Wavelet Modeling of Control Stochastic Systems at Complex Shock Disturbances

This article is devoted to the development of methodological supports and experimental software tools for accuracy analysis and information processing in control stochastic systems (CStS) with complex shock disturbances (ShD) by means of wavelet Haar–Galerkin technologies. Basic new results include methods and algorithms of stochastic covariance analysis and modeling on the basis of the Galerkin method and wavelet expansion for linear, linear with parametric noises, and quasilinear CStS with ShD. Results are illustrated by an information-control system at ShD. New stochastic effects accumulation for systematic and random errors are detected and investigated.

A Haar Wavelet Series Solution of Heat Equation with Involution

It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations.

A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an M×M collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and Volterra nonlinear integral equations and achieves the desired absolute error of 0×10−31 for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation.

Identification of time-varying systems with partial acceleration measurements by synthesis of wavelet decomposition and Kalman filter

Structural systems often exhibit time-varying dynamic characteristics during their service life due to serve hazards and environmental erosion, so the identification of time-varying structural systems is an important research topic. Among the previous methodologies, wavelet multiresolution analysis for time-varying structural systems has gained increasing attention in the past decades. However, most of the existing wavelet-based identification approaches request the full measurements of structural responses including acceleration, velocity, and displacement responses at all dynamic degrees of freedom. In this article, an improved algorithm is proposed for the identification of time-varying structural parameters using only partial measurements of structural acceleration responses. The proposed algorithm is based on the synthesis of wavelet multiresolution decomposition and the Kalman filter approach. The time-varying structural stiffness and damping parameters are expanded at multi-scale profile by wavelet multiresolution decomposition, so the time-varying parametric identification problem is converted into a time-invariant one. Structural full responses are estimated by Kalman filter using partial observations of structural acceleration responses. The scale coefficients by the wavelet expansion are estimated via the solution of a nonlinear optimization problem of minimizing the errors between estimated and observed accelerations. Finally, the original time-varying parameters can be reconstructed. To demonstrate the efficiency of the proposed algorithm, the identification of several numerical examples with various time-varying scenarios is studied.

ON ESTIMATES OF COEFFICIENTS OF GENERALIZED ATOMIC WAVELETS EXPANSIONS AND THEIR APPLICATION TO DATA PROCESSING

Discrete atomic compression (DAC) of digital images is considered. It is a lossy compression algorithm. The aim of this paper is to obtain a mechanism for control of quality loss. Among a large number of different metrics, which are used to assess loss of quality, the maximum absolute deviation or the MAD-metric is chosen, since it is the most sensitive to any even the most minor changes of processed data. In DAC, the main loss of quality is got in the process of quantizing atomic wavelet coefficients that is the subject matter of this paper. The goal is to investigate the effect of the quantization procedure on atomic wavelet coefficients. We solve the following task: to obtain estimates of these coefficients. In the current research, we use the methods of atomic function theory and digital image processing. Using the properties of the generalized atomic wavelets, we get estimates of generalized atomic wavelet expansion coefficients. These inequalities provide dependence of quality loss measured by the MAD-metric on the parameters of quantization in the form of upper bounds. They are confirmed by the DAC-processing of the test images. Also, loss of quality measured by root mean square (RMS) and peak signal to noise ratio (PSNR) is computed. Analyzing the results of experiments, which are carried out using the computer program "Discrete Atomic Compression: Research Kit", we obtain the following results: 1) the deviation of the expected value of MAD from its real value in some cases is large; 2) accuracy of the estimates depends on parameters of quantization, as well as depth of atomic wavelet expansion and type of the digital image (full color or grayscale); 3) discrepancies can be reduced by applying a correction coefficient; 4) the ratio of the expected value of MAD to its real value behaves relatively constant and the ratio of the expected value of MAD to RMS and PSNR do not. Conclusions: discrete atomic compression of digital images in combination with the proposed method of quality loss control provide obtaining results of the desired quality and its further development, research and application are promising.

Polynomial and Wavelet-Type Transfer Function Models to Improve Fisheries’ Landing Forecasting with Exogenous Variables

It is well known that environmental fluctuations and fishing efforts modify fishing patterns in various parts of the world. One of the most affected areas is northern Chile. The reduction of the gaps in the implementation of national fisheries’ management policies and the basic knowledge that supports the making of such decisions are crucial. That is why in this research, a transfer function method with variable coefficients is proposed to forecast monthly disembarkation of anchovies and sardines in northern Chile, taking into account the incidence of large-scale climatic variables on landings. The method uses a least squares procedure and wavelets to expand the coefficients of the transfer function. Linear estimators of the time varying coefficients are proposed, followed by a truncation of the wavelet expansion up to an appropriate scale. Finally, the estimators for the transfer function coefficients are obtained by using the inverse wavelet transformation. Research results suggest that the transfer function models with variable coefficients fit the behavior of the anchovies’ landing with great accuracy, while the use of transfer function models with constant coefficients fits sardines’ landings better. Both fisheries’ landings could be explained to a large extent from the large scale climatic variables.

On the Rate of Convergence of Wavelet Expansions

In this paper we estimate the rate of convergence of wavelet expansion of functions <em>f</em> ∈ <em>L^p</em>, 1 ≤ <em>p</em> ≤ ∞ at a point x. The pointwise and <em>L^p</em> results were obtained by Kelly, S. [4]. Our result generalizes her result.

Investigation of the basic Haar wavelet-transformations in the problem of decryption of space images on detection of waste disposal fields

In this paper, we study the use of orthogonal transformations, namely, the basic Haar wavelet transforms, for data processing of the Earth remote sensing. The internal structure of orthogonal Haar transforms is considered. The Haar matrix is divided into blocks of the same type, so that parallelization of the computations is possible. The expediency of replacing the spectral components corresponding to the whole block (or several blocks) of the original matrix with zeros is asserted. Theoretical and experimental studies are carried out to improve the results of image classification (on the example of cluster analysis). The Haar wavelet expansion coefficients are used as indicators when decoding space images for the presence of waste disposal sites. The aim of this paper is to describe the approach, on the basis of which an optimal method is established on a class of vectors with real components, application of two-dimensional discrete Haar wavelet transformations in the problem of recognition of space images for the presence of waste disposal sites. General methodology of research. The paper uses elements of mathematical analysis, wavelet analysis, the theory of discrete orthogonal transformations, and methods for decoding cosmic images. Scientific novelty. Encoding by means of conversion is an indirect method, especially effective in processing of two-dimensional signals, in particular, space images used for remote sensing of the Earth. We propose the approach that takes into account the structure of the wavelet-Haar matrix, while recognizing waste disposal fields by means of space images. The article comprises the result of the experimental application of wavelet-Haar transformations for decoding of space images. We consider this case, both with and without the technique of taking into account the structure of the wavelet-Haar matrices.