Photo by Samuli Siltanen

Inverse problems are about interpreting indirect measurements. The scientific study of inverse problems is an interdisciplinary field combining mathematics, physics, signal processing, and engineering.

Examples of inverse problems include:
- Three-dimensional X-ray imaging
- Recovering the inner structure of the Earth based on earthquake measurements
- Sharpening a misfocused photograph
- Reconstructing electric conductivity from current-to-voltage boundary measurements
- Finding cracks inside solid structures
- Prospecting for oil and minerals
- Monitoring underground contaminants
- Finding the shape of asteroids based on light-curve data

The common features of all this problems are the need to understand indirect measurements and to overcome extreme sensitivity to noise and modelling inaccuracies.

The topic of this course is statistical inverse problems. Statistical approach to inverse problems aims to quantify how uncertainty in the data or model affect the solutions obtained in problems such as the ones listed above? This course provides main numerical and theoretical tools to understand uncertainty quantification in inverse problems.

During the period IV it is also possible to make a project work ("Inverse problems project work (MAST31405)" (5cr)) related to the course material. More details are given during the lectures.

11.12.2017 klo 09:00 - 2.5.2018 klo 23:59


Tapio Helin

Julkaistu, 4.4.2018 klo 15:25

Dear all,

Please, answer to this email by Wednesday, April 11, if you want to participate the Inverse problems project work during April and May.

This year Inverse problems project work (MAST31405, 5cr) is related to the Bayesian inversion course. The idea is to study a Bayesian inverse problem both theoretically and computationally in teams of two students. The end product is a scientific poster that the team will present in a poster session in late May (the date will be set later).  The poster can be printed using the large-scale printer of the Industrial Mathematics Laboratory. 

Project work assistants are Jalo Nousiainen and Alexander Meaney.

The recommended measurement context of the project is X-ray tomography. You can measure a dataset yourself in the X-ray facility of the Industrial Mathematics Laboratory. In case you have already completed a project work using X-ray data, find another topic together with the lecturer.

The classical table of contents is recommended for structuring the poster:
1 Introduction
2 Materials and methods
3 Results
4 Discussion
Section 2 is for describing the data and the inversion methods used. In section 3 those methods are applied to the data and the results are reported with no interpretation; just facts and outcomes of computations are described. Section 4 is the place for discussing the results and drawing conclusions.

The project has the following workflow:

1) Get familiar with the material provided on the course web page for the project (X-ray tomography problem and codes). You can contact Jalo Nousiainen ( who give you a tutorial on the codes and ideas for computational task (this is optional).

2) Alexander Meaney ( will supervise the team at the X-ray facility. Contact him to agree on a measurement session during April. Also, choose an interesting object to be measured by X-ray at the lab. For suitability of your object, ask more details from Alexander.

3) First goal (deadline April 27) consists of two things: (a) two first sections should be preliminary written in LaTeX (not necessarily in poster format yet) and (b) the Matlab codes related to the measurement should be run and studied. Two things will be graded in the meeting about the first goal: (a) the draft of project work and (b) your understanding of the available Matlab codes relevant to your topic. The grade represents 30% of the final grade of the project work. Please agree on a meeting time with the lecturer for reviewing and grading the first goal. It is not necessary to finish 2) before 3)

4) Second and final goal (deadline late May): poster is presented in the poster session. The poster will be printed in size A1. You may create your own poster (from scratch), or you can use e.g. the template as a starting point and edit its layout, colors, fonts, etc. as much as you like.

See the bottom of the page,+sprin...
for examples of posters.

Sufficient codes are provided on the page

Template for the poster

(Links are also available on the course webpage)

Tapio Helin

Julkaistu, 27.3.2018 klo 15:01

Dear all,

all the guest lectures have now been given. Please, prepare lecture diaries of the fourth (Marko Laine) and fifth (Stratos Staboulis) talk by end of next week (6.4.) and send/bring them to the course assistant (Jalo Nousiainen). Related to that, we will soon publish the criteria of the course grade.

