# Inverse Problems

### 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.

Education

MSc Programme in Physics

MSc Programme in Physics w. 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 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.

Competences
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.

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.

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
Censorship form
No external censorship
Several internal examiners
##### Criteria for exam assessment

see "learning outcome"

Single subject courses (day)

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