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   -> Volume 5, Issue 2


Thesis: Fast matrix products in wavelet bases
 
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Philippe Charton (Philippe.Charton@univ-reunion.fr)
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PostPosted: Wed Dec 04, 2002 9:49 am    
Subject: Thesis: Fast matrix products in wavelet bases
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Thesis: Fast matrix products in wavelet bases


Fast matrix products in wavelet bases: application to the
numerical resolution of PDE.

The following work was completed in January 1996 and is publicly
available via anonymous ftp at ftp.lmd.ens.fr in directory
/MFGA/pub/wavelets.

The author can be reached at philippe.charton@univ-reunion.fr.

Ph.D. DISSERTATION, January 1996:
Ph. Charton
Laboratoire de meteorologie dynamique du CNRS
ENS, Paris, France

Produits de matrices rapides en bases d'ondelettes : application a la
resolution numerique d'equations aux derivees partielles (120pp).

Wavelets are known to be zero-mean functions trying to fit the best
compromise between localization in physical space and localization in
spectral Fourier space. This property allows to characterize and to
economically represent functions with localized singularities as well
as some operators, especially Calderon-Zygmund operators. Thus, this
can be used to fast calculate the action of an operator by using
wavelets bases. For this purpose, we develop data structures and
algorithms adapted to sparse matrices which allow fast matrix-vector
and matrix-matrix products; we then optimized them for translation
invariant operators. Two methods of wavelets decomposition are
available for operators: the standard and non standard methods; we
compare them by applying them to the resolution of two evolution
problems: the heat equation and the advection equations. Numerical
results for both equations show lower computation time for the
standard method, since this method is the only one able to take into
account the compression of thesolution. The standard method is then
extended to 2D problems, thanks to the alternate direction method.
This allows us to define and use a ``pseudo wavelet" scheme, in order
to solve the Navier-Stokes incompressible equations in vorticity /
current function formulation. The non-linear term of the equation is
treated by a collocation method. The number of degrees of freedom
necessary for the ``pseudo wavelet" method is, for a given accuracy,
much smaller than the one of finite differences methods, or spectral
methods. This may be used to solve, at a given computational cost,
problems with a range of scales much larger than the one of classical
methods.

Keywords: Wavelets, Numerical solutions, PDE, sparse matrices,
adaptative methods, Calderon-Zygmund operator.

Philippe CHARTON (ATER) tel : (+262) 93 82 81
IREMIA secretaire :(+262) 93 82 82
Universite de La Reunion Reunion Time = TU + 4
BP 7151 fax : (+262) 93 82 60
15, av. Rene Cassin
F-97715 SAINT-DENIS MESSAG Cedex 9
All times are GMT + 1 Hour
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