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   -> Volume 6, Issue 7

Preprint: Filter Bank Methods for Hyperbolic PDEs
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Johan Walden (

PostPosted: Fri Jul 11, 1997 10:53 am    
Subject: Preprint: Filter Bank Methods for Hyperbolic PDEs
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#4 Preprint: Filter Bank Methods for Hyperbolic PDEs

Title: Filter Bank Methods for Hyperbolic PDEs

Abstract: We use biorthogonal filter banks to solve hyperbolic PDEs
adaptively with a sparse multilevel representation of the signal. The
methods described are of finite difference type, and the filter banks
are used to give a sparse representation of signals, and to transform
between grids on different scales. We derive bounds for the error and
number of coefficients in the sparse representation. These bounds also
apply for filter banks that are not associated with any wavelets. We
develop algorithms for fast differentiation
and multiplication in detail. The strength of the method is shown in
various test problems.


Title: Filter Bank Subdivisions of Bounded Domains

Abstract: We construct filter bank transforms that are adapted to
bounded domains. The transforms are constructed with the aim of
solving PDEs together with finite difference methods, and the
properties important for this kind of application are analyzed. As the
filter banks do not need to correspond to wavelets, short filters can
be used. The price paid is extra growth factors in the number of
coefficients and in the error, compared to the multiresolution
analysis approach. However, these factors are small. We give upper
bounds of the growth factors, and show numerical examples for an
interval, a three-dimensional box, and a triangle. We also show
examples of fast differentiation and other operations on thresholded



Title: Filter Bank Preconditioners for Finite Difference Discretizations of PDEs

Abstract: We study preconditioners that are based on filter bank
methods. Filter banks are more general than biorthogonal wavelets, and
are easier to adapt to boundaries. The filter bank transform is shown
to efficiently decompose operators coming from the discretization of
PDEs with finite difference methods into two parts; one that has a
diagonal preconditioner, and one small part that can be directly
inverted. This feature is shown both for problems with periodic and
non-periodic boundary conditions; the change being small due to the
locality of the filter bank transform. In contrast to earlier work on
the topic, the ``right" transform is chosen for each problem, meaning
that both non-periodicity, and higher-dimensional tensor product
operators are taken into account. This is shown to improve the
method. For a one-dimensional problem, the condition numbers of the
problems involved are shown to be bounded by a constant, independently
of the problem size. The algorithmic complexity of!
the method is also analyzed.


Johan Walden
Uppsala University
Department of Scientific Computing
P.O. Box 120
S-75104, Uppsala
All times are GMT + 1 Hour
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