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