Wavelet Concepts Research Articles

Legendre Wavelet Collocation Solution for System of Linear and Nonlinear Delay Differential Equations

The proposed article is to describe the study of the system of linear and nonlinear delay differential equations subjected to initial-interval conditions. The existence and uniqueness theorem for solution of the proposed problem has been provided. The Legendre wavelet collocation method has been used in solution. We are presenting certain powerful tools as wavelet concepts, properties and description of this method. The main objective of the proposed method is to achieve high accuracy in minimum computation. The convergence analysis of Legendre wavelet collocation method is also presented. Some numerical examples of linear and non-linear system of delay differential equations are discussed in detail. The comparative study of Legendre wavelet collocation method, Exact, Runge–Kutta fourth order, dde23 (MATLAB solver) and Taylor’s series method solutions are also provided.

Cyclicality of real wages in the USA and Germany: New insights from wavelet analysis

This article provides new insights into the cyclical behavior of consumer and producer real wages in the USA and Germany. We apply two methods for the estimation of the cyclical components from the data: the approach based on the structural time series models and the ARIMA-model-based approach combined with the canonical decomposition and a band-pass filter. We examine the extracted cycles drawing on two wavelet concepts: wavelet coherence and wavelet phase angle. In contrast to the analysis in the time or frequency domains, wavelet analysis allows for the identification of possible changes in cyclical patterns over time. From the findings of our study, we can infer that the USA and Germany differ with respect to the lead–lag relationship of real wages and the business cycle. In the USA, both real wages are leading the business cycle in the entire time interval. The German consumer real wage is, on the other hand, lagging the business cycle. For the German producer real wage, the lead–lag pattern changes over time. We also find that real wages in the USA as well in Germany are procyclical or acyclical until 1980 and countercyclical thereafter.

Wavelet concepts for electrocardiographic signal analysis

Wavelet concepts for electrocardiographic signal analysis

Wavelet based correlation coefficient of time series of Saudi Meteorological Data

In this paper, wavelet concepts are used to study a correlation between pairs of time series of meteorological parameters such as pressure, temperature, rainfall, relative humidity and wind speed. The study utilized the daily average values of meteorological parameters of nine meteorological stations of Saudi Arabia located at different strategic locations. The data used in this study cover a period of 16 years between 1990 and 2005. Besides obtaining wavelet spectra, we also computed the wavelet correlation coefficients between two same parameters from two different locations and show that strong correlation or strong anti-correlation depends on scale. The cross-correlation coefficients of meteorological parameters between two stations were also calculated using statistical function. For coastal to costal pair of stations, pressure time series was found to be strongly correlated. In general, the temperature data were found to be strongly correlated for all pairs of stations and the rainfall data the least.

Analysis of surface degradation of high density polyethylene (HDPE) insulation material due to tracking

In the present work, tracking phenomena has been studied with HDPE material under a.c. voltage, with ammonium chloride as the contaminant. It is noticed that the tracking time depends on the conductivity and flow rate of the contaminant. The diffusion coefficient of the material was obtained. The thermal and chemical stability of the material were identified by carrying out a methodical experimental study. The physico-chemical analyses viz. wide angle X-ray diffraction (WAXD), differential scanning calorimetry (DSC) and dynamic mechanical analysis (DMA), were carried out and it was concluded that the mechanism of tracking process is due to the surface degradation. The surface condition of the insulation structure was characterized for any surface discharges or tracking, using the leakage current measurement, utilizing the wavelet concepts.

COMBINING MULTIGRID AND WAVELET IDEAS TO CONSTRUCT MORE EFFICIENT MULTISCALE ALGORITHMS

It is shown how various ideas that are well established for the solution of Poisson's equation using plane wave and multigrid methods, can be combined with wavelet concepts. The combination of wavelet concepts and multigrid techniques turns out to be particularly fruitful. We propose a modified multigrid V cycle scheme that is not only much simpler, but also more efficient than the standard V cycle. Whereas in the traditional V cycle the residue is passed to the coarser grid levels, this new scheme does not require the calculation of a residue. Instead it works with copies of the charge density on the different grid levels that were obtained from the underlying charge density on the finest grid by wavelet transformations. This scheme is not limited to the pure wavelet setting, where it is faster than the preconditioned conjugate gradient method, but equally well applicable for finite difference discretizations.

Wavelet correlation coefficient of ‘strongly correlated’ time series

In this paper, wavelet concepts are used to study two ‘strongly correlated’ financial time series. Apart from obtaining wavelet spectra, we also calculate the wavelet correlation coefficient and show that strong correlation or strong anti-correlation depends on scale.

