Polynomial Wavelet Research Articles

A COMPARATIVE STUDY OF A CLASS OF LINEAR AND NONLINEAR PANTOGRAPH DIFFERENTIAL EQUATIONS VIA DIFFERENT ORTHOGONAL POLYNOMIAL WAVELETS

We propose a wavelet approach on different orthogonal polynomials for solving linear and nonlinear pantograph equations with stretch kind. The pantograph differential equation is a unique proportional delay functional differential equation class. It has been used to deal with numerous physics, mathematics, and engineering applications, such as quantum mechanics, control systems, electrodynamics, and number theory. This scheme is based on constructing the operational matrix for integration via different wavelets with their collocation nodes. This study aims to examine the numerical dynamics of the pantograph equation under stretch kind through different orthogonal polynomial wavelets. Illustrative examples are presented to highlight the flexibility of this scheme, and comparisons are made between the mentioned scheme and other existing schemes using tables and graphs. These numerical results correctly predict the applicability and effectiveness of the mentioned scheme.

A New Algorithm Based on Bernstein Polynomials Multiwavelets for the Solution of Differential Equations Governing AC Circuits

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

Epileptic seizures detection based on some new Laguerre polynomial wavelets, artificial neural networks and support vector machines

Abstract Electroencephalography (EEG) is noninvasive and it is an effective tool to understand the complex nature of the brain. Its analysis by visual inspection is tedious and costly, and that is why many researchers in recent years resort to the use of computer-assisted diagnosis methods. In this work, we propose a new basis of wavelets constructed with Laguerre polynomials inspired by the similarities and differences between the Mexican hat wavelets family and the Gaussian wavelets family. The constructed wavelets are applied in epileptic seizure detection of EEG signals with the help of artificial neural networks and support vector machines for classification. We use a benchmark database, which consists of five hundred signals divided into five different classes, to solve eight clinically relevant classification problems with temporal and spectral features extracted from a continuous wavelet transform (CWT) of the EEG signals and the constructed wavelets. The spectral representation shows an intensive intermediary frequency (10–30 Hz) activity for seizure signals, while the other signals are characterized by a small high frequency (30–40 Hz) activity. We verify that, for any EEG classification problem, the model made up of CWT-PCA-SVM with a quadratic kernel is a classic case of overfitting irrespective of the wavelets used. The results obtained with wavelets constructed by Laguerre polynomials with a three-, five- and ten-fold cross-validation are better than several state-of-the-art classification methods in terms of classification accuracy.

Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations

In this article, a new method is generated to solve nonlinear Lane–Emden type equations using Legendre, Hermite and Laguerre wavelets. We are interested to note that these wavelets will give same solutions with good accuracy. Theorems on convergence analysis are stated and proved on the spaces, which are created by Legendre, Hermite and Laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis. The main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients. A very high level of accuracy reflects the reliability of this scheme for such problems.

Polynomial wavelets on the unit circle

In this paper, we have considered polynomial wavelets on unit circle. The scaling functions are considered to be the fundamental polynomials of the Lagrange interpolants on the equally spaced nodes different from the n roots of unity, which satisfy certain interpolatory conditions.

Multidimensional wavelet neural networks Based on polynomial powers of sigmoid

Wavelet functions have been used as the activation function in feed forward neural networks. An abundance of R&D has been produced on wavelet neural network area. Some successful algorithms and applications in wavelet neural network have been developed and reported in the literature. However, most of the aforementioned reports impose many restrictions in the classical back propagation algorithm, such as low dimensionality, tensor product of wavelets, parameters initialization, and, in general, the output is one dimensional, etc. In order to remove some of these restrictions, a family of polynomial wavelets generated from powers of sigmoid functions is presented. We described how a multidimensional wavelet neural networks based on these functions can be constructed, trained and applied in pattern recognition tasks. As examples of applications for the method proposed a framework for face verfication is presented.

