MAT-52506 Inverse Problems, autumn 2011, 6 credits

Starts 2 Sep

Introduction

Examples of inverse problems:
- sharpening a blurred photograph (more),
- imaging inner organs of patients using X-ray images taken from different directions,
- determining the structure of the Earth using seismic data,
- locating cracks inside materials using electric surface probes,
- finding the shape of a distant asteroid from light intensity measurements (more).
The common feature of all these problems is that they are very sensitive to measurement noise, and their solution is not straightforward.

This course concentrates on how to recognize an inverse problem and how to solve it in practice even when the data are noisy and the number of unknowns is large.

The first part of the course (period I) consists of lectures, and the latter part (period II) of lectures and project work. The project work can be done either individually or in teams of two or three students.
Emphasis is placed on the practical solution of real-world problems; theory is introduced only to the extent needed to understand and implement solution methods.

Central methodological themes of the course are singular value decomposition (SVD) of a matrix, Tikhonov regularization, total variation regularization, and statistical inversion. A case study in generalized projections (lectures and project work) is then used to illustrate the central concepts of uniqueness, stability, model parametrization, and regularization.

Lectures

Lecturer´s office hour (room TD 321): Friday 11-12.

Approximate schedule:
1. period:
1. Intro, convolution, Dirac-delta sequences
2. Tomography, other discretization examples
3. Inverse crimes, ill-posedness, SVD
4. Regularization: truncated SVD, Tikhonov
5. Regularization weight, generalized Tikhonov
6. Total variation, large-scale computations
7. Statistical inversion, Bayesian inference, Fisher information matrix
8. MCMC, Gibbs sampler, Metropolis-Hastings
9. Generalized projections, L2-functions on unit sphere
10. Generalized geometric tomography, convex mapping, uniqueness, stability
11. Project work description
12. Nonlinear formulations and algorithms
2. period:
13. Radar-type data, uniqueness, nonconvex parametrizations
14. Project work discussion, intermediate results
15. Profile contours, multidata, maximum compatibility estimate
16. Examples, current research topics, problem fields
17. Project work reports
18. Reserve lecture, discussion, conclusion

Prerequisites