# Méthodes d'ondelettes pour l'analyse numérique d'intégrales oscillantes

Abstract : We have worked with three families of functions known for their localisation in space and in frequency: the wavelet basis, the wavelet packet basis, and the Malvar local cosine basis. We have constructed and implemented two algorithms: --- for the equation $\Delta(u)+\1e^(c*u)=f$ with $f$ having a tough singularity, we have proved sharp complexity estimates for the algorithm in 1D and 2D, in the wavelet basis, and have reached high precisions in 1D numerical results. --- for the oscillating boundary integral equation known as Combined Field Integral Equation, related to the Helmholtz diffraction problem around a 2D smooth obstacle, in the high frequency context. All above three function families have been tested, and we make a sharp analysis of the compressibility of the resulting system asymptotically for a decreasing wavelength. We have proved an original result, fully justifying the use of N degrees of freedom per wavelength. We have explained and proved the graphical repartitions in the thresholded matrix.
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https://tel.archives-ouvertes.fr/tel-00008353
Contributor : Daniel Goujot <>
Submitted on : Wednesday, April 6, 2005 - 11:04:29 AM
Last modification on : Thursday, November 19, 2020 - 4:22:03 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:31:06 PM

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• HAL Id : tel-00008353, version 1

### Citation

Daniel Goujot. Méthodes d'ondelettes pour l'analyse numérique d'intégrales oscillantes. Mathématiques [math]. Université d'Evry-Val d'Essonne, 2004. Français. ⟨tel-00008353⟩

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