Tapio Helin

Julkaistu, 26.3.2018 klo 16:04

Dear all,

tomorrow on Tuesday 27.3. we will have the last guest lecture. Notice that the talk starts 1pm (no lecture before that) in the usual class room.

This week's speaker is Dr. Stratos Staboulis from Eniram Ltd and he will give a talk with title "Grey-box modelling of marine vessels". This talk is simultaneously part of Inverse problems seminar so the audience includes also personnel of the math department.

Tapio Helin

Julkaistu, 27.2.2018 klo 16:50

Dear all,

due to the strike of the university tomorrow (Wed, Feb 28) the exercise sessions are cancelled and the exercise session related to fifth exercises will be held on Wednesday, March 14.

Tapio Helin

Julkaistu, 21.2.2018 klo 10:13

Dear all,

the room for exercise session has changed from C129 to C128 (next room). The change takes place starting today so that we can use university computers for the exercises. (UPDATE: the room numbers were wrong in the original message)

Tapio Helin

Julkaistu, 20.2.2018 klo 9:47

Dear all,

please notice that lectures on Tuesday 27.2. and Wednesday 28.2. are CANCELLED due to work travel by the lecturer.


Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

Ma 15.1.2018
10:15 - 12:00
Ti 16.1.2018
12:15 - 14:00
Ke 17.1.2018
12:15 - 14:00
Ma 22.1.2018
10:15 - 12:00
Ti 23.1.2018
12:15 - 14:00
Ke 24.1.2018
12:15 - 14:00
Ma 29.1.2018
10:15 - 12:00
Ti 30.1.2018
12:15 - 14:00
Ke 31.1.2018
12:15 - 14:00
Ma 5.2.2018
10:15 - 12:00
Ti 6.2.2018
12:15 - 14:00
Ke 7.2.2018
12:15 - 14:00
Ma 12.2.2018
10:15 - 12:00
Ti 13.2.2018
12:15 - 14:00
Ke 14.2.2018
12:15 - 14:00
Ma 19.2.2018
10:15 - 12:00
Ti 20.2.2018
12:15 - 14:00
Ke 21.2.2018
12:15 - 14:00
Ma 26.2.2018
10:15 - 12:00
Ti 27.2.2018
12:15 - 14:00
Ke 28.2.2018
12:15 - 14:00
Ma 12.3.2018
10:15 - 12:00
Ti 13.3.2018
12:15 - 14:00
Ke 14.3.2018
12:15 - 14:00
Ma 19.3.2018
10:15 - 12:00
Ti 20.3.2018
12:15 - 14:00
Ke 21.3.2018
12:15 - 14:00
Ma 26.3.2018
10:15 - 12:00
Ti 27.3.2018
12:15 - 14:00
Ke 28.3.2018
12:15 - 14:00
Ma 9.4.2018
10:15 - 12:00
Ti 10.4.2018
12:15 - 14:00
Ke 11.4.2018
12:15 - 14:00

Muu opetus

24.01. - 11.04.2018 Ke 10.15-12.00
24.01. - 21.02.2018 Ke 10.15-12.00
Tapio Helin
Opetuskieli: englanti


Lecture notes




Exercise 1

Solutions 1

Exercise 2

There is a typo in the last line of the problem 5. There is no "s" in the problem, so you can just neglect the line "when s>t".

Solutions 2

Exercise 3

Solutions 3

Exercise 4

Solutions 4

Solutions 5

Exercise 6

Solutions 6

Kurssin suorittaminen

Course will be graded with grades 1-5 based on points obtained from home assignments and lecture diary. The additional 5cr project work is graded based on a poster presentation.


Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Probability theory, Measure and integration theory

Master studies

Learning the framework for statistical Bayesian inverse problems. Understand main computational and theoretical ideas of uncertainty quantification via the Bayes formula. Learn efficient computational methods for uncertainty quantification.

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Bayesian approach to ill-posed inverse problems. Computational methods for exploring the posterior distribution. Theory of well-posedness and stability properties of Bayesian inversion. Practical project work in the Industrial Mathematics Laboratory.

Lecture notes

Lectures and exercise classes, including Matlab programming in computer class

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later