Inverse Problems

Course content

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.


MSc Programme in Physics

MSc Programme in Physics with a minor subject

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.

Lectures, exercises and projects.

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.

7,5 ECTS
Type of assessment
Oral examination, 20 minutes
Continuous assessment, during course
Type of assessment details
Oral exam is without preparation, and counts 50% of the grade.
3 individual projects count for 50% of the grade.
The parts of the exam do not have to be passed separately - a total grade is given.
Without aids

All aids allowed for the projects, no aids allowed for the oral exam.

Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Oral exam withot preparation time. The student can choose to hand in new projects  no later than 2 weeks before the oral re-exam, or re-use points from the ones handed in during the course.

Criteria for exam assessment

see "learning outcome"

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 27
  • Preparation
  • 73
  • Practical exercises
  • 16
  • Project work
  • 50
  • Guidance
  • 40
  • English
  • 206


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 2
No restriction
The number of seats may be reduced in the late registration period
Study Board of Physics, Chemistry and Nanoscience
Contracting department
  • The Niels Bohr Institute
Contracting faculty
  • Faculty of Science
Course Coordinator
  • Klaus Mosegaard   (9-707276686a646475674371656c316e7831676e)

Klaus Mosegaard

Saved on the 04-05-2023

Are you BA- or KA-student?

Are you bachelor- or kandidat-student, then find the course in the course catalog for students:

Courseinformation of students