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   -> Volume 3, Issue 9

Preprint: Fast wavelet transforms for matrices arising from BEMs.
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Author Message (David M. Bond)

PostPosted: Mon Dec 02, 2002 1:08 pm    
Subject: Preprint: Fast wavelet transforms for matrices arising from BEMs.
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Preprint: Fast wavelet transforms for matrices arising from BEMs.

The following preprint is available through an anonymous ftp to
"" in the directory "pub/tech.reports". The file
name is (The form is uncompressed PostScript). A list of all
available online CTC technical reports is located in the file
"tr.list", and the corresponding abstracts are located in the file
"tr.abstracts" in the same directory.

Cornell Theory Center Technical Report CTC94TR174 :


Dave M. Bond, Stephen A. Vavasis

Cornell Theory Center
Cornell University
Ithaca, New York, 14853

For many boundary element methods applied to Laplace's equation in
two dimensions, the resulting integral equation has both an integral
with a logarithmic kernel and an integral with a discontinuous kernel.
If standard collocation methods are used to discretize the integral
equation we are left with two dense matrices. We consider expressing
these matrices in terms of wavelet bases with compact support via a
fast wavelet transform as in Beylkin, Coifman and Rokhlin. Upper
bounds on the size of the wavelet transform elements are obtained.
These bounds are then used to show that if the original matrices are
of size $N imes N$, the resulting transformed matrices are sparse,
having only $O(N log N)$ significant entries. Some numerical results
will also be presented.

Unlike Beylkin, Coifman and Rokhlin who use the fast wavelet
transform as a numerical approximation to a continuous operator
already expressed in a full wavelet basis of $L2(R)$, we think of
the fast wavelet transform as a change of basis matrix for a finite
dimension, and apply it to a discretized function or matrix. As a
result, we can use this fast wavelet transform as a ``black box"
transformation in existing boundary element codes.

Sincerely, Dave Bond

719 Engineering & Theory Center Cornell University Ithaca, N.Y. 14853
e-mail :, phone : (607) - 254 - 8872
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