MAT 280 Laplacian Eigenfunctions Home
Page (Spring 2007)
Course: MAT 280
CRN: 44770
Title: Laplacian Eigenfunctions: Theory, Applications, and Computations
Class: TTh 4:40pm-6:00pm, 3106 Math. Sci. Bldg.
Instructor: Naoki Saito
Office: 2142 MSB
Email: saito@math dot ucdavis dot edu
Office Hours: By appointment
Synopsis:
This course is an introduction to the ubiquitous basis functions
that can be computed on a domain of general shape or on a graph, i.e.,
the eigenfunctions of Laplacian defined on such domains.
The importance of such basis functions can be immediately recognized if we view
that sines, cosines, complex exponentials, the Bessel functions, the spherical
harmonics, and the prolate spheroidal wave functions are the examples of
the Laplacian eigenfunctions for the domains of specific geometry (an interval,
disc, sphere, ellipsoid, etc.) with appropriate boundary conditions.
After covering the basics of the Laplacian eigenfunctions/eigenvalues, we will
discuss a vast number of applications such as image analysis, computer
graphics, shape characterization, heat equation, high dimensional data
analysis, and analysis of graphs. The course also surveys several numerical
methods to compute such eigenfunctions and the corresponding eigenvalues
including a strategy based on the commuting integral operator developed by me
and my group recently. The course will also emphasize the connection between
the continuum (i.e., analog) world and the discrete world.
Prerequisite:
MAT 118AB, 129, 201, or their equivalents, or consent of the instructor.
Topics:
- Overture: motivations, scope and structure of the course
- Laplacian eigenvalue problems in the continuum, i.e., some domain Ω ⊂ Rd of finite volume
- Vibrations of a 1D string
- Vibrations of 2D/3D membranes
- Necessary functional analysis basics
- Vibrations and heat conductions in Ω
- Applications: data analysis, spectral geometry, shape recognition
- Laplacian eigenvalue problems in the discrete settings
- Basics of graph theory
- Laplacian and its eigenvalues of a graph
- Random walk and heat conduction on a graphy
- Diffusion maps
- Applications: clustering, image segmentation, statistical learning theory, etc.
- Fast algorithms to compute Laplacian eigenvalues/eigenfunctions
- A method of particular solutions
- A wavelet-based method
- An FMM-based method
Textbooks:
No textbook is required. Many journal papers will be discussed
and distributed in the class. As a general introductory reference in this field,
the following books may be useful.
- W. A. Strauss: Partial Differential Equations: An Introduction,
Chap. 10 & 11, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol.I, Chap. V, VI, & VII, Wiley-Interscience, 1953.
- E. B. Davies: Spectral Theory and Differential Operators,
Cambridge Univ. Press, 1995.
- F. R. K. Chung: Spectral Graph Theory, AMS, 1997.
Class Web Page:
I will maintain the Web pages for this course (one of which you are
looking at now). In particular, please read the comments, handouts, and reference page often.
After each class, I will put relevant comments and references as well as
most of my handouts in class in this page that should
serve as a guide to further understanding of the class material.
Class Mailing List:
The class mailing list was created.
Important announcement will be communicated through this mailing list.
You can also submit your public comments, suggestions, and questions on HW,
and/or some useful information related to the class to this mailing
list. Once you send your email to this list, however, everyone will receive it.
So, please use this wisely and politely. Its name is: mat280lapeig-s07@ucdavis.edu.
Grading Scheme:
- 50% Attendance
- 50% Scribing class notes and producing their latex document (one or two
lectures/student)
Please email me if you
have any comments or questions!
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