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Differencing operators of arbitrarily high order can be constructed by interpolating a polynomial through a set of data followed by differentiation of this polynomial and finally evaluation of the polynomial at the point where a derivative approximation is desired. Furthermore, the interpolating polynomial can be constructed from algebraic, trigonometric, or, perhaps exponential polynomials. This paper begins with a comparison of such differencing operator construction. Next, the issue of proper grids for high order polynomials is addressed. Finally, an adaptive numerical method is introduced which adapts the numerical grid and the order of the differencing operator depending on the data. The numerical grid adaptation is performed on a Chebyshev grid. That is, at each level of refinement the grid is a Chebvshev grid and this grid is refined locally based on wavelet analysis. Jameson, Leland Langley Research Center NAS1-19480; RTOP 505-90-52-01...
Constructing numerical schemes which are both adaptive and suitable for parallel architectures is very challenging. The challenge lies in the need to maintain a balanced load across the processing elements using a method that is both efficient and scalable. Here we propose a method which is adaptive, load balanced, absolutely efficient and scalable offering significant speedup over lower order adaptive schemes. The ability of wavelets to accurately and efficiently represent functions with localized features has spawned intensive research into applying wavelets for the solution of partial differential equations with the promise of significantly reducing the necessary computational effort and memory requirements. Traditionally, this effort has been centered around using wavelets as an orthogonal and complete basis, spanning a space in which to seek approximate solutions satisfying the equation in a Galerkin sense. Besides from the well known difficulties associated with such an approach for non-linear problems, one is also faced with the problem of dealing with non-trivial boundary conditions in an accurate and stable manner. Such restrictions on the applicability of wavelet based methods for the solution of problems of more general interest have, in recent years, induced significant interest into grid-based collocation wavelet methods, with various different approaches being taken. The formulation and implementation of multi-dimensional pure wavelet collocation methods, however, remains a challenging task and many issues require attention. In the present work we take a somewhat different approach to arrive at a grid based method utilizing the unique properties of wavelets. Rather than using the wavelets as a basis, we utilize the ability of wavelets to not only detect the existence of high-frequency information but also to supply information about the spatial location of such strongly inhomogeneous regions.
This book provides comprehensive information on the conceptual basis of wavelet theory and it applications. Maintaining an essential balance between mathematical rigour and the practical applications of wavelet theory, the book is closely linked to the wavelet MATLAB toolbox, which is accompanied, wherever applicable, by relevant MATLAB codes. The book is divided into four parts, the first of which is devoted to the mathematical foundations. The second part offers a basic introduction to wavelets. The third part discusses wavelet-based numerical methods for differential equations, while the last part highlights applications of wavelets in other fields. The book is ideally suited as a text for undergraduate and graduate students of mathematics and engineering.
This volume contains papers selected from the Wavelet Analysis and Multiresolution Methods Session of the AMS meeting held at the University of Illinois at Urbana-Champaign. The contributions cover: construction, analysis, computation and application of multiwavelets; scaling vectors; nonhomogenous refinement; mulivariate orthogonal and biorthogonal wavelets; and other related topics.
This collection of independent case studies demonstrates how wavelet techniques have been used to solve open problems and develop insight into the nature of the systems under study. Each case begins with a description of the problem and points to the specific properties of wavelets and techniques used for determining a solution. The cases range from a very simple wavelet-based technique for reducing noise in laboratory data to complex work on two-dimensional geographical data display conducted at the Earthquake Research Institute in Japan. One case study shows how wavelet analysis is used in the development of a Japanese text-to-speech system for personal computers and another presents new wavelet techniques developed for and applied to the study of atmospheric wind, turbulent fluid, and seismic acceleration data. Although calculus and some junior and senior mathematics courses for scientists and engineers will suffice, a solid background in undergraduate mathematics, particularly analysis and numerical analysis, and some familiarity with the basics of wavelets are helpful for reading this book.
This is the first book to present a systematic review of applications of the Haar wavelet method for solving Calculus and Structural Mechanics problems. Haar wavelet-based solutions for a wide range of problems, such as various differential and integral equations, fractional equations, optimal control theory, buckling, bending and vibrations of elastic beams are considered. Numerical examples demonstrating the efficiency and accuracy of the Haar method are provided for all solutions.
Computational Mechanics of Composite Materials lays stress on the advantages of combining theoretical advancements in applied mathematics and mechanics with the probabilistic approach to experimental data in meeting the practical needs of engineers. Features: Programs for the probabilistic homogenisation of composite structures with finite numbers of components allow composites to be treated as homogeneous materials with simpler behaviours. Treatment of defects in the interfaces within heterogeneous materials and those arising in composite objects as a whole by stochastic modelling. New models for the reliability of composite structures. Novel numerical algorithms for effective Monte-Carlo simulation. Computational Mechanics of Composite Materials will be of interest to academic and practising civil, mechanical, electronic and aerospatial engineers, to materials scientists and to applied mathematicians requiring accurate and usable models of the behaviour of composite materials.