From: Bruce Johnson
Subject: Re: Help with wavelets
Date: 27 Dec 1999 16:48:04 GMT
Newsgroups: sci.math.research,sci.physics.research
In sci.physics.research Tommi Hoynalanmaa wrote:
> I'm doing research on wavelets (solving the Schroedinger equation using
> wavelet basis functions). I need help for the following topics:
I'll give it a shot:
> 1. One and three-dimensional wavelet bases.
Standard use of wavelets refers to 1D Cartesian coordinates. Product
bases in 3 Cartesian coordinates can be used, where the scale and location
indices run separately. Alternatively, as is frequently the case in
image processing (2D) one can take tensor products of wavelets that are
all at the same scale (see, e.g., Daubechies' book Ten Lectures on
Wavelets). This may be appropriate if one expects behavior in x, y,
and z to be similar in scales, but is not always appropriate.
> 2. Representing the Laplacian operator in a wavelet basis in 1 and 3
> dimensions.
Depends on the type of wavelet basis. If you're referring to orthogonal
compact support Daubechies wavelets, see Beylkin, SIAM J. Numer. Anal.
vol 6, p 1716 (1992).
> 3. Calculating the values of wavelet basis functions in arbitrary points
> (ie. calculating the function itself when it is known as a linear
> combination of wavelets).
The Cascade Algorithm discussed by Daubechies converges to give scaling
functions and wavelets on the dyadic rationals k/2^j, and arbitrary
points can be obtained by interpolation. In general, the scaling function
values at the integers can be obtained by solving a small eigensystem,
and then the recursion relations give you the precise values at the dyadic
rationals. I'm sure you can read about this in many places, but I have
one ref, Burrus, et al., Introduction to Wavelets and Wavelet Transforms,
Prentice-Hall, 1996, close at hand.
> 3. Representing the multiplication of wavelet basis functions by an
> arbitrary function (the potential function) in the wavelet basis. That
> is, when the wave function is represented as a linear combination of
> orthogonal wavelets, and their coefficients form a vector, I need to
> represent the multiplication of the wave function by the potential
> function as a matrix that operates on that vector.
This again depends on the type of wavelet you're using. If you're
referring to orthogonal wavelets, you can use a few methods, one of
which is an adaptive quadrature technique, Johnson et al, J. Chem. Phys.,
vol. 110, p. 8309 (1999). This extends to biorthogonal wavelets and
multiwavelets. If you're using more general (not necessarily
orthogonal) wavelets, take a look in Vasilyev and Paolucci, J. Comp. Phys.,
vol. 125, p. 498 (1996); Goedecker and Ivanov, Sol. State Comm., vol 105,
p. 665 (1998); Arias, Rev. Mod. Phys., 1999, and their references. (Sorry,
don't have the vol and p for the last at the moment.)
Hope that's useful,
Bruce
==============================================================================
From: Bruce Johnson
Subject: Re: Wavelets in Numerical Analysis
Date: 4 May 1999 06:12:53 GMT
Newsgroups: sci.math.num-analysis
Keywords: How are wavelets used in practice?
Joe Lindula wrote:
> Hello, I've heard that wavelets are being used in numerical analysis.
> Could anyone briefly give an example or explain how they are used.
> I'll appreciate any suggestions.
> Joe
> jjlindula@netscape.net
A general reply:
There are now a huge number of books and papers on wavelets, dominated mostly
by applications in image/signal processing. Another link you can follow for
tons of information is http://www.wavelet.org, where you will find the
Wavelet Digest maintained by Wim Sweldens.
Construction of compact support orthogonal wavelet families by Daubechies
in the 80's really opened the door to digital wavelet algorithms. These
functions provided localized expansion bases that could be used as
alternatives to the sines and cosines of Fourier analysis. These wavelets
are actually orthogonal between different scales of resolution, so one
is allowed to add resolution in localized areas without forcing it
everywhere in a signal, image, solution to a differential equation, etc.
The basic digital transform to wavelet space is O(N), so fast algorithms
are possible. There are many variations beyond the Daubechies wavelets,
but that's too big a field to talk about here.
The compressional advantages have been slow to make an impact on differential
equations, but many of us hope that they eventually will. There are
several papers by now that have probed their use in Galerkin or Ritz
or other calculations. There has been more development in finite element
and multigrid methods, but orthogonal wavelets (or a generalization) have
great promise with regard to allowing localized multiresolution methods
that are easily adaptive. Regardless of whether wavelets ultimately lead
to the most efficient numerical methods, the idea of incorporating scale
as an explicit parameter has a strong appeal to a lot of people.
Hmmm, references. One of the very influential numerical analysis papers
is Beylkin, Coifman and Rokhlin, Comm. Pure App. Math 44, 141 (1991).
There are some recent collocation method papers for wave equation type
stuff by Vasilyev & Paolucci in J. Comput. Phys. over the last 3 years.
I have a wavelet quadrature paper coming out in the May 1 J. Chem. Phys.
that has references for earlier applications to quantum mechanics. Stefan
Goedecker has a book on use of wavelets for partial differential equations
that is either out or about to come out. I can't hit all bases here, but
these will give some hooks into the literature.
Regards,
Bruce Johnson
==============================================================================
From: Bruce Johnson
Subject: Re: Wavelets in Numerical Analysis
Date: 4 May 1999 21:46:29 GMT
Newsgroups: sci.math.num-analysis
Stuart C. Schaffner wrote:
> On 4 May 1999 06:12:53 GMT, Bruce Johnson
> wrote:
>>[snip]
>>... Stefan
>>Goedecker has a book on use of wavelets for partial differential equations
>>that is either out or about to come out.
>>
> Neither amazon.com nor barnesandnoble.com list any books by Goedecker.
> If you find out anything more, I'd like to know.
> Stu Schaffner
> The MITRE Corp.
I remembered that there was an entry for this in the Wavelet Digest
from late last year. Finally found it, hope it's ok to reproduce it.
Goedecker just came to visit at Rice a couple of weeks ago and did
a good job of explaining the subject matter to a non-expert
audience (I haven't seen the book myself, though).
Bruce
-----------------------------------
S. Goedecker:
"Wavelets and their application for the solution of differential
equations",
Presses Polytechniques Universitaires et Romandes, Lausanne,
Switzerland 1998, (ISBN 2-88074-398-2)
http://ppur.epfl.ch
This book is based on a postgraduate course given by the author at
EPFL in January/February 1998. The motivation for teaching this course
as well as for writing this book was to make this fascinating and
highly useful field of wavelets accessible to non-mathematicians. The
overwhelming part of the literature on wavelets is in the classical
mathematical style of theorems followed by proofs. The threshold for
entering the subject for non-mathematicians is therefore rather
high. In addition computational scientists, who just want to apply
this theory, are more interested in an intuitive understanding of the
important features instead of the formal mathematical framework. This
book is intended to make the theory of wavelets understandable to this
audience. In addition to a self-contained and intuitive presentation
of the theory of wavelets, extensive tables with the basic filter
coefficients of differential operators in several wavelet families can
be found in this book. After working through this book anyone who
wants to numerically solve partial differential equations in physics,
chemistry or engineering using wavelets should be able to do
so. Wavelets are a basis set with extraordinary properties for the
solution of differential equations. Their flexibility and efficiency
allows us to attack problems which are very hard or even impossible to
tackle with conventional methods. It is to be expected that the theory
of wavelets will soon be part of any science and engineering
curriculum in the same way as Fourier analysis is nowadays.