Program Overview

Course Information

The course MATH 76.02: Computational Methods for Inverse Problems is offered by Professor Anne Gelb, Mathematics Department, Dartmouth College. The course details are as follows:

• Instructor: Professor Anne Gelb
• Course Time: 10A T-Th 10:10pm-12:00pm (x-hour F 3:30pm-4:20pm)
• Course Location: Haldeman 028
• Office: Kemeny 207
• Office Hours: T W 2:00-3:00 and by appointment

Course Description

Inverse problems are ubiquitous in scientific research, and occur in applications ranging from medical imaging to radar sensing. The input data are often under-sampled, noisy and may additionally be blurry. Physical obstructions may also prevent accurate data acquisition. Recovering an underlying signal or image can be critical for diagnosis, classification, or inference. This course describes fundamental aspects of inverse problems and various computational approaches for solving them. Importantly, the students will learn how to choose the appropriate methodology for the particular challenges presented by the given application, and moreover how to critically analyze the quality of their results. Specifically, students will analyze accuracy, efficiency and convergence properties of the computational techniques for various classes of problems and when possible to quantify the uncertainty of their results. Although programming will not be formally taught as part of the course, students will write numerical code in languages such as MATLAB or Python to compute their solutions. Resources will be provided to help students learn to write MATLAB code.

Prerequisites

• Math 11 or Math 8(9) and 13
• Math 22 (Math 24)
• Math 20(60) recommended
• Some experience in MATLAB or another programming language is highly beneficial

Textbooks

• Hansen, P. Discrete Inverse Problems: Insight and Algorithms, SIAM (required)
• Ascher, Uri M. and Greif, Chen. (2011) A First Course in Numerical Methods, SIAM (required)
• Hansen, P., Nagy, J. and O'Leary, D. Deblurring Images: Matrices, Spectra, and Filtering, SIAM (recommended)

Grading

Grades in the class will be based on five homework sets which will ensure mastery of theoretical and computational skills. Students will have the option of doing the final homework or an approved project. Students may (and are encouraged to) work together on the first four homework sets, but will need to turn in their own assignments. Students may not work together on the final homework. In some cases the approved project may be collaborative, as long as it is clear how the work is divided. It is strongly recommended that all homework assignments, especially those involving programming problems, be started early.

Final Project Option

Students must have their project approved by August 5. Students may propose their own project. The instructor is also happy to discuss suitable research projects, with the goal being further exploration of a topic introduced in class. Students will produce a well-written short report (3-5 pages) by August 30.

Grading Formula

• Four homework sets (75%)
• Final homework/project (15%)
• Participation & Attendance (10%)

Important Dates and Grading Information

• The homework problem sets will be made available and due approximately every ten days
• X Hours will not likely be used unless needed to finish covering material
• Participation & attendance: Students are expected to attend class. Two unexcused absences are permitted, with each additional absence resulting in a loss of 5% of your attendance grade
• Last day of class: August 26
• Last homework due August 30

Syllabus

TENTATIVE Lecture Plan

Week | Lecture
---|---
Weeks 1 | Introduction and Motivation. Reading: Hansen Chapter 1; 2.1 and 2.2
Weeks 2-3 | Basic ideas in numerical linear algebra. Reading: Ascher and Greif Chapters 4-6
Week 4-5 | Integral problems and discretization. Reading: Hansen Chapters 2.4, 2.5, 3
Weeks 6-7 | Regularization methods. Reading: Hansen Chapter 4
Weeks 8 | Choosing parameters for regularization methods. Reading: Hansen Chapter 5
Week 9-10 | Special topics: Compressive Sensing

Course Policies

Honor Principle

Students are encouraged to work together to understand course material. However, each student is responsible for his/her own assignment, and any homework problem solution that appears to result from a team effort will result in zero points awarded for all parties involved. Homework that is prominently influenced by AI will also result in zero points awarded.

Accessibility Policy

Students needing special accommodations are encouraged to make an office appointment with the instructor prior to the end of the second week of the term. At this time, students should provide copies of disability registration forms, which list the particular accommodations recommended.

Student Religious Observances

Some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with your instructor before the end of the second week of the term to discuss appropriate accommodations.

Late Policy

Homework due dates are strictly enforced for full credit. Each day homework is late results in a 10% penalty. Students requesting special accommodations should inform the instructors well in advance so that the instructors will have sufficient time to work with Student Accessibility Services to ensure appropriate accommodation.