Use Of Spectral Methods Research Articles

An extended difference of coherence test for comparing and combining several independent coherence estimates: theory and application to the study of motor units and physiological tremor

Recently there has been an increase in the use of spectral methods for the analysis of experimental data. These analytical methods allow the study of interactions between simultaneously recorded signals and are particularly suited to the study of systems displaying rhythmic behaviour. A useful parameter in this context is the coherence function which provides a bounded measure of linear association between two signals. In this report we introduce two new techniques for dealing with an arbitrary number of independent coherence estimates. The first technique provides a test to compare the coherence estimates for statistically significant differences. The second allows the original coherence estimates to be combined, or `pooled' into a single representative estimate. These two measures, taken together, provide a powerful tool for characterising and summarising the correlations within data sets. Applications of the techniques are illustrated by analysing the interactions between single motor unit discharges and finger tremor, and between pairs of motor unit discharges in human subjects.

Global Modelling of the Ocean and Atmosphere Using the Spectral Element Method

ABSTRACT The use of spectral methods now has a long history in global atmospheric modelling wherein the attractive properties of Fourier series on spheres, including higher-order convergence rates and efficient implementation via the transform method, have proven advantageous. Partially offsetting these advantages, however, are several competing disadvantages. Two of these, the appearance of Gibbs oscillations for localized processes (e.g., orographic interactions) and the difficulty of mapping spectral techniques onto parallel computer architectures, are inherent to the global nature of these techniques. A third drawback, the restriction of these methods to regular geometries, has severely limited their application to the modelling of the large-scale ocean circulation. We describe a global circulation model that has, in principle, none of these limitations. The model utilizes the spectral element method that combines the geometrical flexibility of traditional finite element methods with the rapid convergence rates of spectral approximation techniques. Simple test problems drawn from both oceanic and atmospheric modelling are used to demonstrate that the resulting model is exponentially convergent, yet allows effective representation of irregular geometry and efficient grid refinement in regions of dynamical interest. Lastly, performance characteristics on the nCUBE/2 and Cray T3D architectures confirm that the element model is ideally suited to the parallel computing environment.

Thermal convection in vertical cylinders. A method based on potentials of velocity

Any solenoidal vector field can be written in terms of two scalar fields. Solving the Navier-Stokes equation for these two so-called potentials is a widely used method, although difficulties have been reported in using this method to describe non-Cartesian confined flows, in particular cylindrical geometries. Difficulties arise because the potentials are coupled at the boundaries but also because of possible numerical instabilities associated with the high order of the equations. This paper shows how all these difficulties can be circumvented: firstly by properly deriving the required boundary conditions, but also through an adequate discretization, which is the final requirement. The use of spectral methods together with a reduction of order for the axisymmetric modes is sufficient for this purpose. In this paper, as a test, a velocity potentials method is used to solve the stability problem for the onset of convection in a fluid confined in a cylindrical container.

Chebyshev and Legendre Spectral Methods in Algebraic Manipulation Languages

In this tutorial, we show how to combine Galerkin and collocation methods with algebraic manipulation languages. Because polynomial and trigonometric manipulations are a special strength of REDUCE, Maple, and other symbolic languages, it is easy to use Chebyshev and Legendre polynomials and Fourier series as the basis sets. However, because the truncation is small and calculations are performed symbolically in rational arithmetic, many of the standard precepts for numerical use of spectral methods must be modified or reversed. We offer many guidelines and suggestions. Six examples programmed in REDUCE and Maple—eigenproblems, bifurcations, and nonlinear diffusion—illustrate the possibilities.

Use of spectral methods of analysis and evaluation of the biological action of industrial pollution

Use of spectral methods of analysis and evaluation of the biological action of industrial pollution

Use of spectral methods to study branching processes with diffusion in a noncompact phase space

Use of spectral methods to study branching processes with diffusion in a noncompact phase space

Acoustic wave propagation in 2-D cylindrical coordinates

SUMMARY We present a spectral method for solving the 2-D acoustic wave equation in cylindrical coordinates. The method is based on discretization of the wavefield into a grid of r and θ where r is the distance from the centre, and θ is the radial angle. A Chebychev expansion is used to perform derivatives along the r coordinate and a Fourier expansion is used for calculating θ coordinate derivatives. The use of spectral methods in a cylindrical coordinate system enables us to calculate wave propagation in cylindrical type geometries very accurately. The algorithm is tested against problems with known analytical solutions.

