for a single population independently to that for multiple populations simultaneously, by developing a new framework for coherent modelling and forecasting of mortality rates for multiple subpopulations within one large population. We treat the mortality of subpopulations as multilevel functional data and then a weighted multilevel functional principal component approach is proposed and used for modelling and forecasting the mortality rates. The proposed model is applied to sex-specific data for nine developed countries, and the forecasting results suggest that, in terms of overall accuracy, the model outperforms the independent model (Hyndman and Ullah 2007) and is comparable to the Product-Ratio model (Hyndman et al 2013) but with several advantages. Finally, in the third topic, we introduce a clustering method based on principal curves for clustering of human mortality as functional data. And this innovative clustering method is applied to French total mortality data for exploring its potential features.2017-01-16T10:46:09ZDiscontinuous Galerkin Methods on Polytopic MeshesDong, Zhaonanhttp://hdl.handle.net/2381/391402017-01-14T04:08:36Z2017-01-13T15:44:19ZTitle: Discontinuous Galerkin Methods on Polytopic Meshes
Authors: Dong, Zhaonan
Abstract: This thesis is concerned with the analysis and implementation of the hp-version
interior penalty discontinuous Galerkin finite element method (DGFEM) on computational
meshes consisting of general polygonal/polyhedral (polytopic) elements.
Two model problems are considered: general advection-diffusion-reaction boundary
value problems and time dependent parabolic problems. New hp-version a
priori error bounds are derived based on a specific choice of the interior penalty
parameter which allows for edge/face-degeneration as well as an arbitrary number
of faces and hanging nodes per element.
The proposed method employs elemental polynomial bases of total degree p (Pp-
bases) defined in the physical coordinate system, without requiring mapping from
a given reference or canonical frame. A series of numerical experiments highlighting
the performance of the proposed DGFEM are presented. In particular,
we study the competitiveness of the p-version DGFEM employing a Pp-basis on
both polytopic and tensor-product elements with a (standard) DGFEM and FEM
employing a (mapped) Qp-basis. Moreover, a careful theoretical analysis of optimal
convergence rate in p for Pp-basis is derived for several commonly used
projectors, which leads to sharp bounds of exponential convergence with respect
to degrees of freedom (dof) for the Pp-basis.
Description: File under embargo until 3rd June 2017.2017-01-13T15:44:19ZEfficient Option Pricing under Levy Processes, with CVA and FVAShek, C. K.Law, J.Levendorskiĭ, Sergeihttp://hdl.handle.net/2381/391102017-01-11T03:20:36Z2017-01-10T10:09:37ZTitle: Efficient Option Pricing under Levy Processes, with CVA and FVA
Authors: Shek, C. K.; Law, J.; Levendorskiĭ, Sergei
Abstract: We generalize the Piterbarg [1] model to include (1) bilateral default risk as in Burgard and Kjaer [2], and (2) jumps in the dynamics of the underlying asset using general classes of Lévy processes of exponential type. We develop an efficient explicit-implicit scheme for European options and barrier options taking CVA-FVA into account. We highlight the importance of this work in the context of trading, pricing and management a derivative portfolio given the trajectory of regulations.2017-01-10T10:09:37ZQuantifying non-Newtonian effects in rotating boundary-layer flowsGriffiths, P. T.Garrett, S. J.Stephen, S. O.Hussain, Z.http://hdl.handle.net/2381/390892017-01-10T03:20:43Z2017-01-09T14:39:47ZTitle: Quantifying non-Newtonian effects in rotating boundary-layer flows
Authors: Griffiths, P. T.; Garrett, S. J.; Stephen, S. O.; Hussain, Z.
Abstract: The stability of the boundary-layer on a rotating disk is considered for fluids that adhere to a non-Newtonian governing viscosity relationship. For fluids with shear-rate dependent viscosity the base flow is no longer an exact solution of the Navier–Stokes equations, however, in the limit of large Reynolds number the flow inside the three-dimensional boundary-layer can be determined via a similarity solution. The convective instabilities associated with flows of this nature are described both asymptotically and numerically via separate linear stability analyses. Akin to previous Newtonian studies it is found that there exists two primary modes of instability; the upper-branch type I modes, and the lower-branch type II modes. Results show that both these modes can be stabilised or destabilised depending on the choice of non-Newtonian viscosity model. A number of comments are made regarding the suitability of some of the more well-known non-Newtonian constitutive relationships within the context of the rotating disk model. Such a study is presented with a view to suggesting potential control mechanisms for flows that are practically relevant to the turbo-machinery industry.
