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Spring 2007, general information

The lecturer of the course is Erkki Somersalo.

Time and place: Wednesday 12-14, room U356 and Thursday 12-14, room U356.

Extent: This is a 5 credit ECTS (Europen Credit Transfer System) course. Attending the short course Computational models in cell biology (see below) adds one extra credit .
The course focuses on computational and statistical methods applied to inverse problems. In inverse problems, the goal is to retrieve information of unknowns that are not directly accessible for measurements, but instead have an indirect effect on observable quantities. Examples of typical inverse problems can be found, e.g., in biomedical non-invasive methods, where the internal state of the human body is assessed by external measurements (MEG, EEG, MRI or optical tomography). In the statistical approach, the unknowns are modelled as random variables and the inverse problem is recast in the form of statistical inference of exploring the probability distributions of these variables. The emphasis in this course is on Markov Chain Monte Carlo (MCMC) methods that are useful, e.g., in statistical physics. Examples from medical imaging, biomathematics, image and signal processing are discussed.
In 2007, a short course "Computational models in cell biology" will be lectured by a visiting professor Daniela Calvetti. For details of the contents of the course, click here . The schedule of the short course is the following:

14.3. (Wednesday) 14-15
12.4. (Thursday) 12-14 ( Note: the normal Thurday lecture is moved to Wednesday before the Easter break.
25.4. (Wednesday) 14-16
26.4. (Thurday) 13-14

Literature

J. Kaipio and E. Somersalo: Statistical and Computational Inverse Problems , Springer Verlag 2004
J. S. Liu: Monte Carlo Strategies in Scientific Computing, Springer Verlag 2001.

Requirements

The emphasis is on computational methods, and basic skill of using MATLAB is required.
No regular weekly homework assignments are given. Instead, a few larger homework projects are required. These projects are discussed in the class and graded. These grades, together with a small final exam determine the final grade.

Weekly program

15.1.-19.1. Introduction to the concept of probability in the context of inverse problems. Section 1 of the book manuscript Subjective Computing, Introduction to Bayesian Methods in Computational Science by D. Calvetti and E. Somersalo (To appear in the Springer Verlag series Surveys and Tutorials in the Applied Mathematical Sciences. Material distributed in class room only for copyright reasons, I apologize for the inconvenience).

22.1.-26.1. Continuation, Sections 2 and 3 of the book manuscript.

29.1.-2.2. Introduction to sampling and Markov Chains. Sections 5 and 9 of the above book manuscript, distributed in the class. Additional material: Antonietta Mira: Introduction to Monte Carlo and MCMC Methods , Lecture Notes of the Summer School on Bayesian methods in Inverse Problems, Kuopio, June 21-24,2004. This material and MUCH MORE can be found on the web page of the Finnish Inverse Problems Society , http://venda.uku.fi/research/FIPS/BMIP/

5.2.-9.2. The Metropolis-Hastings algorithm, continuing with the material from Section 9.

12.2.-16.2. Simple low-dimensional examples of random walk Metropolis-Hastings algorithm.

19.2.-23.2. Example : an inverse problem in (bio)chemical engineering. Diagnostics of the samplers.

26.2.-2.3. Preliminaries, and implementation of Gibbs sampler.
More about diagnostics of the sampler.

5.3.-9.3. Exam period; no lectures this week.

12.3.-16.3. Adaptation. Adaptive Metropolis-Hastings sampler. (AM) .

19.1.-23.3. Adaptation using a moving window. Some useful variations of Gibbs sampler.

26.1.-30.3. Dynamic inverse problems. Introduction to particle filters and Kalman filters.

2.4.-6.4. and 9.4.-13.4. (Notice the Easter holiday 5.-11.4) Lectures only on Wenesday, March 4. At least part of the time used for the discussion of home assignments (both first and second). An example of the use of Kalman filters in positioning.

16.4.-20.4. No lectures.

23.1.-27.4. Dynamic inverse problems and particle filtering. Wrapping all up.

Home assignments

1. The first home assignment can be downloaded from here . The Matlab file of the deconvolution example shown in the classroom can be found here .
The deadline for this assignment is Friday, February 23.

2. The second assignment can be downloaded from here .
The deadline for this assignment is Friday, March 23.

3. The third assignment can be downloaded from here .
The deadline for this assignment is Friday, May 4.

Material from previous years

The homepage of the 1996 inverse problems course can be found here.
The homepage of the 2002 inverse problems course can be found here.
The homepage of the 2006 inverse problems course can be found here.
This page was created by <erkki.somersalo@tkk.fi>
Last update 18 Jan. 07