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Wavelets are a base set with extraordinary properties for the solution of differential equations. They allow to deal with problems wich conventional methods fail to resolve. This book is intended to make the theory of wavelets understandable to numerical practitioners interested in solving partial differential equations with wavelets. In addition to a self-contained and intuitive presentation of the theory of wavelets, this book contains extensive tables with the basic filter coefficients of differential operators in several wavelets families. Anyone should be able to solve partial differential equations numerically in physics, chemistry or engineering using wavelets. It is expected that this theory will soon become part of science and engineering curricula like Fourier analysis is nowadays.
this book allows non-mathematicans to understand the latest research results reached with techniques of wavelets
this book is mainly education oriented and will be accessible to anybody with standard mathematical knowledge of an engineer or a scientist
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Wavelets, an optimal basis set. A first tour of some wavelet families. Forming a basis set. The Haar wavelet. The concept of Multi-Resolution Analysis. The fast wavelet transform. How to plot wavelets. Interpretation of a wavelet spectrum. Interpolating wavelets. Expanding polynomials in a wavelet basis. Orthogonal versus biorthogonal wavelets. Expanding functions in a wavelet basis. Dynamic versus static data compression. Expanding nonuniform data in wavelets. Lifted wavelets. Second generation wavelets. Wavelet based smoothing of a function. The Fourier spectrum of wavelets. Wavelets in 2 and 3 dimensions. Wavelet grids in higher dimensions. The standard operator form. The non-standard operator form. Calculation of differential operators. Differential operators in higher dimensions. Transforming between wavelet families. Scalar products. The solution of Poisson's equation. The solution of Schrödingers's equation. Efficient implementation of filter operations. Outlook and conclusions. Appendix: Various filter coefficients.Solution of selected exercices. References.
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"Ce livre fait partie d'une collection qui publie les meilleurs cours dispensés en Suisse Romande au niveau du troisième cycle, pouvant intéresser les chercheurs en physique, les enseignants ou leurs étudiants à partir de la licence. En plus de l'intérêt du sujet traité, et de son niveau élevé, il semble que les ouvrages de cette collection privilégient les qualités de clarté et de concision."
Bulletin de l'Union des Physiciens. Avril 1999. http://www.cnam.fr/hebergement/udp
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