Math 6645, Spring 2021
Office Hours: W
10-11AM or by appointment on Microsoft Teams, Some additional
office hours are TBA.
Lectures: M W
12:30-1:45PM on Microsoft Teams
Course webpage: https://people.math.gatech.edu/~wliao60/21Spring_Math6645.html
General course information
- Prerequisites: Math 4317 and Math 4640 (or
description: theoretical and computational aspects of
polynomial, spline, wavelet and frame approximation, and
regression in statistical learning
Some references (not required):
- Numerical Analysis Mathematics of Scientific
Computing (3rd) by Kincaid and Cheney, Brooks/Cole.
- Mathematics of Approximation by Johan De
Villiers, Springer Link (GT Library).
- Interpolation and approximation by
polynomials, by G.M. Phillips, (GT Library, on-line).
- Numerical solution of partial differential
equations by the finite element method, by Claes Johnson,
- The Mathematical Theory of Finite Element
Methods, by Susanne C. Brenner, L. Ridgway Scott, Springer (GT
- A Basis Theory Primer by Christopher Heil,
Birkhauser Link. (GT Library, on-line).
- Applied Mathematics Data Compression,
Spectral Methods, Fourier Analysis, Wavelets, and Applications
by Chui and Jiang, (GT Library, on-line).
- Ten Lectures on Wavelets, by Ingrid
Daubechies, SIAM Link
- Introduction to Nonparametric Estimation by
Alexandre B. Tsybakov, (GT Library, on-line)
- A Distribution-Free Theory of Nonparametric
Regression by Laszlo Gyorfi,
Michael Kohler, Adam Krzyzak and Harro Walk, (GT Library)
- "Universal algorithms for learning theory part
i: piecewise constant functions", by
Binev, Cohen, Dahmen, DeVore and Temlyakov, Journal of Machine
Learning Research, 2005.
- "Universal algorithms for learning theory.
part ii: Piecewise polynomial functions", by Binev, Cohen, Dahmen and DeVore,
Constructive approximation, 2007.
- "Nonlinear approximation", by Devore, Acta
- Homework (50%): Students are strongly
encouraged to solve all the written homework problems. Exams
will be based on these homework problems. Only part of the
homework will be turned in. Homework are due
tentatively on February 10, March 24 and April 21.
- Exams (50%=25%+25%): There will be two exams
(a midterm and a Final) during the semester which will be
based on the homework/lecture materials.
- A midterm is tentatively on March 3rd. The
midterm will be take-home, which lasts for 24 hours.
- The Final is in the exam week (April 29-May
6, specific time and location is TBA).
- No make up exams are allowed. In case of
serious illness, a doctor's note is required. For excused
absences, your adviser's notice is required.
- Presentation: If a student gives a
presentation on the material related with the topics of this
course, he/she is exempted from Final. The presentation will
occur at the end of the semester (around April 7-26). The
professor will select some papers in February.
Students can present either one of those papers or something
else related to his or her research. This decision should be
made and consulted with the professor before March 1,
and the student will give a mini 5 minute presentation
around March 15-31 (the date is to be determined).
HONOR CODE: All
students are expected to comply with the Georgia Tech Honor
code. Please review the student code of conduct.
- Polynomial approximations: Classical
polynomial interpolation, Chebyshev, orthogonal polynomial,
Spline, Least squares approximation --
Kincaid and Cheney Section 6.1 - 6.10
- Trigonometric polynomial approximation:
Fourier series and the FFT, transformation and their
application -- Heil Chapter 8,9,10
- Wavelet and frame approximation -- Heil Chapter 8,9,12
- Regression in statistical learning: Curse of
dimensionality, kernel methods, polynomial partitioning
estimates, adaptive methods -- Gyorfi, Kohler, Krzyzak and Walk, Chapter
1,2 and Tsybakov, Chapter 1
Tentative schedule and homework
- Week 1, January 20
- Kincaid and Cheney, Section 6.1: Polynomial
approximation, Lagrange form, Newton form, Error analysis,
Chebyshev Polynomials and Chebyshev nodes
- Kincaid and Cheney, Section 6.1: