# MATH 76.02: Computational Methods for Inverse Problems

Course Information:
• Instructor: Professor Anne Gelb, Mathematics Department, Dartmouth College
• Course Time: 9L MWF 8:50am-9:55am (x-hour Thu 9:05am-9:55am)
• Course Location: Kemeny 006
• Office: Kemeny 207 Office Hours: x-hour; WF 10:00-11:00 and by appointment.

Course Description: Inverse problems are ubiquitous in scientific research, and occur in appli- cations 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 par- ticular challenges presented by the given application, and moreover how to critically analyze the quality of their results. Specifically, students will an- alyze 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 22 or Math 24. Math 20 or Math 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 (required).
• Vogel, C. Computational Methods for Inverse Problems, SIAM (suggested).
• Kaipio, J. and Somersalo, E. Statistical and Computational Inverse problems (suggested).
The SIAM books are available as e-books through the Dartmouth library. A good reference for MATLAB coding is D.J. Higham and N.J. Higham (2005) MATLAB Guide, Second Edition. SIAM. There are also many other resources online as well as the MATLAB tutorial available through MATLAB.

Grading: Grades in the class will be based on homework sets which will ensure mastery of theoretical and computational skills. There will also be one take home exam around the fifth week of class which will not have computational components. Students will have the option of doing a take home final exam, which will have computational components, or doing a team project (up to five members on a team). Students may work together on the homework, but will need to turn in their own assignments. Students may not work together on take home exams. It is strongly recommended that all homework assignments, especially those involving programming problems, be started early.

Final project information: During the last three weeks students will work intensively on a research project. Some suggestions will be provided at the very beginning of the term, but students may also propose their own project, which must be approved by the instructor by the sixth week of class. Each student or team will prepare a 15 minute presentation and turn in a short written report (3-5 pages).

Grading formula: (i) Four homework sets (60%); (ii) take home midterm exam (15%); (iii) final project or exam (15%); (iv) Participation & Attendance (10%).

## Important dates and grading information

• First day of term: Thursday June 23 2022. We will NOT meet in person and there will be no office hours.
• The homework problem sets will be made available on CANVAS and due approximately every ten days. Students should turn homework in electronically on CANVAS. Due to the varying complexity of the material, some homework sets will naturally be more challenging than others. Regardless, each homework set is weighted the same for the final grade.
• Midterm exam: Hand out: TBD. Due: TBD
• X Hours will not likely be used unless needed to finish covering material. However, students may choose to informally congregate during this time to discuss homework and projects. I will also be on hand in the classroom during that time for office hours.
• Participation & attendance: Students are expected to attend most classes. I understand that intership interviews often occur during the summer. Please try not to schedule them during class time whenever possible. From time to time during X hours students will have the opportunity to lead discussion on how to approach algorithmic development and/or MATLAB programming. Volunteers are always appreciated, and it's a great way to test your skills.
• Last day of class: August 24 2022. Students opting to take the final must turn it in by the end of the day.

## Syllabus

### TENTATIVE lecture plan which WILL be subject to further changes.

Week Lecture
Weeks 1 Introduction and Motivation.
Weeks 2-3 Basic ideas in numerical analysis: function representation; quadrature, (square) linear systems, least squares methods, singular value decomposition.
Week 4 Regularization. The effects of noise and ill-conditioning.
Weeks 5 Special topics: Topological data analysis.
Week 6 Fourier transforms.
Week 7 Compressive sensing.
Week 8-9 Applications using compressive sensing: medical imaging, deblurring, multiple measurements.

## Course Policies

### Honor Principle

Students are encouraged to work together to understand course material. This includes helping each other by providing insight into homework problems. 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.

### Accessibility Policy

Students needing special accommodations are encouraged to make an office appointment with Professors Gelb and Fu 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 Accessibility Services within the Academic Skills Center. The Director of Student Accessibility is Ward Newmeyer. Office 205 Collis Center; Phone (603) 646-9900.

### 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.