Methods for Estimating The Diagonal of Matrix Functions

Many applications such as path integral evaluation in Lattice Quantum Chromodynamics (LQCD), variance estimation of least square solutions and spline ts, and centrality measures in network analysis, require computing the diagonal of a function of a matrix, Diag(f(A)) where A is sparse matrix, and f is some function. Unfortunately, when A is large, this can be computationally prohibitive. Because of this, many applications resort to Monte Carlo methods. However, Monte Carlo methods tend to converge slowly. One method for dealing with this shortcoming is probing. Probing assumes that nodes that have a large distance between them in the graph of A, have only a small weight connection in f(A). to determine the distances between nodes, probing forms Ak. Coloring the graph of this matrix will group nodes that have a high distance between them together, and thus a small connection in f(A). This enables the construction of certain vectors, called probing vectors, that can capture the diagonals of f(A). One drawback of probing is in many cases it is too expensive to compute and store A^k for the k that adequately determines which nodes have a strong connection in f(A). Additionally, it is unlikely that the set of probing vectors required for A^k is a subset of the probing vectors needed for Ak+1. This means that if more accuracy in the estimation is required, all previously computed work must be discarded. In the case where the underlying problem arises from a discretization of a partial dierential equation (PDE) onto a lattice, we can make use of our knowledge of the geometry of the lattice to quickly create hierarchical colorings for the graph of A^k. A hierarchical coloring is one in which colors for A^{k+1} are created by splitting groups of nodes sharing a color in A^k. The hierarchical property ensures that the probing vectors used to estimate Diag(f(A)) are nested subsets, so if the results are inaccurate the estimate can be improved without discarding the previous work. If we do not have knowledge of the intrinsic geometry of the matrix, we propose two new classes of methods that improve on the results of probing. One method seeks to determine structural properties of the matrix f(A) by obtaining random samples of the columns of f(A). The other method leverages ideas arising from similar problems in graph partitioning, and makes use of the eigenvectors of f(A) to form effective hierarchical colorings. Our methods have thus far seen successful use in computational physics, where they have been applied to compute observables arising in LQCD. We hope that the renements presented in this work will enable interesting applications in many other elds

Newton flows for elliptic functions

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e., doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions $f$ of fixed order $r$ ($\geq$ 2) we prove: For almost all functions $f$, the corresponding Newton flows are structurally stable i.e., topologically invariant under small perturbations of the zeros and poles for $f$ [genericity]. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded, graph $G(f)$ on $T$ with $r$ vertices, 2$r$ edges and $r$ faces that fulfil certain combinatorial properties (Euler, Hall) on some of its subgraphs. The graph $G(f)$ determines the conjugacy class of the flow [characterization]. A connected, cellularly embedded toroidal graph $G$ with the above Euler and Hall properties, is called a Newton graph. Any Newton graph $G$ can be realized as the graph $G(f)$ of the structurally stable Newton flow for some function $f$ [classification]. This leads to: up till conjugacy between flows and(topological) equivalency between graphs, there is a 1-1 correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order $r$ of the underlying functions $f$ [representation]. In particular, it follows that in case $r$ = 2, there is only one (up to conjugacy) structurally stabe elliptic Newton flow, whereas in case $r$ = 3, we find a list of nine graphs, determining all possibilities. Moreover, we pay attention to the so-called nuclear Newton flows of order $r$, and indicate how - by a bifurcation procedure - any structurally stable elliptic Newton flow of order $r$ can be obtained from such a nuclear flow. Finally, we show that the detection of elliptic Newton flows is possible in polynomial time. The proofs of the above results rely on Peixoto's characterization/classication theorems for structurally stable dynamical systems on compact 2-dimensional manifolds, Stiemke's theorem of the alternatives, Hall's theorem of distinct representatives, the Heter-Edmonds-Ringer rotation principle for embedded graphs, an existence theorem on gradient dynamical systems by Smale, and an interpretation of Newton flows as steady streams

Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

Gas turbine sub-idle performance modelling : groundstart altitude relight, and windmilling

