Wavelets and Partial Differential Equations
Martin J. Mohlenkamp
(Department of Mathematics,
College of Arts & Sciences,
Ohio University)
A course presented at the II
PanAmerican Advanced Studies Institute in Computational
Science and Engineering, Universidad Nacional Autónoma de
Honduras, Tegucigalpa, Honduras, June 1418, 2004.
Course Description
In this course we will discuss the basics of
wavelets. Including multiresolution analysis, lifting techniques,
and basic applications to data compression, denoising, and signal
and image processing. We will discuss the interplay between
wavelets and differentiation. In particular we will discuss
efficient representation in terms of wavelets of derivatives and
products, and how to handle boundary terms as well as irregular
data. These are all cornerstones of many nonlinear PDE solvers.
Course Outline
The official course outline, which was created (by Cristina Pereyra)
some months earlier, is:
 Lecture 1: Time/Frequency Analysis

 Fourier analysis.
 Windowed Fourier transform.
 Wavelet transform.
 Lecture 2: Fast Algorithms and Applications

 Multiresolution Analysis.
 Filter banks.
 Lifting schemes.
 Signal/Image compression, denoising.
 Lecture 3: Main Characters

 Prewavelets: splines, orthogonal polynomials, etc.
 Wavelets: Haar, Meyer, Daubechies, Coiflets, symmlets, etc.
 Postwavelets: brushlets, edgelets, ridgelets, etclets.
 Lecture 4: Variations over a Theme

 Wavelet packets and local cosine bases.
 Biorthogonal wavelets.
 Wavelets on the interval.
 Multiwavelets.
 Lecture 5: Applications to Signal/Image Processing

 Representation of d/dx in wavelet bases.
 Pointwise products.
 Interpolation of sample values.
 Characterization of Sobolev spaces.
The outline of what was actually taught is:
 Day 1: Background
 Lecture on:
 Fourier analysis
 Time/Frequency Analysis
 Local cosine bases
 Day 2: Basic Wavelets
 Lecture on:
 Multiresolution Analysis
 Haar Wavelets
followed by demonstrations in the computer lab using the
Matlab wavelet toolbox.
 Day 3: General Wavelets
 Lecture on:
 Fast Wavelet transform
 Filtering in digital signal processing
 Vanishing moments and other properties
 How to choose the correct wavelet.
 Wavelet Packets
 Day 4: Multiwavelets and PDEs
 Lecture on:
 Polynomial Multiwavelets
 Interpolating version, scaling space adaptive
version, and matching boundary values
 PDEs with multiwavelets (following
Alpert, Beylkin, Gines, Vosovoi 2002 (.pdf)): semigroup formulation
and operator calculus
 Day 5: Finale

 Lecture on the derivative operator in multiwavelets
 Test
 Lab using the Matlab wavelet toolbox to explore 2D
wavelets, compression, and denoising
Course Materials
Cristina Pereyra and
I produced a set of lecture notes (.pdf).
Some slides used were from a tutorial on Wavelets and their
Applications (.pdf) that I gave
in 2002.
There is also a lot of freely available information on the web. Here
is a selection of general resources:
 .html
A general introductory online text from the "Seminaire ParisBerlin 
Seminar BerlinParis".
 .html
A selection of notes in postscript (seeming to focus on image
processing) entitled "TUTORIAL: Multiscale Methods and
Applications".
 .html
Lecture notes in pdf from John Hopkins University for
"Introduction to Wavelets".
 .html
Includes lecture notes in postscript from the University of
Texas for "Wavelets and Signal Processing".
 .html
Slides and handouts in pdf from the MIT course "Wavelets, Filter
Banks and Applications".
 .html
Course notes in html from Dalhouse University for "Wavelets
and Filter Banks".
 .pdf
Lecture notes and background material (at 287 pages,
essentially a book) in pdf from the University of Minnesota
for "Introduction to the Mathematics of Wavelets".
 .html
Slides in pdf from George Mason University for "Wavelet Theory".
Martin J. Mohlenkamp
Last modified: Sun Jul 4 14:12:50 EDT 2004