NFYA04034U Inverse Problems

Volume 2021/2022

MSc Programme in Physics

MSc Programme in Physics with a minor subject


The objective of the course is to provide theory and methods for solving and analyzing inverse problems in physical sciences. Inverse problem theory will be formulated as a probabilistic data integration problem, and a number of analytical/numerical methods for solution of inverse problems will be presented. The role and interplay between problem formulation, uncertainties in data and model parameters, and prior knowledge are important themes in the course. A significant part of the course involves work with projects where inverse problems from physics, astrophysics, geoscience or engineering will be analyzed.

Learning Outcome


This course aims to provide the student with skills to

  • Formulate a complex data analysis problem as an inverse problem
  • Describe and quantify data uncertainties and modeling errors
  • Describe available prior (external) information using probabilistic models and methods
  • Find probabilistic solutions to
    - Linear and weakly non-linear, Gaussian inverse problems
    - General (non-linear, non-Gaussian) inverse problems
  • Analyze and validate solutions to inverse problems



This course will give the students a mathematical description of inverse problems as they

appear in connection with measurements and experiments in physical sciences. It teaches them to solve inverse problems with analytical and numerical methods. The students will study illposedness, numerical instability, non-uniqueness of solutions, and how noise propagates into uncertainty of the solutions.


Through the course the students will be able to identify inverse problems in various fields of physical sciences, classify them, and choose appropriate solution methods. The students will be able to find appropriate parameterizations, treat data uncertainties, and to evaluate the accuracy and resolution of the inverse solution.

See Absalon for final course material.

An introductory programming course is recommended.
Knowledge of linear algebra, mathematical analysis, and differential equations (ordinary and partial) corresponding to a B.Sc. in physics or mathematics is expected.

In general, academic qualifications equivalent to a BSc degree is recommended.
Lectures, exercises and projects.
  • Category
  • Hours
  • Lectures
  • 27
  • Preparation
  • 73
  • Practical exercises
  • 16
  • Project work
  • 50
  • Guidance
  • 40
  • Total
  • 206
7,5 ECTS
Type of assessment
Continuous assessment
Oral examination, 20 minutes
3 projects (group or individual) [weighed by 12.5%, 12.5% and 25%] followed by 1 individual oral examination [weighed by 50%]. Both the continuous evaluation and the oral examintation should be passed separately.
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Same as ordinary exam. The student can choose to re-use points from projects handed in during the course, or make new projects, which must be handed in no later than 2 weeks before the oral re-exam.

Criteria for exam assesment

see "learning outcome"