MAT-62006 Inverse Problems, spring 2015, 7 credits

Note the lecture hall: TD308. The course starts Fri 20 Mar. NO LECTURES 18&25 Mar, 3&8 Apr (Easter vacation).


Inverse problems research is an active area of contemporary mathematics, see TUT inverse problems group .
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,
- 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
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 consists of lectures, and the latter part 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. Case studies (lectures and project work) are then used to illustrate the central concepts of uniqueness, stability, model parametrization, and regularization.


Lecturers: professor Mikko Kaasalainen (Part I) and Dr. Sampsa Pursiainen (Part II). Lectures: Wednesday 10-12 (hall TD308) and Friday 12-14 (hall TD308).
The course is for graduate and advanced undergraduate studies.

Lecturer´s office hour (room TD 321): Friday 11-12.


Students entering the course are expected to know
  • basic linear algebra and matrix computations,
  • 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.

    How to pass the course

  • Pass course exam (24 points),
  • Strongly recommended: project work (up to 15 extra points).
    Grading is based on the following score:
    (exam points + project work points)/24.

    Note that project work points are pure extra, so accomplishing anything in the project work always improves the grade.
    You can return the project work (email pdf) any time in 2015.

    Course material

    Overview and examples of inverse problems:Mathematics makes the unseen visible (and vice versa).
    Lecture notes, Part I (Siltanen): Notes, version 12. Exam coverage: 1, 2.1-2.3,3, 4.1-4.2
    Part II: Lectures 1-7 (part_II_slides.pdf or part_II_summary.pdf) except definitions 4-8. A large part of the material presented in the lecture slides of Part II can also be found in the book by Kaipio and Somersalo on pages 49-114, see below.
    Lecture notes & related material, Part II:
  • Lecture 1: Slides, p_posterior.m
  • Lecture 2: Slides, marginal.m, structural_covariances.m
  • Lecture 3: Slides, ground_layers.m, ground_layers_matrix.mat
  • Lecture 4: Slides, deblur_hierarchical.m, blur_matrix.m, inverse_gamma_samples.m, gamma_samples.m
  • Lecture 5: Slides, random_walk_unit_disk.m
  • Lecture 6: Slides, one_dimensional_sampling.m, gibbs_sampler_unit_disk.m
  • Lecture 7: Slides, deblur_L1_gibbs.m, deblur_hierarchical_gibbs.m
  • Lectures 1-7: Slides, Printable
  • Project work: moving_camera.m (moving camera blur)

    Project work description

    Also, chapters 1-3 in the book Kaipio and Somersalo: Statistical and Computational Inverse Problems (Springer 2005) cover most of the course.
    Additional material is available at

  • Somersalo (in Finnish):
  • Tan, Fox and Nicholls (in English):
    This page was last updated 7 May 2015.