# Inverse problems, spring 2015

**The course is lectured in English.**

Inverse problems are about interpreting indirect measurements. The scientific study of inverse problems is an interdisciplinary field combining mathematics, physics, signal processing, and engineering. Examples of inverse problems include

- Three-dimensional X-ray imaging (more information)
- Recovering the inner structure of the Earth based on earthquake measurements
- Sharpening a misfocused photograph (more information )
- Reconstructing electric conductivity from current-to-voltage boundary measurements (see this page and this page)
- Finding cracks inside solid structures
- Prospecting for oil and minerals
- Monitoring underground contaminants
- Finding the shape of asteroids based on light-curve data (see this page)

The common features of all this problems are the need to understand indirect measurements and to overcome extreme sensitivity to noise and modelling inaccuracies.

### What does the course contain?

The goals of the course are

- introduce discrete matrix models of some widely used measurements, such as tomography and convolution
- show how to detect ill-posedness (sensitivity to measurement noise) in matrix models using Singular Value Decomposition
- compute noise-robust reconstructions using
*regularization* - write Matlab algorithms for sharpening photographs and computing tomographic reconstructions
- discussion of nonlinear inverse problems, with Electrical Impedance Tomography as an example

The lectures make up 10 credit units. In addition to lectures the course involves a **project work**. It is done in teams of two and gives 5 credit units to each student.

The course is in total **15 credit units**.

### Lecturer

### Scope

15 sp.

### Type

Advanced studies

### Prerequisites

Recommended courses to take before this course: Linear algebra 1 and 2, Applications of matrix computations.

Some previous experience with Matlab programming is very helpful.

### Lectures

**Period III:** Lectures as follows:

Tuesday 10-12 in room D123

Wednesday 12-14 in room D123

Friday 12-14 in room C123.

Two hours of exercise classes per week.

**Period IV:** Lectures and exercises in the beginning of the period. Later project work, which is reported as a poster in a poster session.

### Exams

There will be an exam after the lecture part of the course.

### Bibliography

**Mueller J L and Siltanen S: ***Linear and Nonlinear Inverse Problems with Practical Applications.* SIAM 2012.

### Registration

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### Exercises