Computational methods for inverse problems, January 2009, 3-5 credits

Minicourse given at Helsinki University of Technology in January 2009.
Results are available here.


Inverse problems is an active area of contemporary mathematics.
Applications include signal processing, nondestructive testing, 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 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 (3 credit units) consists of lectures and exercises, and the latter part (additional 2 credit units) is a project work done in teams of two 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.

Lectures are given by professor Samuli Siltanen, Department of Mathematics, Tampere University of Technology.
The course is in part based on the lecturer's experience in medical imaging industry (Instrumentarium Imaging, General Electric, Palodex Group).
You can reach the lecturer by email to .

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

Course schedule

Location of all lectures is hall U322 in the Department of Mathematics at Helsinki University of Technology.

Wednesday January 7, 9-12 First lecture, introduction to inverse problems, discrete measurement models and singular value decomposition (SVD).

Thursday January 8, 9-12 Truncated SVD, Tikhonov regularization and total variation regularization.
Thursday January 8, 14-16 Exercise session 1 in hall Y342a, assistant Juho Linna (email Juho.Linna (at)
Exercise 1. Matlab routines: ex_conv1Ddata_comp.m, ex_conv1D_naive.m, ex_conv1D_SVD.m.
Matlab files with solutions: H1_4.m, H1_5.m, H1_6.m, H1_4_rec.m.

Friday January 9, 9-12 Statistical inversion: Bayes formula, posteriori distribution, estimation methods.
Friday January 9, 14-16 Exercise session 2 in hall Y342a
Exercise 2. Matlab routines: ex_conv1D_Tikhonov_Lcurve.m, ex_conv1D_stackedform.m.
Matlab files with solutions: H2_4.m, H2_5.m.

Thursday January 15, 13-17 Exercise session (theory and Matlab), assistant Lauri Harhanen (email loharhan (at)
Extensive Exercise 3.
Matlab routines for exercise set 1: oblur.m written by Per Christian Hansen, ex_conv_2D.m, ex_conv_2D_SVD.m and ex_conv_2D_Tikhonov_CG.m.
Matlab routines for exercise set 2: conv1D_07_stackedform.m.
Matlab routines for exercise set 3: deblur_BarzilaiBorwein.m, deblur_aTV_fgrad.m, deblur_misfit_grad.m, deblur_aTV.m, deblur_aTV_grad.m, deblur_aTV_feval.m, deblur_misfit.m.
The idea is that you solve as many problems as you can and report on Thursday the list of problems you solved. Please have written solutions with you at the exercise session (and prints of important Matlab plots), and give them to the assistant who will check that you have completed the problems. The assistant may ask you to present your solution in the session. If you have questions about the exercise, or if you find bugs in the Matlab codes, please do not hesitate to contact me by email. I will inform the whole group about corrections or updates then.

Friday January 16, 10-12 in hall U322. Instructions about project works, division into two-student teams, distribution of project work topics
Friday January 16, 14-16 in hall K. Final exam.

*** Four weeks time to complete the project work, return electronically *** Instructions on how to prepare the project work can be found on the Project work page.


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 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?

    Officially, this course is an implementation of the course
    Mat-1.3658 Special course in numeric analysis (3-5 cr)
    of Helsinki University of Technology (HUT). So for HUT students it is simple to get credits.
    There are so many students from University of Helsinki that I will find a collective way to accept the course there.
    Students from elsewhere need to get the course accepted in their home university, please contact the lecturer for details.

    The course can be either as a 3 credit version (lectures, exercises, exam), or as a 5 credit version by completing a project work.
    The course is graded on the scale from 1 (passed) to 5 (excellent).

    The grade of the 3 credit version consists of
    1. Attending the lectures and exercises on January 7-9 (20%),
    2. Return completed exercises on January 15 (40%), and
    3. Final exam (40%).

    Instructions on how to prepare the project work can be found on the Project work page.

    Course material

    This is the most recent version of lecture notes: Lecture Notes, version 10. Note that it is still in very preliminary state. Please send comments and suggestions.

    Additional material is available at

  • Chapters 1-3 in the book Kaipio and Somersalo: Statistical and Computational Inverse Problems (Springer 2005)
  • Kaipio (in Finnish):
  • Somersalo (in Finnish):
  • Tan, Fox and Nicholls (in English):

    Matlab resources

    The one-dimensional convolution example is demonstrated in the lectures using the following routines:
    This page was last updated on February 11, 2009.