News (updated 16.9.2022)

  • The slides of the twelfth lecture are available here (recording).
  • A bonus exercise sheet is available here. Please note that these exercises will not be graded and they do not need to be returned.


Lectures Mondays 10:15-12:00 T9/046
Exercises Mondays 12:15-14:00 T9/046
Oral exam Monday August 1, 2022 A6/212
Make-up oral exam Tuesday September 27, 2022 A6/212

Please contact in advance to organize a personal exam appointment.

General information


Mathematical measurement models describe the causal effects of physical systems based on their material properties, initial conditions or other model parameters. In many practical problems, we have measurement data of the outcomes of these so-called "forward models" and we wish to infer the model parameters which caused the observations. This is an inverse problem.

Inverse problems are intrinsically ill-posed: the reconstruction of the unknown quantity may be highly sensitive to noise in the measurements, or a unique solution may not exist. For these reasons, regularization is an essential tool in order to find solutions to inverse problems. In this course, we will consider both deterministic regularization methods and statistical Bayesian inference. We will discuss the main challenges related to inverse problems as well as the main solution techniques.

Target audience

The course is intended for mathematics students at the Master's level.


Multivariable calculus, linear algebra, basic probability theory, and MATLAB (or some other programming language).

Completing the course

Passing the course exam and completing weekly exercises.

Lecture notes

Week 1: Introduction, Fredholm equation and its solvability, truncated SVD (files: recording)
Week 2: Morozov discrepancy principle, Tikhonov regularization, and backward heat equation (files: week2.m)
Week 3: Introduction to X-ray tomography, Landweber-Fridman iteration (files: tomodemo.m)
Week 4: Conjugate gradient method (files: cgdemo.m)
Week 5: Brief introduction to probability theory and Bayes' formula for inverse problems (files: week5.m)
Week 6: Deconvolution example, Bayesian estimators, and well-posedness of Bayesian inverse problems (files: week6.m)
Week 7: Sampling from Gaussian distributions, inverse transform sampling, prior modeling, and the linear Gaussian setting (files: priormodeling.m, week7.m)
Week 8: Small noise limit of the posterior distribution, Monte Carlo, and importance sampling
Week 9: Markov Chain Monte Carlo (files: mh.m, gibbs.m, autocovariance.m)
Week 10: Optimization perspective and Gaussian approximation
Week 11: Brief overview of Bayesian and Kalman filters (files: kfdemo.m)
Week 12: Total variation regularization for X-ray tomography (files: recording, tvdemo.m, sinogram.mat)


Exercise 1
Exercise 2
Exercise 3 (files: week3.mat)
Exercise 4 (files: week3.mat)
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11 (files: kfdemo.m)
Bonus exercises (files: pde.mat)

Please note that the bonus exercises will not be graded and do not need to be returned.


Dr. Vesa Kaarnioja

Arnimallee 6, Room 212
Consulting hours: By appointment


The course will mainly follow the following texts:

  • J. Kaipio and E. Somersalo (2005). Statistical and Computational Inverse Problems. Springer, New York, NY.
  • D. Sanz-Alonso,  A. M. Stuart, and A. Taeb (2018). Inverse Problems and Data Assimilation.
  • D. Calvetti and E. Somersalo (2007). Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing. Springer, New York, NY.

Further topics:

  • O. P. Le Maître and O. M. Knio. Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Scientific Computation. Springer, New York, 2010.
  • R. C. Smith. Uncertainty Quantification: Theory, Implementation, and Applications, volume 12 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.
  • T. J. Sullivan. Introduction to Uncertainty Quantification. Springer, New York, in press.
  • D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton, NJ, 2010.