Webpage of the Course MAT.501UB WS 2022/23
Welcome to the Practical Course of Inverse Problems MAT.501UB for the Master Degree in Mathematics at the University of Graz. This is the companion module to the theoretical Course of Inverse Problems MAT.500UB. Exercise sheets will be released every two weeks which will cover practical aspects of the topics covered in MAT.500UB. There will also be coding assignments in MatLab or Python.
Please do not hesitate to email me with any questions you have regarding the module or the exercises.
Note: I thank my predecessors Martin Holler, Richard Huber and Francisco Romero for providing lots of high quality teaching material.
There will be 6 classes in Room 11.34 from 12:00-13:30. These are the dates:
To each Lecture will correspond one Exercise Sheet. This will be uploaded 2 weeks before the Lecture, and is due for Crossing and Presentation on the day
of the Lecture.
Crossing: At the beginning of each Lecture, a form will be handed out in which you should declare the problems you solved. This is referred to as Crossing. You will be awarded points for each cross, according to the amount specified on the Exercise Sheet for the corresponding problem. The total is always 100 points. The final Crossing Percentage is then computed by averaging on the 6 Exercise Sheets.
Presentation: Based on the crossing, some students will be called at the blackboard to solve one of the exercises they declared. A presentation is given a grade between 0 and 5, with 5 being highest. Ideally each student should present at least 2 times during the course, one of which a numerical exercise. The final Presentation Percentage is averaged and scaled to a number out of 100. The numerical exercises, to be solved in MatLab or Python, can be presented using my laptop and the room projector: either send the solution to me by email, or I can provide a USB drive in class.
The Final Percentage is computed by averaging Presentation Percentage and Crossing Percentage. The Final Percentage will be converted into a Final Grade according to the table below. A pass will be granted for a grade of 4 or better.
| Percentage | 0-49% | 50-59% | 60-74% | 75-89% | 90-100% |
|---|---|---|---|---|---|
| Grade | 5 | 4 | 3 | 2 | 1 |
Introduction: Differentiation, Deconvolution, and Radon transform. Ill-posedness of inverse Problems. Compact linear operators. Singular value decomposition. Moore-Penrose inverse
Regulatrisation of linear inverse problems: Linear methods of filtering (TSVD), Tikhonov regularisation. Source condition and convergence rates. Choice of parameters (a priori, a posteriori, heuristic). Optional: projection methods (e.g. Galerkin), iterative regularisation (e.g. Landweber, CG)
Nonlinear inverse problems: Ill-posedness (relation to linear problems). Nonlinear Tikhonov regularisation. Optional: Electrical impedance tomography (EIT), Calderon problem, Inverse scattering problems (Helmholtz)
Assumed knowledge: understanding of Functional Analysis, Numerical Analysis and PDEs
| Due date | Exercises | Topics |
|---|---|---|
| 11 Oct | Sheet 1 | Differentiation. Deconvolution |
| 25 Oct | Sheet 2 | Compact operators |
| 15 Nov | Sheet 3 | Minimal norm elements. Moore-Penrose inverse |
| 22 Nov | Sheet 4, Coding 4 | Radon Transform (RT) and its numerical implementation |
| 13 Dec | Sheet 5, Coding 5 | Singular values of RT. Limited angle problem |
| 24 Jan | Sheet 6, Coding 6 | Tikhonov regularization. Primal-Dual Algorithm |