MAT-52506 Inverse Problems, fall 2008, 6 credits
MAT-52500 Inversio-ongelmat, syksy 2008, 6 opintopistettä

Introduction

Examples of inverse problems:
- sharpening a blurred photograph (more),
- imaging inner organs of patients using X-ray images taken from different directions (more),
- determining the structure of the Earth using seismic data,
- locating cracks inside materials using electric surface probes,
- finding the shape of a distant asteroid from light intensity measurements (more).
The common feature of all these problems is that they are very sensitive to measurement noise, and their solution is not straightforward.

This course teaches how to recognize an inverse problem and how to solve it in practice even when the data is noisy and the number of unknowns is very large.

The first part of the course (period I) consists of lectures and exercises, and the latter part (period II) is a project work. The project work can be done either individually or in teams of two or three students.
Main emphasis is on practical solution of problems arising in applications;
theory is introduced only to the extent needed to understand and implement solution methods.

Central themes of the course are singular value decomposition (SVD) of a matrix, Tikhonov regularization, total variation regularization, and statistical inversion.
These solution methods will be demonstrated in detail in the cases of deblurring and tomography.

The course is in part based on the lecturer's experience in medical imaging industry (Instrumentarium Imaging, General Electric, Palodex Group).

All course material, including lectures, are given in English.

Weekly exercises

Exercises are held in hall SJ202 on Fridays at 12:15-14:00. The first exercise session is 12.9.2008.

Exercise 1, Friday 12.9.2008, is here: Laskari01.pdf. You also need these Matlab routines: ex_conv1Ddata_comp.m and ex_conv1D_naive.m. Here are the solutions to the Matlab exercises: H1_1.m and H1_2.m.
Exercise 2, Friday 19.9.2008, is here: Laskari02.pdf. You may need some of the Matlab routines given below.
Here are the solutions to the Matlab exercises: H2.zip.
Exercise 3, Friday 26.9.2008, is here: Laskari03.pdf. You may need some of the Matlab routines given below.
Here are the solutions to the Matlab exercises: H3.zip.
Exercise 4, Friday 3.10.2008, is here: Laskari04.pdf. Here are some Matlab routines needed: bb_deblur.m, db_aTV.m, db_aTV_feval.m, db_aTV_fgrad.m, db_aTV_grad.m, db_misfit.m and db_misfit_grad.m.
Here are the solutions to the Matlab exercises: H4.zip.

Lectures

Lecturer´s office hour Tuesday 16-17 (room TD 321).

The basic example: One-dimensional deconvolution

Student feedback and development of the course

Prerequisites

basic linear algebra and matrix computations (equivalent to the course MAT-31090 Matriisilaskenta),

the concept of least squares solution of a set of linear equations,

some Matlab programming skills,

rudiments of probability theory (concepts of probability distribution and random variable).

This course is part of the activity of the Finnish Centre of Excellence in Inverse Problems Research.

Information on the lecturer´s research interests can be found here.

How to pass the course?

First alternative has three parts:

Pass interim exam (24 points),

collect exercise points (80% of the exercises done gives the maximum 12 points), and

complete the project work (24 points). The project work is mandatory.
For students who returned an acceptable project work, grading is based on the following score:
(exam points + exercise points + project work points)/60.
It is possible to do the interim exam on 22.10.2008 at 10-12 in hall TB220 or as subset of final exam on 26.11.2008.

Second alternative: Pass one final exam on 26.11.2008 or later.

Here are some old exams: Interim exam October 10, 2007, Final exam November 27, 2007, and Final exam January 7, 2008.

Course material

This is the most recent version of lecture notes: Notes, version 6. Note that it is still in very preliminary state. Please send comments and suggestions.

Also, chapters 1-3 in the book Kaipio and Somersalo: Statistical and Computational Inverse Problems (Springer 2005) covers most of the course.
However, the book is quite condensed. More accessible material is available at

Kaipio (in Finnish): http://venda.uku.fi/studies/virtual/KON/KON1/lectures/main.pdf

Somersalo (in Finnish): http://www.math.hut.fi/teaching/invvanha/indexvanha.html.fi

Tan, Fox and Nicholls (in English): http://home.comcast.net/~szemengtan/

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Web page of the 2007 course is available here.
Web page of the 2006 course is available here.

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This page was last updated on March 12, 2009.

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