Engine performance modelling is a major part of the engine design process, in which specialist solvers are employed to predict, understand and analyse the engine’s behaviour at various operating conditions. Sub-idle whole engine performance synthesis solvers are not as reliable and accurate as design point solvers. Lack of knowledge and data result in component characteristics being reverse-engineered or extrapolated from above-idle data. More stringent requirements on groundstart and relight capabilities, has prompted the need to advance the knowledge on low-speed engine performance, thereby requiring more robust sub-idle performance synthesis solvers. The objective of this study, was to improve the accuracy and reliability of a current aero gas turbine sub-idle performance solver by studying each component in isolation through numerical simulations. Areas researched were: low-speed and locked-rotor com- pressor characteristics, low-power combustion efficiency, air blast atomizer and combustor performance at sub-idle, torque-based whole engine sub-idle performance synthesis, and mixer performance at far off-design conditions. The observations and results from the numerical simulations form the contribution to knowledge of this research. Numerical simulations of compressor blades under highly negative incidence angles show the complex nature of the flow, with the results used to determine a suitable flow deviation model, a method to extract blade aerodynamic char- acteristics in highly separated flows, and measure the blockage caused by highly separated flow with operating condition and blade geometry. The study also concluded that the use of Blade Element Theory is not accurate enough to be used at such far off-design con- ditions. The linearised parameter-based whole engine performance solver was converted to used torque-based parameters, which validated against engine test data, shows that it is suitable for low-power simulations with the advantage of having the potential to start engine simulations from static conditions. A study of air-blast atomization at windmilling relight conditions has shown that current established correlations used to predict spray characteristics are not suitable for altitude relight studies, tending to overestimate the atomization quality. Also discovered is the highly influential interaction of compressor wakes with the combustor and atomizer under altitude relight conditions, resulting in more favourable lighting conditions than previous assumptions and models have shown. This is a completely new discovery which will result in a change in the way combustors are designed and sized for relight conditions, and the way combustion rig tests are conducted. The study also has valuable industrial contributions. The locked-rotor numerical data was used within a stage-stacking compressible flow code to estimate the compressor sub- idle map, of which results were used within a whole engine performance solver and results validated against actual engine test data. The atomization studies at relight were used to factor in the insensitivity of current spray correlations, which together with a newly de- veloped sub-idle combustion efficiency sub-routine, are used to determine the combustion efficiency at low-power settings. The interaction of compressor wakes with the atomizer showed that atomizer performance at relight is underestimated, resulting in oversized combustors. By using the knowledge gained within this research, combustor size can be reduced, resulting in lower NOx at take-off and a smaller and lighter core, with a com- bustor requiring less cooling air. The component research has advanced the knowledge and modelling capability of sub-idle performance solvers, increasing their reliability and encouraging their use for future aero gas turbine engines

Current and rotation driven instabilities in Couette flows

Couette flows are an invaluable tool in understanding some of the most important processes in astrophysical and geophysical fluid dynamics. When driven by rotation or electric currents, instability in such flows can be a key ingredient in a number of fundamental processes. For example, Taylor-Couette flow can give rise to the magnetorotational instabilities widely thought to be responsible for angular momentum transport in astrophysical disks. In this thesis, we add to the understanding of instabilities in Couette flows by focussing on two specific applications; inductionless magnetorotational instability in Taylor-Couette flow, and narrow-gap spherical Couette flows. After providing an overview of the numerical methods utilised throughout, we perform a linear stability analysis on inductionless magnetorotational instability, allowing for a fully generalised set-up which allows for every possible combination of imposed currents. A full exploration of the relevant parameter space is given. We then introduce the generalised quasilinear approximation, which serves to restrict nonlinear interaction between modes. In doing so, we are able to ascertain which modal interactions are essential to the key flow dynamics. More importantly, this is formally equivalent to the cumulant expansions utilised in the growing field of direct statistical simulation. Therefore, we are able to posit the future usefulness of statistical simulations to general wall-bounded flows. Finally, we utilise direct numerical simulation to probe the existence of axisymmetric pulse train solutions in narrow-gap spherical Couette flow, in which computations have been, until now, unable to utilise sufficiently narrow gap widths. As such, the only detailed prior solutions consist of asymptotic studies. We numerically confirm the existence of these pulse trains, and chart their initial bifurcations from the steady state solution. We also examine their magnetohydrodynamic equivalents, which have so far not been considered

Research reports: 1985 NASA/ASEE Summer Faculty Fellowship Program

A compilation of 40 technical reports on research conducted by participants in the 1985 NASA/ASEE Summer Faculty Fellowship Program at Marshall Space Flight Center (MSFC) is given. Weibull density functions, reliability analysis, directional solidification, space stations, jet stream, fracture mechanics, composite materials, orbital maneuvering vehicles, stellar winds and gamma ray bursts are among the topics discussed

