MATH 76.02/146: Computational Methods for Inverse Problems

Course Information:

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 (Math 24). Math 20 (Math 60) recommended. Some experience in MATLAB or another programming language is highly beneficial. (The textbook codes are all in MATLAB.)

Textbooks:

The SIAM books are available as e-books through the Dartmouth library. A good (hardcopy) 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 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 November 5. Students may propose their own project. I am 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 November 26 (end of final exam period).

Grading formula: (i) Four homework sets (75%); (ii) final homework/project (15%); (iv) Participation & Attendance (10%).

Important dates and grading information

  • 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.
  • 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.
  • Participation & attendance: Students are expected to attend most classes. From time to time during X hours students will have the opportunity to lead discussion on how to approach algorithmic development, programming (MATLAB/Python), or to generate new discussion regarding computational inverse problems, which might inspire some final projects. Volunteers are always appreciated, and it's a great way to test your skills.
  • Last day of class: November 15. We will hold X hour on November 12 since there will be no class on November 18.

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: Fourier and inverse Fourier transforms.
Week 6 Special topics: Compressive Sensing.
Week 7 Special topics: Bayesian inference.
Weeks 8-9 Special topics: Sampling.

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.