MULTIRESOLUTION ON <b><i>n</i></b>-DIMENSIONAL SPHERES

In the present paper, multiscale systems of polynomial wavelets on an n-dimensional sphere are constructed. Scaling functions and wavelets are investigated,and their reproducing and localization properties and positive definiteness are examined. Decomposition and reconstruction algorithms for the wavelet transform are presented. Formulae for variances in space and momentum domain, as well as for the uncertainty product, of zonal functions over n-dimensional spheres are derived and applied to the scaling functions.

Local Polynomial Wavelet Neural Network with a Nonlinear Structured Parameter Optimization Method

This paper presents a Local Polynomial Wavelet Neural Network with a Structured Nonlinear Parameter Optimization Method (LPWNN-SNPOM). The LPWNN-SNPOM is an improvement of the Wavelet Neural Network with a Hybrid Learning Approach (WNN-HLA). These two models have mainly three differences: (i) The LPWNN-SNPOM method contains a bias, whose main contribution is to shift the output by mapping all points to the mean of the target points and leaving the hidden neurons and polynomial weight model the differences from that point. The bias may ameliorate the overall performance of the network to a certain degree; (ii) The single parameter weights connecting the hidden layer with the output in the WNN-HLA are replaced by polynomial functions of the inputs in the LPWNN-SNPOM, allowing the weights to vary with the changes in the input and share the dynamics with the wavelet compartment; (iii) Unlike the WNN-HLA, which uses an online optimization approach, the LPWNN-SNPOM makes usage of an offline optimization approach known as the Structured Nonlinear Parameter Optimization Method (SNPOM) to avoid any failure that may occur during online optimization. The performance and effectiveness of the proposed model are illustrated using several examples, whose results show the feasibility of the proposed model and demonstrate that it improved upon the WNN-HLA and performed better than some other well-known models.

Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions

Space-time variational formulations and adaptive Wiener–Hermite polynomial chaos Galerkin discretizations of Kolmogorov equations in infinite dimensions, such as Fokker–Planck and Ornstein–Uhlenbeck equations for functions defined on an infinite-dimensional separable Hilbert space $$H$$ , are developed. The well-posedness of these equations in the Hilbert space $$\mathrm{L}^2(H,\mu )$$ of functions on the infinite-dimensional domain $$H$$ , which are square-integrable with respect to a Gaussian measure $$\mu $$ with trace class covariance operator $$Q$$ on $$H$$ , is proved. Specifically, for the infinite-dimensional Fokker–Planck equation, adaptive space-time Galerkin discretizations, based on a wavelet polynomial chaos Riesz basis obtained by tensorization of biorthogonal piecewise polynomial wavelet bases in time with a spatial Wiener–Hermite polynomial chaos arising from the Wiener–Itô decomposition of $$\mathrm{L}^2(H,\mu )$$ , are introduced. The resulting space-time adaptive Wiener–Hermite polynomial Galerkin discretization algorithms of the infinite-dimensional PDE are proved to converge quasioptimally in the sense that they produce sequences of finite-dimensional approximations that attain the best possible convergence rates afforded by best $$N$$ -term approximations of the solution from tensor-products of multiresolution (wavelet) time-discretizations and the Wiener–Hermite polynomial chaos in $$\mathrm{L}^2(H,\mu )$$ . As a consequence, the proposed adaptive Galerkin solution algorithms exhibit dimension-independent performance, which is optimal with respect to the algebraic best $$N$$ -term rate afforded by the solution and the polynomial degree and regularity of the multiresolution (wavelet) time-discretizations in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of “active” coordinates identified by the proposed adaptive Galerkin approximation algorithms. The computational work and memory required by the proposed algorithms scale linearly with the support size of the coefficient vectors that arise in the approximations, with dimension-independent constants.

A uniqueness theorem for series by Stromberg’s system

The paper is devoted to the Stromberg’s polynomial wavelets. A uniqueness theorem is proved for series by this system. For Paley function of this system a weak estimate of type (1,1) is obtained. Counterexamples are also discussed.

