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The Wavelet Digest
   -> Volume 2, Issue 6

Correction on WD 2.5 #5: Abstract from Wolfgang Dahmen's talk.
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PostPosted: Fri Nov 29, 2002 2:30 pm    
Subject: Correction on WD 2.5 #5: Abstract from Wolfgang Dahmen's talk.
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Correction on WD 2.5 #5: Abstract from Wolfgang Dahmen's talk.

By mistake the abstract of Wolfgang Dahmen's talk was not included
in the announcement of the wavelet minisymposium at the Dutch
Mathematical Conference in Amsterdam. It is included here (latex version).

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{large {f Computing inner products of wavelets}}\
W. Dahmen (joint work with C.A. Micchelli)

Employing wavelet-type bases for Galerkin discretizations of elliptic
boundary value problems leads to very good preconditioners and hence
to very efficient conjugate gradient solvers
for the resulting linear systems. Practical implementations of such
concepts, however, require the computation of inner products of wavelets.
It is pointed out that essentially all computational task needed for setting
up the linear systems can be reduced to the computation of quantities of
the form
intlimits_{R^s}varphi_0(x)prod_{j=1}^m (D^{mu^j}varphi_j)(x-alpha^j)dx~,
where $alpha^jin ^s, ~mu^jin ^s_+$ and the $varphi_j$'s are
are refinable functions, i.e., they satisfy two-scale relations
varphi_j(x)=sum_{alpha in ^s}a^j_{alpha}varphi_j(2x-alpha )
for some masks ${a^j_{alpha}}_{alpha in ^s}$. One should note that
for differential operators with variable coefficients one may indeed have
to handle more than two factors in ( ef{1}) and that it is very important
to admit different refinable functions occuring in such integrals.
It is shown that the exact (up to roundoff) computation of these integrals
can be reduced to an eigenvector/moment problem whose size depends only
on the support of the scaling functions $varphi_j$. The starting point is
the observation that ( ef{1}) may be viewed as the restriction of derivatives
of a certain refinable function of $ms$ variables to lattice points.
The main issue is to show that the above mentioned moment conditions,
which come from polynomial reproduction properties of scaling functions,
determine certain eigenvectors uniquely. Using results about convergence
of stationary subdivision schemes one can prove that this is indeed the case
when the initial scaling functions $varphi_j$ are stable. One should note
that when $s$ is larger than one, i.e., when dealing with partial differential
equations the uniqueness problem is much more involved than for $s=1$.
If time permits
some comments on current implementations will be made.
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