NFYA04034U Inverse Problems

Volume 2015/2016
M.Sc. Physics

The purpose of the course is to provide insight into the mathematical description of indirect measurement problems (inverse problems) as seen in physical, astrophysical and geophysical laboratory-, observatory- and field investigations.

The course covers classical, linear inverse theory as well as classical and modern nonlinear inverse theory. A number of analytical/numerical methods for solution of linear and nonlinear inverse problems are presented. The propagation of noise on the data to uncertainty on the solution is an important theme in the course.

Learning Outcome

The course aims at giving the students the skills to

  • solve linear inverse problems with analytical methods,
  • solve weakly nonlinear inverse problems with iterative methods, based on linearization,
  • solve strongly nonlinear inverse problems with Monte Carlo methods,
  • investigate how noise on data propagates into uncertainty on the solutions,
  • define overdetermination and underdetermination of an inverse problem, and explain the connection between these properties and numerical instability,
  • define an ill-posed problem, and
  • explain the role of a priori information in a Bayesian formulation of an inverse problem.

This course will give the student a mathematical description of inverse problems as they appear in connection with measurements and experiments in geophysics, astrophysics and other areas of physics. It teaches them to solve linear inverse problems with analytical and numerical methods and non-linear problems with Monte Carlo methods. The students will study the propagation of noise in data to uncertainty in the solutions.



Through the course the student will be able to identify an inverse problems in various fields of physics, classify it and choose an appropriate method to solve it. The student will be able to treat data uncertainties to evaluate the accuracy and stability of the inverse solution.



Published scientific papers and Richard Aster, Brian Borchers, Clifford Thurber: "Parameter estimation and inverse problems", International Geophysics Series, Elsevier 2013, 2nd edition

Undergraduate classical physics (including electromagnetism), mathematical analysis and linear algebra.
Lectures and computer exercises
Restricted elective for specialisation "Physics".
It is expected that the student brings a laptop.
Necessary software:
Windows: Xming
.Xming:http:/​/​​projects/​xming/​ & http:/​/​​XmingNotes/​
For support please contact SCIENCE IT, e-mail:, 35 32 21 00
Linux:X11 runs automatically
MAC: For all systems since OS 10.5 you can use X11, which you can download for free at http:/​/​​landing/​.
X11 is a part of OS X in Leopard and Lion.
  • Category
  • Hours
  • Lectures
  • 32
  • Practical exercises
  • 16
  • Preparation
  • 68
  • Project work
  • 90
  • Total
  • 206
7,5 ECTS
Type of assessment
Continuous assessment
3 larger computer excercises and accompanying reports. The two first exercises count 25% each, the third counts 50%.
Marking scale
7-point grading scale
Censorship form
No external censorship
More internal examiners
New project reports must be submitted.
Criteria for exam assesment

Grade 12 is given for the outstanding performance demonstrating complete fulfilment of the goals described inSkills, with no or few, unimportant shortcomings.