MAT-52506 Inverse Problems, autumn 2009, 6 credits
MAT-52500 Inversio-ongelmat, syksy 2009, 6 opintopistettä

Introduction

Inverse problems research is an active area of contemporary mathematics.
Applications include signal processing, nondestructive testing, modelling of astrophysical data, shape optimization in mechanical engineering, monitoring cell function in biology, geophysical remote sensing, medical imaging, and option pricing in mathematical finance.

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 concentrates on how to recognize an inverse problem and how to solve it in practice even when the data are noisy and the number of unknowns is large.

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

Central methodological themes of the course are singular value decomposition (SVD) of a matrix, Tikhonov regularization, total variation regularization, and statistical inversion. A detailed case study in generalized projections (lectures and project work) is then used to illustrate the central concepts of uniqueness, stability, model parametrization, and regularization.
Course material (lecture notes and project work description) is in English.

Lectures

Project work instructions are online. Discussed 16 Oct (lecture).
Lecture notes, Part II are online.
No lecture on 30 Oct (opening of Tampere Math Center at TUT).
06 Nov: lecture will be given by course assistant Juho Linna (MK travelling). Questions/practical advice on project work.
Exam reading: Part I: 1, 2.1, 2.2, 3, 4.1, 4.2, 5; Part II: 1, 2.1, 2.2, 2.3, 3.1, 3.3, 4.1, 7
Dig inverse problems? Need a thesis topic (master/doctor)? Plenty of project topics on IPs and mathematical modelling available, ask MK.

Lecturer: professor Mikko Kaasalainen. Lectures in period 1: Monday 10-12 (hall TB219) and Friday 12-14 (hall TB219). Period 2: Friday 12-14 (hall TB219).
Lectures are given in Finnish or English depending on the audience. The course is for graduate and advanced undergraduate studies.

Lecturer´s office hour (room TD 321): 1. period Monday 12-13, 2. period Friday 11-12.

Approximate schedule for 2009:
1. period:
1. Intro, convolution, Dirac-delta sequences
2. Tomography, other discretization examples
3. Inverse crimes, ill-posedness, SVD
4. Regularization: truncated SVD, Tikhonov
5. Regularization weight, generalized Tikhonov
6. Total variation, large-scale computations
7. Statistical inversion, Bayesian inference, Fisher information matrix
8. MCMC, Gibbs sampler, Metropolis-Hastings
9. Generalized projections, L2-functions on unit sphere
10. Generalized geometric tomography, convex mapping, uniqueness, stability
11. Project work description
12. Minkowski inverse problem, nonlinear formulation and algorithms
2. period:
13. Radar-type data, uniqueness, nonconvex parametrizations
14. Project work discussion, intermediate results
15. Profile contours, multidata, maximum compatibility estimate
16. Examples, current research topics, problem fields
17. Project work reports
18. Reserve lecture, discussion, conclusion

Prerequisites

Students entering the course are expected to know
  • 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 activities 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 two parts:
  • Pass course exam (24 points; to be held after 2.period lectures),
  • complete the project work (24 points + several extra points).
    For students who return an acceptable project work report, grading is based on the following score:
    (exam points + project work points)/48.

    Second alternative: Pass a final exam later (and miss the project work extra points option)

    Course material

    Lecture notes from 2008 (Siltanen): Notes, version 12.
    Additional lecture notes for 2009 (Kaasalainen): Lecture notes, Part II .
    Project work description and instructions

    Also, chapters 1-3 in the book Kaipio and Somersalo: Statistical and Computational Inverse Problems (Springer 2005) cover 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/
    Web page of the 2008 course is available here.
    This page was last updated 4 Dec 2009.