Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

Inverse Problems

Practical information
Study year 2016/2017
Block 2
Programme level Full Degree Master
Course responsibles
  • Klaus Mosegaard (9-6f71756769636374664270646b306d7730666d)
  • Thomas Mejer Hansen (5-857e767972517f737a3f7c863f757c)
  • The Niels Bohr Institute
Course number: NFYA04034U

Course content

The objective of the course is to provide theory and methods for solving and analyzing inverse problems in physics and geosciences. Inverse problem theory will be formulated as a probabilistic data integration problem, and a number of analytical/numerical methods for solution of linear and nonlinear inverse problems will be presented. The role and interplay between uncertainties in data, model and prior knowledge is an important theme in the course.

A significant part of the course involves work with projects where inverse problems from physics and geosciences will be analyzed,for example analysis of seismic data or climate data.

Learning outcome

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

This course will give the student a mathematical description of inverse problems as they appear in connection with measurements and experiments in physics and geosciences. 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 inverse problems in various fields of physics and geosciences, 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.



Recommended prerequisites

Throughout the course Matlab will be used extensively, and therefore Dat F (NFYA06018U) or equivalent will be useful.
Linear Algebra corresponding to the course LinAlgNat or equivalent.

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M.Sc. Physics


Study Board of Physics, Chemistry and Nanoscience

Course type

Part time Master and Diploma courses


1 block


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Teaching and learning methods

Lectures, exercises (using Matlab), and projects.


No restriction to number of participants




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


Category Hours
Lectures 27
Practical exercises 16
Project work 90
Preparation 73
English 206


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

Criteria for exam assessment

see "skills"

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 handen in no later than 2 weeks before the oral re-exam.

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