Design study of a horizontal axis tidal turbine blade

Tidal current power generation offers a prospect of renewable energy which is predictable, and has lower CO2 emissions than traditional energy generation sources. It also has the potential to fulfil a significant part of the energy requirements of the UK and the rest of the world. The horizontal axis tidal turbine (HATT) acts as one of the means to convert the kinetic energy available in seawater into mechanical energy, and this research explores the hydrodynamics and the Computational Fluid Analysis (CFD) based design study of this. The first aim of this research was to develop a novel HATT blade shape through bio-mimicking a curved caudal fin shape to produce improved power coefficient. A second aim was to compare two different turbulence modelling techniques to enable the comparison of the power coefficients with the standard HATT models in tidal turbine blade literature. There were two types of numerical approaches used: The SST model and a more complex mathematical model, LES-Smagorinsky, to perform steady state and transient CFD analysis respectively on the designed blades using ANSYS CFX. The initial default HATT was designed, parameterised, and represented as a straight blade following to the standard HATT literature. The airfoil centres of the straight blade are built around the centreline, where the centreline acts as the master, and a novel third order polynomial function was integrated on the centreline to model the Blue Marlin fish caudal fin look-alike target shape. This approach was used to model the further 3 sets of curved blade shapes in percentage wise chord lengths. The CFD analysis of the two dimensional airfoils was conducted using ANSYS CFX, and compared against the literature. A further comparative analysis was performed with different mesh settings, and using the SST turbulence model. The comparative analysis formed an integral part of the CFD analysis to define the boundary conditions and the verification of the three dimensional CFD based HATT design study. The design strategy to move the curved blade backwards to the straight blade was also developed. The results obtained from the three dimensional comparative CFD analysis show good agreement between the two different turbulence modelling techniques used also producing an improved curved blade shape achieving the power coefficient of 0.5073% for SST simulations and 0.5178% for the LES-Smagorinsky CFD simulations. It is seen that LES-Smagorinsky CFD results produce slightly greater efficiency than the SST simulations, but the computational overhead required is massive. Finally, after comparing the improved efficiency of the bio-mimicked curved blade with the standard HATT models in the literature, it can proved that bio-mimicking the caudal fin look-alike blade produces a higher power coefficient than the standard HATT blade

Ames research center publications, 1975

This bibliography cites 851 documents by Ames Research Center personnel and contractors which appeared in formal NASA publications, journals, books, patents, and contractor reports in 1975, or not included in previous annual bibliographies. An author index is provided

Theoretical study of ion toroidal rotation in the presence of lower hybrid current drive in a tokamak

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Nuclear Science and Engineering, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 175-182).In this thesis, the effect of the lower hybrid current drive on ion toroidal rotation in a tokamak is investigated theoretically. Lower hybrid frequency waves are utilized to drive non-inductive current for steady state tokamaks and ion toroidal rotation is used to control disruptions and improve confinement. It has been observed in many tokamaks that lower hybrid waves can change the ion toroidal rotation. These measurements indicate that it may be possible to control rotation with lower hybrid waves, but to do it, it is necessary to understand the mechanisms underlying the rotation change. The toroidal angular momentum injected by the lower hybrid waves initiates acceleration in the the counter-current direction. The parallel and perpendicular components of the toroidal angular momentum are transferred from the waves to ions through electrons via two different channels, and the ions obtain the full toroidal angular momentum injected by the lower hybrid waves after several ion collision times. The momentum transferred to the ions is transported out by turbulent radial transport. The radial transport of toroidal angular momentum is evaluated using gyrokinetics corrected to the higher order in poloidal rhostar. The higher order corrections lead to momentum redistribution even in the absence of rotation, which is called intrinsic momentum transport. The intrinsic momentum transport due to diamagnetic effects is an important piece of the radial momentum transport. The change in the steady state rotation due to lower hybrid waves is estimated theoretically by evaluating the momentum source, the momentum pinch and diffusion, and the intrinsic momentum transport. The effect of the current profile on the intrinsic momentum transport, which is modified by the lower hybrid wave, may explain the reversal of the rotation change from counter-current direction to co-current direction observed in low plasma current discharges in Alcator C-Mod.by Jungpyo Lee.Ph.D