Lectures Mon 10:15-12:00 A6/025/026 Dr. Vesa Kaarnioja
Exercises Tue 10:15-12:00 A6/007/008 Dr. Vesa Kaarnioja
Oral exam Wed July 26, 2023 A6/213  

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

The conditions for completing this course are successfully completing and submitting at least 60% of the course's exercises and successfully passing the course exam.


  • Please register to the course via Campus Management (CM), then you will be automatically registered in MyCampus/Whiteboard as well. Please note the deadlines indicated there. For further information and in case of any problems, please consult the Campus Management's Help for Students.
  • Non-FU students should register to the course in KVV (Whiteboard)

Lecture notes

Lecture notes will be published here after each week's lecture.

Exercise sheets

Weekly exercises will be published here after each lecture.

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.