Preliminary investigations on the use of spectral methods for flow calculations

Preliminary investigations on the use of spectral methods for flow calculations

Preliminary investigations on the use of spectral methods for flow calculations

Numerical simulation of fluid flows is increasing in popularity and utility in the study of wind effects on structures. Many different schemes, such as finite difference and finite elements, have been used. For this study, sponsored by the Federal Highway Administration, many of these methods were studied and compared. A Chebyshev-collocation Spectral method was eventually chosen for its extreme computational efficiency. Spectral Methods have been used in a variety of flow calculation problems. Others have applied it to channel flows and compressible flow problems. The current work is an attempt toward applying the spectral method to flow around bluff bodies, in particular rectangular prisms and flat plates. The first test of the technique was a model of bar deflecting along it's axis due only to it's own weight. Then, a two dimensional Poisson solver was implemented with the method. Next, a uni-domain Navier-Stokes solver was attempted. Currently, a multidomain two dimensional incompressible Navier Stokes solver is under development. In addition, a microcomputer flow visualization program has been developed using a steady state finite difference flow modeler to provide sample data. Finally, some conclusions and areas of future work will be discussed.

Spectral methods in numerical plasma simulation

An introduction is given to the use of spectral methods in numerical plasma simulation. As examples of the use of spectral methods, solutions to the two-dimensional Euler equations in both a simple, doubly periodic region, and on an annulus will be shown. In the first case, the solution is expanded in a two-dimensional Fourier series, while a Chebyshev-Fourier expansion is employed in the second case. A new, efficient algorithm for the solution of Poisson's equation on an annulus is introduced. Problems connected to aliasing and to short wavelength noise generated by gradient steepening are discussed.

On the use of spectral methods for the determination of fractal dimension

This paper discusses the use of fractal theory for quantitative analysis of profiles from two‐dimensional surfaces. The theoretical relationship between spectral parameters and fractal dimension is examined, including the conditions under which the derived relationship between spectral parameters and fractal dimension by Voss (1985) is valid, and the limitations in making inferences from spectral parameters. Applications of fractal theory to geophysical data are also discussed. In particular, it is shown that an amplitude spectrum with a decay corresponding to a fractal dimension of 1.5 can result from the concatenation of time series with decays corresponding to different fractal dimensions. It has been shown that the spectral density of a fractal distribution will be characterized by a power‐law decay, but this paper illustrates that a power‐law decay is not sufficient to identify a fractal distribution. Although this paper discusses applications to the study of Earth topography, the results are applicable to the study of any one‐dimensional profile.

On the use of spectral methods for the numerical solution of stiff problems

This paper describes some aspects of the use of spectral methods for the numerical solution of systems of stiff partial differential equations. It is shown that despite the high spatial precision of these methods, a reasonable accuracy can only be attained with a large number of number and therefore, some kind of adaptive ‘gridding’ is necessary. A way to implement this adaptation based on the computation of a norm of the solution is proposed. Numerical examples concerning flame propagation problems and Burges' equation are presented and discussed.

Spectral methods and their implementation to solution of aerodynamic and fluid mechanic problems

AbstractFundamental concepts underlying spectral collocation methods, especially pertaining to their use in the solution of partial differential equations, are outlined. Theoretical accuracy results are reviewed and compared with results from test problems. A number of practical aspects of the construction and use of spectral methods are detailed, along with several solution schemes which have found utility in applications of spectral methods to practical problems. Results from a few of the successful applications of spectral methods to problems of aerodynamic and fluid mechanic interest are then outlined, followed by a discussion of the problem areas in spectral methods and the current research under way to overcome these difficulties.

Time Series Analysis in Pooled Cross-Sections

This article proposes the use of spectral methods to pool cross-sectional replications (N) of time series data (T) for time series analysis. Spectral representations readily suggest a weighting scheme to pool the data. The asymptotically desirable properties of the resulting estimators seem to translate satisfactorily into samples as small as T = 25 with N = 5. Simulation results, Monte Carlo results, and an empirical example help confirm this finding. The article concludes that there are many empirical situations where spectral methods canbe used where they were previously eschewed.

