Inverse Problems

Course content

Inverse problems are problems where physical data from indirect measurements are used to infer information about unknown parameters of physical systems. Noise‐contaminated data and prior information on model parameters are the basic elements of any inverse problem. Using probability theory, we seek a consistent formulation of inverse problems, and from our fully probabilistic results we can, in principle, answer any question pertaining our state of information about the system when all information has been integrated. The objective of the course is to provide theory and methods for solving and analyzing inverse problems. A significant part of the course involves work with projects where inverse problems from physicical disciplines will be analyzed.

Education

MSc Programme in Physics

MSc Programme in Physics with a minor subject

Learning outcome

Skills
This course aims to provide the student with skills to

  • Describe and quantify data uncertainties and modeling errors.

  • Describe available prior (external) information using probabilistic/statistical models and methods

  • Solve inverse problems

    • Linear and weakly non-linear Gaussian inverse problems

      • Probabilistic least squares inversion

      • Classical parameter estimation methods and regularization

    • Non-linear non-Gaussian inverse problem

      • Importance sampling (rejection, Metropolis, extended Metropolis)

  • Analyze and validate solutions to inverse problems

Knowledge
This course will give the student a mathematical description of inverse problems as they appear in connection with measurements and experiments in physical sciences. 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.

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

Lectures, exercises (using Matlab), and projects.

See Absalon for final course material. The following is an example of expected course litterature.

 

Tarantola (2005) Inverse Problem Theory, and Lecture notes.

Throughout the course Matlab will be used extensively, and therefore an introductory programming course in MatLab is recommended.
Knowledge of Linear Algebra corresponding to the B.Sc. in physics or mathematics is expected.

Academic qualifications equivalent to a BSc degree is recommended.

ECTS
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 pased separately.
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners
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