Inverse Problems (Fall, 2008)

CPMA 591/MATH 491

Contact:

Professor: Dr. Carl Toews
E-mail: toewsc * duq.edu
Office: College Hall 418
Office Hours: Tuesday, 10:45am-1pm, Wednesday, 12pm-1pm

Course Materials:

Lectures: T 6:00pm-8:40pm, College Hall 220
Textbook: C. Groetsch, Inverse Problems in the Mathematical Sciences, available in bound photocopy through the bookstore.

Grading:

The course grade is a uniformly weighted average of participation, homework, a project, and a final. A total score of 90% (resp., 80%, 70%, 60%) or more guarantees a final grade of A (resp., B or higher, C or higher, D or higher).

Homework policy:

Homework will be assigned weekly at the end of each class, and collected the next week at the beginning of class.

Exam policy:

The course will conclude with a take-home final exam.

The Big Picture

Background and goals:

Introduction to the fundamental ideas of mathematical modeling, along with regularization strategies for solving ill-posed inverse problems.

Prerequisites:

A solid background in calculus and linear algebra. Differential equations will help, but is not strictly necessary.

Content:

We will cover the following topics:

1. Condition numbers and (in)stability
2. Banach spaces of functions
3. The singular value decomposition, regularization
4. The normal equations, linear regression, least squares
5. Variance, covariance, and statistical estimation
6. Principal component analysis
7. Tikhonov regularization
8. Maximum entropy regularization
9. Total variation regularization

Resources:

Introductory textbooks on inverse problems:

• Aster, Borchers, and Thurber Parameter Estimation and Inverse Problems , Elsevier Academic Press, 2005.

• www.mathworks.com
• http://www.math.ufl.edu/help/matlab-tutorial/
• http://www.gnu.org/software/octave (Octave is the freeware version of Matlab)

Schedule

Handouts, etc.

• Syllabus