Robust Padé approximation via SVD

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity

Inspection of nuclear assets with limited access

Traditionally the inspection of nuclear containers and their welds, is highly challenging, time-consuming and expensive due to the complexities and logistics of the environment and process. This problem is further compounded when the asset lifetime is increased beyond their original design intent, and when accessibility to the asset is obstructed. One such problem being faced by Sellafield LTD in the UK is the storage of spent nuclear fuel for reprocessing, with government policy changing from favouring reprocessing to long term storage [1]. These assets are contained in thin 1.4404 stainless steel canisters with a Resistance Seam Weld (RSW) sealing the canister body to the lid. Additionally, the storage facilities only allow for partial circumferential access. This work is concerned with the development of an ultrasonic screening method of the RSW located on these canisters whilst in storage (“in-situ”). The work aims to make use of a Feature Guided Wave (FGW) that will transmit an ultrasonic wave confined only to the RSW that will allow screening of the full circumference from partial circumferential access. Feature Guided Waves were initially discovered experimentally from work conducted at by BAE and Imperial College London [2], and further studied analytically and experimentally by Imperial College London & Nanyang Technical University [3]–[5]. This work leveraged the use of the Semi-Analytical-Finite-Element Method (SAFEM) to predict all wave modes that can exist within the cross section of a component. If the cross section of the component, contained a feature where the geometry is significantly different (i.e. a weld, a bend in a CFRP stiffener or a fin), filtering based on the properties of the wave modes predicted (e.g Axial Power Flow, Kinetic Energy etc.) can be performed to predict the existence of wave modes confined to this feature. Moreover, the wave modes can be classified based on their mode shape, to inform transduction choice. In this work, the SAFEM has been employed in conjunction with filtering to predict modes confined to the RSW in within the canister. An automatic classification technique has been developed to trace the dispersion relationships of the wave modes so that experimental validation can be obtained, and a screening method can be developed. Understanding the dispersion relationship of guided waves is of utmost importance, so that reliable and repeatable experimental results are obtained. The SAFEM was employed on the resistance seam weld structure shown in Figure 1. The model was run from 50-1000kHz in 10kHz increments with over 8,000 wave modes being predicted at each frequency interval. A variation of the filter proposed by Yu et al. & Zuo et al. was employed on the results to reduce the number of wave modes to that only confined within the vicinity of the weld geometry [4], [6]. The resulting data set was traced automatically utilising a combination of orthogonal mode sorting and Padé expansion across the aforementioned frequency range [7], [8]. It is thought that this work is the first to automatically trace the dispersion relationship in this manner for this type of problem. Four modes were successfully traced that appear to offer promise for an initial NDT screening method. Current work is focusing on experimentally validating these modes and deploying an NDT screening regime based the previous simulations in both a lab based and industrial environment

On the complex singularities of the inverse Langevin function

We study the inverse Langevin function $\mathscr{L}^{-1}(x)$ because of its importance in modelling limited-stretch elasticity where the stress and strain energy become infinite as a certain maximum strain is approached, modelled here by $x\to1$. The only real singularities of the inverse Langevin function $\mathscr{L}^{-1}(x)$ are two simple poles at $x=\pm1$ and we see how to remove their effects either multiplicatively or additively. In addition, we find that $\mathscr{L}^{-1}(x)$ has an infinity of complex singularities. Examination of the Taylor series about the origin of $\mathscr{L}^{-1}(x)$ shows that the four complex singularities nearest the origin are equidistant from the origin and have the same strength; we develop a new algorithm for finding these four complex singularities. Graphical illustration seems to point to these complex singularities being of a square root nature. An exact analysis then proves these are square root branch points.Comment: 25 pages, 10 figures, 4 tables, 50 equations, 28 reference

Sparse Modelling and Multi-exponential Analysis

The research fields of harmonic analysis, approximation theory and computer algebra are seemingly different domains and are studied by seemingly separated research communities. However, all of these are connected to each other in many ways. The connection between harmonic analysis and approximation theory is not accidental: several constructions among which wavelets and Fourier series, provide major insights into central problems in approximation theory. And the intimate connection between approximation theory and computer algebra exists even longer: polynomial interpolation is a long-studied and important problem in both symbolic and numeric computing, in the former to counter expression swell and in the latter to construct a simple data model. A common underlying problem statement in many applications is that of determining the number of components, and for each component the value of the frequency, damping factor, amplitude and phase in a multi-exponential model. It occurs, for instance, in magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, electronic odour recognition, keystroke recognition, nuclear science, music signal processing, transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tolerance testing, to name just a few. The general technique of multi-exponential modeling is closely related to what is commonly known as the Padé-Laplace method in approximation theory, and the technique of sparse interpolation in the field of computer algebra. The problem statement is also solved using a stochastic perturbation method in harmonic analysis. The problem of multi-exponential modeling is an inverse problem and therefore may be severely ill-posed, depending on the relative location of the frequencies and phases. Besides the reliability of the estimated parameters, the sparsity of the multi-exponential representation has become important. A representation is called sparse if it is a combination of only a few elements instead of all available generating elements. In sparse interpolation, the aim is to determine all the parameters from only a small amount of data samples, and with a complexity proportional to the number of terms in the representation. Despite the close connections between these fields, there is a clear lack of communication in the scientific literature. The aim of this seminar is to bring researchers together from the three mentioned fields, with scientists from the varied application domains.Output Type: Meeting Repor

Genèse et diffusion d'un théorème de Robert de Montessus de Ballore sur les fractions continues alg\'ebriques.

En 1902, Robert de Montessus de Ballore démontre la convergence d'une fraction continue algébrique associée à une fonction analytique à l'origine et méromorphe dans un domaine contenant l'origine. Aujourd'hui ce théorème est encore cité. Et le nom Montessus de Ballore sert à nommer des généralisations du résultat. Nous déterminerons le contexte et les différentes étapes qui ont conduit Robert de Montessus à l'élaboration de son résultat. Cette étude s'appuie notamment sur la correspondance scientifique de Robert de Montessus

Numerical Performance of the Holomorphic Embedding Method

abstract: Recently, a novel non-iterative power flow (PF) method known as the Holomorphic Embedding Method (HEM) was applied to the power-flow problem. Its superiority over other traditional iterative methods such as Gauss-Seidel (GS), Newton-Raphson (NR), Fast Decoupled Load Flow (FDLF) and their variants is that it is theoretically guaranteed to find the operable solution, if one exists, and will unequivocally signal if no solution exists. However, while theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not. Numerically, the HEM may require extended precision to converge, especially for heavily-loaded and ill-conditioned power system models. In light of the advantages and disadvantages of the HEM, this report focuses on three topics: 1. Exploring the effect of double and extended precision on the performance of HEM, 2. Investigating the performance of different embedding formulations of HEM, and 3. Estimating the saddle-node bifurcation point (SNBP) from HEM-based Thévenin-like networks using pseudo-measurements. The HEM algorithm consists of three distinct procedures that might accumulate roundoff error and cause precision loss during the calculations: the matrix equation solution calculation, the power series inversion calculation and the Padé approximant calculation. Numerical experiments have been performed to investigate which aspect of the HEM algorithm causes the most precision loss and needs extended precision. It is shown that extended precision must be used for the entire algorithm to improve numerical performance. A comparison of two common embedding formulations, a scalable formulation and a non-scalable formulation, is conducted and it is shown that these two formulations could have extremely different numerical properties on some power systems. The application of HEM to the SNBP estimation using local-measurements is explored. The maximum power transfer theorem (MPTT) obtained for nonlinear Thévenin-like networks is validated with high precision. Different numerical methods based on MPTT are investigated. Numerical results show that the MPTT method works reasonably well for weak buses in the system. The roots method, as an alternative, is also studied. It is shown to be less effective than the MPTT method but the roots of the Padé approximant can be used as a research tool for determining the effects of noisy measurements on the accuracy of SNBP prediction.Dissertation/ThesisMasters Thesis Electrical Engineering 201

Numerically Robust Load Flow Techniques in Power System Planning

Since deregulation of the electric power industry, investment in the sector has not kept up with demand. State grids were interconnected to form vast power networks, which increased the overall system’s complexity. Conventional generation sources have, in some cases, closed under financial stress caused by the growing penetration of renewable sources and unfavourable government measures. The power system must adapt to a more demanding environment to that for which it was conceived. This thesis investigates the robustness of planning and simulation study tools for the determination of bus-voltages and voltage stability limits. It also provides an approach to obtain greater certainty in the determination of voltages where conventional methods fail to be deterministic. Two complementary methods for determining the collapse voltage are developed in this thesis. The first method applies Robust Padé approximations to the holomorphic embedding load flow method; while the second method uses the Newton-Raphson numerical calculation method to obtain both high and low voltage solution branches, and voltage stability limits of power system load buses. The proposed methods have been implemented using MATLAB and been demonstrated through a number of IEEE power system test cases. The robust Padé approximation algorithm improves the reliability of solutions of load flow problems when bus-voltages are presented in Taylor series form by converting the series into optimised rational functions. Differences between the classic Padé approximation algorithm and the new robust version, which is based on singular value decomposition (SVD), are described. The new robust approximation method can determine an optimal rational function approximation using the coefficients of a Taylor series expansion. Consequently, the voltage collapse points, as well as the steady-state voltage stability margin, can be calculated with high reliability. Voltage collapse points (i.e. branching points) are identified by using the locations of poles/zeros of a rational function approximation. Numerical examples are devised to illustrate potential use of the proposed method in practical applications. Use of the Newton-Raphson method, combined with the discrete Fourier transform and robust Padé approximation, enables the calculation of the voltage stability limits and both the high and low voltage solution branches for the load buses of a power system. This can work to a great advantage of existing N-R based software users, as problems of initial guess, multiple solutions and Jacobian matrix conditioning when operating close to the voltage collapse point are avoided. The findings are assessed by comparisons with conventional Newton-Raphson, the holomorphic embedding load flow method, and continuation power flow method. This thesis contains a combination of conventional and publication formats, where some introductory materials are included to ensure that the thesis delivers a consistent narrative. For this reason, the first two chapters provide the required background information, research gap identification and contributions, whilst other chapters are written to provide more detailed work that has not yet been published or to summarise the research outcomes and future research directions. Furthermore, publications are listed in their publication formats, complete with statements of the authors’ contributions.Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 201

