# Wavelets and Partial Differential Equations

A course presented at the II Pan-American Advanced Studies Institute in Computational Science and Engineering, Universidad Nacional Autónoma de Honduras, Tegucigalpa, Honduras, June 14-18, 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 non-linear 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
• Pre-wavelets: splines, orthogonal polynomials, etc.
• Wavelets: Haar, Meyer, Daubechies, Coiflets, symmlets, etc.
• Post-wavelets: 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 Paris-Berlin -- Seminar Berlin-Paris".
• .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