Description: 12 month embargo2017-01-09T14:39:47ZRepresentations of Quantum Conjugacy Classes of Non-Exceptional Quantum GroupsAshton, Thomas Stephenhttp://hdl.handle.net/2381/390242016-12-22T03:21:01Z2016-12-21T16:00:17ZTitle: Representations of Quantum Conjugacy Classes of Non-Exceptional Quantum Groups
Authors: Ashton, Thomas Stephen
Abstract: Let G be a complex non-exceptional simple algebraic group and g its Lie algebra. With every point x of the maximal torus T ʗ G we associate a highest weight module Mx over the Drinfeld-Jimbo quantum group Uq(g) and an equivariant quantization of the conjugacy class of x by operators in End(Mx). These equivariant quantizations are isomorphic for x lying on the same orbit of the Weyl group, and Mx support different exact representations of the same quantum conjugacy class.
This recovers all quantizations of conjugacy classes constructed before, via special x, and also completes the family of conjugacy classes by constructing an equivariant quantization of “borderline" Levi conjugacy classes of the complex orthogonal group SO(N), whose stabilizer contains a Cartesian factor SO(2) SO(P), 1 6 P < N, P Ξ N mod 2.
To achieve this, generators of the Mickelsson algebra Zq(g; g’), where g’ ʗ g is the Lie subalgebra of rank rkg’ = rkg-1 of the same type, were explicitly constructed in terms of Chevalley generators via the R-matrix of Uq(g).2016-12-21T16:00:17ZComparison of the effects of surface roughness and confinement on rotor–stator cavity flowÖzkan, M.Thomas, P. J.Cooper, A. J.Garrett, Stephen Johnhttp://hdl.handle.net/2381/390232016-12-21T03:31:56Z2016-12-20T16:51:40ZTitle: Comparison of the effects of surface roughness and confinement on rotor–stator cavity flow
Authors: Özkan, M.; Thomas, P. J.; Cooper, A. J.; Garrett, Stephen John
Abstract: Results of a computational study are discussed which investigate roughness-induced and geometry-induced (confinement) effects on the steady-state velocity components in 3-D boundary-layer flow over the rotor disc in a rotor–stator flow configuration. It is found that, for the rotor–stator flow investigated, the roughness-induced effects are very similar to geometry-induced effects, both in nature and magnitude. The overall aim was to compare these two types of effects with corresponding roughness-induced effects in the von Kármán boundary-layer flow over a disc spinning freely in an unrestricted fluid environment. The research was conducted in the context of a programme investigating surface roughness as a means of laminar flow control for the development of new passive drag-reduction techniques. The goal was to establish whether it was possible unequivocally to distinguish between roughness-induced and geometry-induced effects on the boundary-layer flow above the rotor disc. The results obtained suggest that, for the type of system discussed here, it must be expected to be difficult to distinguish between these effects in experiments. The similarities regarding the nature and magnitude of results obtained from comparing predictions for three different computational modelling approaches indicate the required sensitivity of measurement technologies aiming to resolve the investigated effects in experimental studies.2016-12-20T16:51:40ZOn the diagonal subalgebra of an Ext algebraGreen, E. L.Snashall, Nicole JaneSolberg, O.Zacharia, D.http://hdl.handle.net/2381/389662016-12-16T03:19:18Z2016-12-15T15:05:06ZTitle: On the diagonal subalgebra of an Ext algebra
Authors: Green, E. L.; Snashall, Nicole Jane; Solberg, O.; Zacharia, D.
Abstract: Let R be a Koszul algebra over a field k and M be a linear R-module. We study a graded subalgebra ΔM of the Ext-algebra ExtR⁎(M,M) called the diagonal subalgebra and its properties. Applications to the Hochschild cohomology ring of R and to periodicity of linear modules are given. Viewing R as a linear module over its enveloping algebra, we also show that ΔR is isomorphic to the graded center of the Koszul dual of R. When R is selfinjective and not necessarily graded, we study connections between periodic modules M, complexity of M and existence of non-nilpotent elements of positive degree in the Ext-algebra of M. Characterizations of periodic algebras are given.2016-12-15T15:05:06Z