Wavelet methods for PDEs — some recent developments

This paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues: adaptive solution concepts and nontrivial domain geometries. After describing a general multiresolution framework the key features of wavelet bases are highlighted, namely locality, norm equivalences and cancellation properties. Assuming first that wavelet bases with these properties are available on the relevant problem domains, the relevance of these features for a wide class of stationary problems is explained in subsequent sections. The main issues are preconditioning and the efficient (adaptive) application of wavelet representations of the involved operators. We indicate then how these ingredients combined with concepts from nonlinear or best N-term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations. These schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal. We conclude this section with some brief remarks on data structures and implementation, interrelations with regularity in a certain scale of Besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems. We address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems. Wavelet bases on nontrivial domains are discussed next. The main issues are the development of Fourier free construction principles and criteria for the validity of norm equivalences. Finally, we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems.

Introduction to wavelet-based compression of medical images.

Medical image compression can significantly enhance the performance of picture archiving and communication systems and may be considered an enabling technology for telemedicine. The wavelet transform is a powerful mathematical tool with many unique qualities that are useful for image compression and processing applications. Although wavelet concepts can be traced back to 1910, the mathematics of wavelets have only recently been formalized. By exploiting spatial and spectral information redundancy in images, wavelet-based methods offer significantly better results for compressing medical images than do compression algorithms based on Fourier methods, such as the discrete cosine transform used by the Joint Photographic Experts Group. Furthermore, wavelet-based compression does not suffer from blocking artifacts, and the restored image quality is generally superior at higher compression rates.

Multiscale Computation with Interpolating Wavelets

Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allowexactrecovery of the multiresolution representation of a function from its sample values on afiniteset of points in space. We present a detailed study of the application of wavelet concepts to physical problems expressed in such bases. The manuscript describes algorithms for the associated transforms which for properly constructed grids of variable resolution compute correctly without having to introduce extra grid points. We demonstrate that for the application of local homogeneous operators in such bases, the nonstandard multiply of Beylkin, Coifman, and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtain less stringent conditions on the grids, we generalize the nonstandard multiply so that communication may proceed between nonadjacent levels. The manuscript concludes with timing comparisons against naı̈ve algorithms and an illustration of the scale-independence of the convergence rate of the conjugate gradient solution of Poisson's equation using a simple preconditioning.

Applied Wavelet Analysis with S-PLUS.

From the Publisher: This book introduces applied wavelet analysis through the S-PLUS software system. Using a visual data analysis approach, wavelet concepts are explained in a way that is intuitive and easy to understand. Wavelets, as well as a whole range of related signal-processing techniques, such as wavelet packets, local cosine analysis, and matching pursuits are covered. Applications of wavelet analysis are illustrated, including nonparametric function estimation, digital image compressing, and time-frequency acoustic signal analysis.

A fast multiresolution moment‐method algorithm using wavelet concepts

AbstractA fast multiresolution moment‐method algorithm is developed based on the spectral‐domain wavelet concept. The computational savings are achieved by taking advantage of the multiscale feature of the spectral‐domain Green's function. It is shown that the multiresolution algorithm leads to reduced computational complexity in the matrix‐vector multiplication operation. © 1995 John Wiley & Sons, Inc.

On the efficient representation of the electrodynamic green's function using multiresolution wavelet concepts

AbstractA multiresolution wavelet concept is applied to efficiently represent the electrodynamic Green's function. In the joint spectral‐spatial representation, the multiscale feature of the spectral‐domain Green's function is clearly revealed. Due to this multiscale feature in the spectral domain, a wavelet application in the spectral domain (KDWT) is more appropriate in representing the Green's function than the wavelet transform applied in the space domain (SDWT). We present a multiresolution implementation of the KDWT and show that the KDWT representation results in a sparser moment impedance matrix than the SDWT representation. © 1995 John Wiley & Sons, Inc.

Utilization of wavelet concepts in finite elements for an efficient solution of Maxwell's equations

The principles of dilation and shift are two important properties that are attributed to wavelets. It is shown that inclusion of such properties in the choice of a basis in Galerkin's method can lead to a slow growth of the condition number of the system matrix obtained from the discretization of the differential form of Maxwell's equations. It is shown that for one‐dimensional problems the system matrix can be diagonalized. For two‐dimensional problems, however, the system matrix can be made mostly diagonal. This paper illustrates the application of the new type of “dilated” basis for a Galerkin's method (or equivalent, for example, finite element method) for the efficient solution of waveguide problems. Typical numerical results are presented to illustrate the concepts.

Measuring wavelet phase from seismic data

Geophysicists are well into their second decade of serious interest in the seismic wavelet. Several requirements — to invert seismic complexes into their causative reflection series, to do stratigraphic modeling, to correct section displays to zero phase form, to standardize data of different vintages for composite structural mapping — put a knowledge of the basic wavelet shape, which can provide the way to satisfy these needs, near the top of every wish list. Ever since the convolutional model, with its basic wavelet and reflectivity series concepts, was proposed more than 30 years ago, there has been some kind of continuous effort to estimate the wavelet in a reliable and convenient manner. In spite of this constant attention, wavelets today are usually determined by directly measuring them in the field, or by estimating them in the processing environment through assuming their phase properties. Phase assumptions always leave the serious wavelet estimator a little queasy. He is always looking for some d...