A new family of extended Gauss quadratures with an interior interval constraint

AbstractStarting from two sequences {Ĝa,c,n} and {Ĝd,b,n} of ordinary Gauss quadrature formulae with an orthogonality measure dσ on the open intervals (a,c) and (d,b), respectively. We construct a new sequence {Ĝa,b,e(n)} of extended Gaussian quadrature formulae for dσ on (a,b), which is based on some preassigned points, the nodes of Ĝa,c,n, Ĝd,b,n and the e(n) zeros contained in (c,d) of a nonclassical orthogonal polynomial on [a,b] with respect to a linear functional. The principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three-term recurrence formulae are related. Thus, a new class of Gaussian quadratures, having some nodes contained in a given interior interval, can be computed directly by standard software for ordinary Gauss quadrature formulae

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction

Voltage fluctuations caused by groups of wind turbines

Wind turbines connected to the distribution network can be the cause of voltage fluctuations and resulting fluctuations in the light intensity emitted by light bulbs. These fluctuations may cause people disturbance. A model has been developed to obtain a flicker prediction which is useful in the design process of a wind farm. The model is based exclusively in the frequency domain (FD). This new approach allows very fast and efficient evaluation. The impact of individual parameters is often easier to recognise and evaluate in a FD-representation. The following factors leading to flicker disturbances from a wind farm have been considered in detail: The wind spectrum: Effects of terrain and wind farm wakes on the wind turbulence spectrum have been considered and existing models have been expanded. The wind coherence: A new coherence model for large separation distances has been derived for use within a wind farm. Effects of the terrain on the coherence of power produced by turbines within a wind farm have been considered. The wind turbine: A simplified dynamic wind turbine model allows the prediction of turbine specific contributions to flicker for a variety of wind turbines using a minimal set of parameters. The flickermeter: Flicker measurements are found to sometimes neglect the impact of low frequency voltage variations. These are found to be very important for the correct flicker prediction. A new FD-flickermeter has been developed. The model has been validated against experimental data and a sensitivity analysis shows which parameters are most likely to influence the voltage flicker and which are best altered to minimise the flicker

Multigrid methods for nonlinear second order partial differential operators

This thesis is concerned with the efficient numerical solution of nonlinear partial differential equations (PDEs) of elliptic and parabolic type. Such PDEs arise frequently in models used to describe many physical phenomena, from the diffusion of a toxin in soil to the flow of viscous fluids. The main focus of this research is to better understand the implementation and performance of nonlinear multigrid methods for the solution of elliptic and parabolic PDEs, following their discretisation. For the most part finite element discretisations are considered, but other techniques are also discussed. Following discretisation of a PDE the two most frequently used nonlinear multigrid methods are Newton-Multigrid and the Full Approximation Scheme (FAS). These are both very efficient algorithms, and have the advantage that when they are applied to practical problems, their execution times scale linearly with the size of the problem being solved. Even though this has yet to be proved in theory for most problems, these methods have been widely adopted in practice in order to solve highly complex nonlinear (systems of) PDEs. Many research groups use either Newton-MG or FAS without much consideration as to which should be preferred, since both algorithms perform satisfactorily. In this thesis we address the question as to which method is likely to be more computationally efficient in practice. As part of this investigation the implementation of the algorithms is considered in a framework which allows the direct comparison of the computational effort of the two iterations. As well as this, the convergence properties of the methods are considered, applied to a variety of model problems. Extensive results are presented in the comparison, which are explained by available theory whenever possible. The strength and range of results presented allows us to confidently conclude that for a practical problem, discretised using a finite element discretisation, an improved efficiency and stability of a Newton-MG iteration, compared to an FAS iteration, is likely to be observed. The relative advantage of a Newton-MG method is likely to be larger the more complex the problem being solved becomes

Strategies for producing fast finite element solutions of the incompressible Navier-Stokes equations on massively parallel architectures

To take advantage of the inherent flexibility of the finite element method in solving for flows within complex geometries, it is necessary to produce efficient implementations of the method. Segregation of the solution scheme and the use of parallel computers are two ways of doing this. Here, the optimisation of a sequential segregated finite element algorithm is discussed, together with the various strategies by which this is done. Furthermore, the implications of parallelising the code onto a massively parallel computer, the MasPar, are explored. This machine is of Single Instruction Multiple Data type and so modifications to the computer code have been necessary. A general methodology for the implementation of finite element programs is presented based on projecting the levels of data within the algorithm into a form which is ideal for parallelisation. Application of this methodology, in a high level language, has resulted in a code which runs at just under 30MFlops (in double precision). The computations are performed with minimal inter-processor communication and this represents an efficiency of 20% of the theoretical peak speed. Even though only high level language constructs have been used, this efficiency is comparable with other work using low level constructs on machines of this architecture. In particular, the use of data parallel arrays and the utilisation of the non-unique machine specific features of the computer architecture have produced an efficient, fast program

The design of multiconfiguration axisymmetric optical systems

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