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Fall 2007, general information

The lecturer of the course is Erkki Somersalo.

Time and place: Wednesday 14-16, room U345 and Thursday 14-16, room U322.

NOTE: First lecture on Wednesday, September 19, 2007!

Extent: This is a 5 credit ECTS (Europen Credit Transfer System) course.
In inverse problems, the goal is to retrieve information of a quantity that is not directly observable. As an example of a typical inverse problem in the field of medical imaging is the problem of mapping the electric potential on the surface of the heart from electric body surface measurements. A characteristic feature of inverse problems is their ill-posedness. Small noise in the data or small errors in the model may produce huge perturbations in the reconstructions of the quantity of interest. A classical cure for the ill-posedness is to use regularization methods: the original ill-posed problem is replaced by a nearby well-posed problem. Regularization techniques constitute the first part of the present course. Another approach to inverse problems is to redefine them in the Bayesian statistical framework as inference problems. The unknown are modelled as random variables and the inverse problem is recast as a quest of information, the primary target of investigation being probability densities of the variables. The second part of the course focusses on this approach.
The weekly program of the course will be posted on this page as it unfolds.

Literature

J. Kaipio and E. Somersalo: Statistical and Computational Inverse Problems , Springer Verlag 2004
D. Calvetti and E. Somersalo: Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing , Springer Verlag 2007 (to appear in November).

Requirements

The emphasis is on computational methods, and basic skills of using MATLAB are required. A short primer to get started with Matlab can be found here .

Regular weekly homework assignments are given.

Weekly program

17.9.-23.9. Preliminaries: examples of inverse problems, ill-posedness. Tools of linear algebra: Singular value decomposition, ranges and null spaces. The notes can be downloaded from here .

24.9.-30.9. Regularization by Truncated Singular Value Decomposition (TSVD). Morozov's discrepancy principle. The notes can be downloaded from here .

1.10.-7.10. Numerical example of TSVD: deconvolution . Tikhonov regularization, selection of the regularization parameter. Notes are here .

8.10.-14.10. Iterative methods for solving linear systems. Krylov subspace methods, Conjugate Gradient (CG) method. Lecture notes of this week are here .

15.10.-21.10. Regularization of the CG method by early stopping of the iterations. Lecture notes on CGLS and a Matlab code for numerical differentiation. Lectures only on Wednesday! .

22.10.-28.10. Exam period; no lectures this week.

29.10.-4.11. Lectures continue on Thursday. Introduction to Bayesian statistics and inverse problems.

5.11.-11.11. Towards the Bayesian formulation of inverse problems. Examples of the construction of likelihoods and priors.

12.11.-18.11. On likelihoods and priors. Maximum likelihood estimators.

19.11.-25.11. An example of Bayesian reasoning in a deconvolution example . The Matlab code is here .

26.11.-2.12. Continuation: hypermodels, examples

3.12.-9.12. No lecture on Thurday (National holiday)

10.12.-16.12. Exam, Tuesday 11.12, 9--12. The place will be announced here later.

Home assignments


The weekly excercises will be held on Fridays, 12--14 in Room U345. The teaching assistant is Dr. Knarik Tunyan.

1. Exercise 1
2. Exercise 2
3. Exercise 3 will be held in the computer classroom Maari-D, Friday 12--14.
4. Exercise 4 will be held in the computer classroom Maari-D, Friday 12--14.
5. Exercise 5 , Room U345.
6. Exercise 6 , Maari-D.
7. Exercise 7 , Maari-D.
7. Exercise 8 , Maari-D. (This is the last excercise.)

This page was created by <erkki.somersalo@tkk.fi>
Last update 21 Nov. 07