# Inverse Problems

### Course content

An Inverse problems is, in its simplest form, a problem of data-fitting. We seek a parameterized model of a physical (or biological, or economical) system, that is able to 'explain' observed data within the measurement errors. Examples are:

• Computing CO2 emission and absorption over a geographical area from observations in a network of measurement stations, using numerical simulations of atmospheric flow

• Computing the infection mechanism of a virus from test data

• Computing parameters of an economic model from statistical data about consumer behavior

• Computing parameters of the earth (e.g., seismic wave propagation velocities) from seismic data, using the wave equation

Data fitting is an optimization problem (the difference between observed data and data computed from the system model must be "small"), but an inverse problem has at least two additional complications:

1. Usually it does not have a unique solution. So, how do we choose a reasonable solution (out of the many)? Observational data are almost always contaminated by noise, and this will make any solution uncertain. How do we compute this uncertainty?

Inverse problem theory provides ways of tackling the above challenges.The objective of the course is to provide theory and methods for solving and analyzing inverse problems. A significant part of the course involves working with projects where inverse problems 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

• Monte Carlo sampling methods

• Analyze and validate solutions to inverse problems

Knowledge
This course will give the student a mathematical description of inverse problems. It teaches him/her to solve linear inverse problems with analytical and numerical methods, and non-linear problems with Monte Carlo methods. The students will study non-uniqueness, resolution and uncertainty of the computed models.

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 quantify 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.

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 passed 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
• Preparation
• 73
• Practical exercises
• 16
• Project work
• 50
• Guidance
• 40
• English
• 206