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.
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.
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.
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