Robust estimation for boundary correction in wavelet regression

For boundary problems present in wavelet regression, two common methods are usually considered: polynomial wavelet regression (PWR) and hybrid local polynomial wavelet regression (LPWR). Normality assumption played a key role for making such choices for the order of the low-order polynomial, the wavelet thresholding value and other calculations involved in LPWR. However, in practice, the normality assumption may not be valid. In this paper, for PWR, we propose three automatic robust methods based on: MM-estimator, bootstrap and robust threshold procedure. For LPWR, the use of a robust local polynomial (RLP) estimator with a robust threshold procedure has been investigated. The proposed methods do not require any knowledge of noise distribution, are easy to implement and achieve high performances when only a small amount of data is in hand. A simulation study is conducted to assess the numerical performance of the proposed methods.

Damage analysis of concrete structures using polynomial wavelets

This paper presents and discusses a hybrid-mixed stress finite element model based on the use of polynomial wavelets for the physically non-linear analysis of concrete structures. The effective stress and the displacement fields in the domain of each element and the displacements on the static boundary are independently approximated. As none of the fundamental equations is locally enforced a priori, the hybrid-mixed stress formulation enables the use of a wide range of functions. In the numerical model reported here, all approximations are defined using complete sets of polynomial wavelets. These bases present some important features. In one hand, the functions are orthogonal, which is an important issue when implementing hybrid-mixed stress elements as it ensures high levels of sparsity. On the other hand, the polynomial wavelet basis is defined through linear combinations of Legendre polynomials. This fact enables the use of closed-form solutions for the computation of the integrations involved in the definition of all linear structural operators. A simple isotropic damage model is adopted and a non-local integral formulation where the strain energy release rate is taken as the non-local variable is considered. The numerical model is both incremental and iterative and is solved with a modified Newton–Raphson method that uses the secant matrix. Classical benchmark tests are chosen to illustrate the use of the model under discussion and to assess its numerical performance.

Robust Wavelet Estimation to Eliminate Simultaneously the Effects of Boundary Problems, Outliers, and Correlated Noise

Classical wavelet thresholding methods suffer from boundary problems caused by the application of the wavelet transformations to a finite signal. As a result, large bias at the edges and artificial wiggles occur when the classical boundary assumptions are not satisfied. Although polynomial wavelet regression and local polynomial wavelet regression effectively reduce the risk of this problem, the estimates from these two methods can be easily affected by the presence of correlated noise and outliers, giving inaccurate estimates. This paper introduces two robust methods in which the effects of boundary problems, outliers, and correlated noise are simultaneously taken into account. The proposed methods combine thresholding estimator with either a local polynomial model or a polynomial model using the generalized least squares method instead of the ordinary one. A primary step that involves removing the outlying observations through a statistical function is considered as well. The practical performance of the proposed methods has been evaluated through simulation experiments and real data examples. The results are strong evidence that the proposed method is extremely effective in terms of correcting the boundary bias and eliminating the effects of outliers and correlated noise.

Hybrid wavelet denoising procedure of discontinuous surfaces

In this study, the authors propose a method for recovering a two-dimensional surface (image) from noisy observations containing significant jumps and discontinuities. The proposed procedure, termed as the segmented polynomial wavelet regression (SPWR) algorithm, combines wavelet regression with polynomial extrapolation and segmentation procedures. The bias that might occur on the discontinuous surface is alleviated through a segmentation process, and at the same time, inhomogeneous multi-scale features of the surface are efficiently treated by wavelet regression. The SPWR algorithm enjoys all the benefits of wavelet regression by implicitly detecting the discontinuities, and it is fast and easy to implement. Through a simulation study, it is demonstrated that the proposed method can produce substantially effective results.