Electronic correlations in inhomogeneous model systems: numerical simulation of spectra and transmission

Many fascinating features in condensed matter systems emerge due to the interaction between electrons. Magnetism is such a paramount consequence, which is explained in terms of the exchange interaction of electrons. Another prime example is the metal-to-Mott-insulator transition, where the energy cost of Coulomb repulsion competes against the kinetic energy, the latter favoring delocalization. While systems of correlated electrons are exciting and show remarkable and technologically promising physical properties, they are difficult to treat theoretically. A single-particle description is insufficient; the quantum many-body problem of interacting electrons has to be solved. In the present thesis, we study physical properties of half-metallic ferromagnets which are used in spintronic devices. Half-metals exhibit a metallic spin channel, while the other spin channel is insulating; they are characterized by a high spin polarization. This thesis contributes to the development of numerical methods and applies them to models of half-metallic ferromagnets. Throughout this work, the single-band Hubbard Hamiltonian is considered, and electronic correlations are treated within dynamical mean-field theory. Instead of directly solving the lattice model, the dynamical mean-field theory amounts to solving a local, effective impurity problem that is determined self-consistently. At finite temperatures, this impurity problem is solved employing continuous-time quantum Monte Carlo algorithms formulated in the action formalism. As these algorithms are formulated in imaginary time, an analytic continuation is required to obtain spectral functions. We formulate a version of the N-point Padé algorithm that calculates the location of the poles in a least-squares sense. To directly obtain spectra for real frequencies, we employ Hamiltonian-based tensor network methods at zero temperature. We also summarize the ideas of the density matrix renormalization group algorithm, and of the time evolution using the time-dependent variational principle, employing a diagrammatic notation. Real materials never display perfect translational symmetry. Thus, realistic models require the inclusion of disorder effects. In this work, we discuss these within a single-site approximation, the coherent potential approximation, and combine it with the dynamical mean-field theory, allowing to treat interacting electrons in multicomponent alloys on a local level. We extend this combined scheme to off-diagonal disorder, that is, disorder in the hopping amplitudes, by employing the Blackman–Esterling–Berk formalism. For this purpose, we illustrate the ideas of this formalism using tensor diagrams and provide an efficient implementation. The structure of the effective medium is discussed, and a concentration scaling is proposed that resolves some of its peculiarities. The limit of vanishing hopping between different components is discussed and solved analytically for the Bethe lattice with a general coordination number. We exemplify the combined algorithm for a Bethe lattice, showing results that exhibit alloy-band-insulator to correlated-metal to Mott-insulator transitions. We study models of half-metallic ferromagnets to elucidate the effects of local electronic correlations on the spectral function. To model half-metallicity, a static spin splitting is used to produce the half-metallic density of states. Applying the Padé analytic continuation to the self-energy instead of the Green’s function produces reliable spectral functions agreeing with the zero-temperature results obtained for real frequencies. To address transport properties, we investigate the interface of a half-metallic layer and a metallic, band insulating, or Mott insulating layer. We observe charge reconstruction which induces metallicity at the interface; quasiparticle states are present in the Mott insulating layer even for a large Hubbard interaction. The transmission through a barrier made of such a single interacting half-metallic layer sandwiched by metallic leads is studied employing the Meir–Wingreen formalism. This allows for a transparent calculation of the transmission in the presence of the Hubbard interaction. For a strong coupling of the central layer to the leads, we identify high intensity bound states which do not contribute to the transmission. For small coupling, on the other hand, we find resonant states which enhance the transmission. In particular, we demonstrate that even for a single half-metallic layer, highly polarized transmissions are achievable