The use of spectral methods for the spatial modelling of gravity data

The use of spectral methods for the spatial modelling of gravity data

A Chebyshev spectral method for boundary-layer models

Boundary-layers in the atmosphere and ocean often take on the structure of mixed layers bounded by narrow, highly stable regions. Simulation of the movement of such stable regions is a challenging task for the numerical modeler. Here we explore the use of Chebyshev spectral methods. We compare the results of spectral and finite difference methods for a simple one-dimensional test problem with known analytic solution. The results indicate the usefulness of the spectral method in obtaining accurate solutions to boundary-layer problems.

Wavenumber domain methods for fast interpretation of potential field data

At this stage of sophistication in the use of spectral methods for the interpretation of potential field data it seems appropriate to analyse critically the many methods that have been derived within the past decade or so. In this paper we take the following principal points of departure: 1. (1) The data which we have to analyse are: 1.1. (a) inaccurate. 1.2. (b) insufficient. 1.3. (c) inconsistent quoting Jackson. This means that: 1.1. (a) errors are attached to each datum. 1.2. (b) too few data have been measured to fully describe their variability. 1.3. (c) the models which we apply for explaining the data are inconsistent with the real field data. 2. (2) Fourier transformation of the data necessarily leads to the propagation of errors other than the original errors of the individual measurements. So, if it is attempted to make a detailed and time consuming analysis of the data, it is advisable to stay in the space domain. On the other hand, if a quick and rather coarse analysis is sufficient, it may be an advantage to switch to the wavenumber domain, where — in the case of simple bodies —the interpretational parameters often may be found from straight line fittings in certain ranges of wavenumbers. In this way some kind of averaging is performed, and the estimated parameters may be conceived as average ones.

Stoichiometric Interaction of a β‐Receptor Blocking Drug with Fraction of Liver Cytochrome P‐450 as a Possible Cause for its Dose‐dependent Availability

Abstract The liver's ability to bind and metabolize “first‐pass” drugs is a major factor responsible for their blood concentrations on oral administration. Therefore we have characterized the binding of such a drug, the adrenergic β‐receptor antagonist alprenolol, to rat liver. The microsomal fraction accounted for almost all the liver binding capacity. On titration of microsomes with alprenolol, the induced type I spectral change, reflecting alprenolol‐cytochrome P‐450 interaction, showed a high and a low affinity binding phase with identical “optical capacity”. The spectral dissociation constant for the high affinity binding was 0.17 μM. By use of spectral methods we found that 13–14 per cent of the total microsomal cytochrome P‐450 participated in the alprenolol binding at the end of the first phase and 27 per cent at the end of the second phase. Spectral titrations showed that SKF 525‐A and imipramine interact with the same sites of cytochrome P‐450 as alprenolol. Binding studies with radioactive alprenolol showed the presence of a minimum of three classes of microsomal binding sites. The last had a very high capacity. The first phase caused a marked non‐linearity in the overall microsomal binding of the drug, and it had almost identical capacity (0.12 mol alprenolol/mol P‐450) and affinity (Ks = 0.21 μM) as the first type I spectral binding phase. This strongly indicates that this first “overall” binding phase reflects selective binding to P‐450, which is further strengthened by the fact that SKF 525‐A and imipramine markedly influenced this phase. Moreover, this phase was not present in plasma membranes, which do not contain cytochrome P‐450. Hence we conclude that a stoichiometric (1:1) interaction of alprenolol with a 12–14 per cent fraction of rat liver microsomal cytochrome P‐450 causes a non‐linear overall microsomal binding. Such a phenomenon probably explains this drug's dose‐dependent availability. We suggest that when this high affinity site is saturated the low affinity site of cytochrome P‐450 oxidizes a large fraction of the alprenolol escaping ‘first pass’ liver extraction.

Use of spectral methods in soil science and agrochemistry in the Soviet Union

The article briefly describes the application of emission spectrographic, flame emission and atomic absorption methods in samples of soils and plants. The discussion is mainly on sources of error and methods of eliminating them, the role of standards and the requirements of soil-agrochemical analyses.

Method of separating sulfur compounds and aromatic hydrocarbons from high-boiling fractions of sour crudes

1. Feasibility has been demonstrated for the application to medium-viscosity distillates (boiling up to 450°C) of a method for recovering sulfur compounds and aromatic hydrocarbons that had been developed for low-viscosity distillates (300–400°C). 2. This method of recovering sulfur compounds and aromatic hydrocarbons, together with the use of spectral methods of analysis, makes it possible to study the structural features of the components in high-boiling petroleum fractions.