Comparison of polynomial fitting and wavelet transform to remove drift in electrochemical noise analysis

Polynomial fitting and wavelet transform, which could be used to remove trends of electrochemical noise (EN), have been compared in this paper. The typical pitting EN records were chosen to study the effects of these two methods on statistical parameters, power spectral density (PSD), whole energy distribution plot (WEDP) and relative energy distribution plot (EDP), while the indication of selecting the appropriate method was put forward. The results show that for the calculation of the statistical parameters, the polynomial with m<5 and wavelet removal methods all could be selected. Power spectral density plots and WEDPs indicate that the higher order polynomial and wavelet removal methods would reduce the low frequency signals more heavily, so lower order (m=1–3) polynomial is recommended. The appearance of plateau in low frequency part of PSD plots could be regarded as the indication of the best selection of m. Combined EDPs with the analysis of the EN time records, the best m could also be selected, and its value might be just as the same as that obtained from PSD results. In general, the polynomial detrending is a more suitable and flexible trend removal method.

On concentration, deviation and Dvoretzky's theorem for Besov, Lizorkin–Triebel and other spaces

We obtain explicit estimates of the constants related to the concentrations of measures and distance, deviation and Dvoretzky's theorem for the finite-dimensional subspaces of a wide class of function and other spaces including, in particular, various anisotropic spaces of Besov, Lizorkin–Triebel and Sobolev types endowed with geometrically friendly norms defined in terms of averaged differences, local polynomial approximations, functional calculus, wavelets and other means. New approaches are shown to be providing better estimates in the abstract setting as well.

Polynomial wavelets in hybrid‐mixed stress finite element models

AbstractThis paper reports the use of polynomial wavelets as approximation functions in hybrid‐mixed stress finite element models applied to the solution of plane elasticity problems. The stress and displacement fields in the domain and the displacements on the static boundary are independently approximated. The kinematic boundary conditions are locally satisfied. All remaining equations are enforced in a weighted residual form so designed as to ensure that the discrete model embodies the relevant properties of continuum systems, namely the static‐kinematic duality and elastic reciprocity. A set of numerical applications is presented to illustrate the use of the hybrid‐mixed model and to assess its efficiency and accuracy. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Synchronous machine model identification using continuous wavelet NARX network

A new wavelet network structure combining polynomial models with continuous wavelet decomposition is introduced for the identification of a synchronous generator model. The proposed structure uses features of polynomial models and wavelet networks to model non-linearities in the system. In this study, a serial-parallel identification model is applied to system modelling. In this structure, real system outputs are exercised for prediction of the future system outputs, so that stability and approximation of the network are guaranteed. This method is applied to identify the model of a synchronous machine from experimental data collected on a physical machine. The results show that the identified model has very good accuracy and may be valid for a broad range of operating conditions.

Bayesian automatic polynomial wavelet regression

This paper considers the problem of Bayesian automatic polynomial wavelet regression (PWR). We propose three different Bayesian methods based on integrated likelihood, conditional empirical Bayes, and reversible jump Markov chain Monte Carlo (MCMC). From the simulation results, we find that the proposed methods are similar to or superior to the existing ones.

A recursive procedure to obtain a class of orthogonal polynomial wavelets

In this paper we present a recursive approach to generate complex orthogonal polynomial systems. The systems belong to a class of polynomial wavelets successfully introduced by Skopina [M. Skopina, Orthogonal polynomial Shauder bases in C [ − 1 , 1 ] with optimal growth of degrees, Sb. Math. 192 (3) (2001) 433–454; M. Skopina, Multiresolution analysis of periodic functions, East J. Approx. 3 (1997) 203–224]. Consequently, by using the obtained recursive-type relation, it is possible to generate a great variety of complex polynomial functions which satisfy useful wavelet-like properties. We prove some additional multiscale results concerning these systems. More precisely, we state a practical two-scale relation and the decomposition and reconstruction formulae which determine the multiresolution analysis framework. From the reconstruction formula, we obtain the recursive approach which provides the Skopina’s systems. Finally, a numerical example in which explicit complex orthogonal polynomials are found recursively is presented.