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   -> Volume 4, Issue 6


Preprint: Papers on the Wavelet Extrapolation method.
 
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Kevin Amaratunga (kevin@phaeton.mit.edu)
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PostPosted: Tue Dec 03, 2002 1:19 pm    
Subject: Preprint: Papers on the Wavelet Extrapolation method.
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Preprint: Papers on the Wavelet Extrapolation method.

The following preprints are now available on line. These papers describe
the Wavelet Extrapolation method for the treatment of finite length data
sequences and for the solution of PDEs on domains of arbitrary shape.
Please see our web page at

http://wavelets.mit.edu/Kevin/Papers/Waveextrap-fwt.pdf

for downloading information and for image examples. (Alternatively, email
us at the addresses given below.)

Title: A Discrete Wavelet Transform Without Edge Effects

IESL Technical Report No: 95-02

Authors: John R. Williams and Kevin Amaratunga

Abstract:
In this paper, we use Daubechies' orthogonal wavelets to develop a
discrete wavelet transform which does not exhibit edge effects. This
work follows on from our earlier work on high order polynomial
extrapolation methods for initial and boundary value problems. The
underlying idea is to extrapolate the data at the boundaries by
determining the coefficients of a best fit polynomial through data
points in the vicinity of the boundary.


Title: Time Integration Using Wavelets

Presented at: SPIE Aerosense `95 - Wavelet Applications II
Vol 2491, 894-902, Orlando, FA, 1995.

Authors: Kevin Amaratunga and John R. Williams

Abstract:
In this work, we describe how wavelets may be used for the temporal
discretization of ODEs and PDEs. A major problem associated with the
use of wavelets in time is that initial conditions are difficult to
impose. A second problem is that a wavelet-based time integration
scheme should be stable. We address both of these problems.

Firstly, we describe a general method of imposing initial conditions,
which follows on from some of our recent work on initial and boundary
value problems. Secondly, we use wavelets of the Daubechies family as
a starting point for the development of stable time integration
schemes. By combining these two ideas we are able to develop schemes
with a high order of accuracy. More specifically, the global error is
O(h^{p-1}), where p is the number of vanishing moments of the
original wavelet. Furthermore, these time integration schemes are
characterized by large regions of absolute stability, comparable to
increasingly high order BDF methods. In particular, Daubechies D4 and
D6 wavelets give rise to A-stable time-stepping schemes.

In the present work we deal with single scale formulations. We note,
however, that the standard multiresolution analysis for orthogonal
wavelets on L^2(R) applies here. This opens up interesting possibilities
for treating BVP's and IVP's at multiple scales.


Title: High Order Wavelet Extrapolation Schemes for Initial Value Problems
and Boundary Value Problems

IESL Technical Report No: 94-07

Authors: John R. Williams and Kevin Amaratunga

Abstract:
One of the main problems with the Wavelet-Galerkin method is the
treatment of boundary conditions. Here, we describe a general method
of imposing boundary conditions based on polynomial extrapolation.
The resulting schemes are exact if the solution is a polynomial of
degree p-1, where p is the number of vanishing moments of the
wavelet. More generally, numerical evidence confirms that the error
decays as O(h^p) for pure Dirichlet boundary conditions and as
O(h^{p-1}) for Neumann boundary conditions. Boundary conditions can
be imposed at points other than mesh points without loss of accuracy.
This makes the method suitable for boundaries of arbitrary shape.

We extend the polynomial extrapolation idea to initial value problems
in two ways. First, we describe how the method may be used to impose
initial conditions. Secondly, the method is used to develop
stable time-stepping schemes for the wavelet coefficients. These
schemes have a global error of O(h^{p-1}), and they have large
regions of absolute stability, comparable to those of increasingly
high order BDF methods. In particular, Daubechies D4 and D6 wavelets
give rise to A-stable time-stepping schemes.

In the present work we deal with single scale formulations. We note,
however, that the standard multiresolution analysis for orthogonal
wavelets on L^2(R) applies here. This leaves us with interesting
possibilities for treating BVP's and IVP's at multiple scales, without
the use of special boundary wavelets.

John R. Williams Kevin Amaratunga
<john@iesl.mit.edu> <kevin@iesl.mit.edu>
Associate professor / Director Graduate student / Research assistant

Intelligent Engineering Systems Laboratory, Room 1-253
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139, USA
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