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 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 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. (The textbook codes are all in MATLAB.)
Textbooks:
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. It can be very useful to use open AI resources to help with coding and analytic problems. There is a difference between having AI solve the problem and reasonably using it for assistance. Please note the honor policy.
Final project option: Students must have their project approved by August 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 August 30 .
Grading formula: (i) Four homework sets (75%); (ii) final homework/project (15%); (iv) Participation & Attendance (10%).
Week | Lecture |
---|---|
Weeks 1 | Introduction and Motivation. Reading: Hansen Chapter 1; 2.1 and 2.2 (this might be difficult, we will review carefully in class). | 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. |