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In a variety of applications (computerized tomography, geology, image
processing etc.) there is a need to solve inverse problems, where the goal
is to extract information about the studied phenomena from the measured
data corrupted by errors (noise). Since these problems are sensitive on
perturbations in the data, it is necessary to solve them using special
approaches, so called regularization methods. The lectures give an insight into the properties of inverse problems,
and
summarize modern regularization approaches including parameter choice
methods.
Last update: T_KNM (07.04.2015)
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To finish the course successfully, it is required to pass the exam covering all presented topics, see "Requirements to the exam".
Furthermore, students will complete assignments during the practicals. Assignments consist of implementing numerical experiments in the MATLAB environment using the regularization toolbox. Results are regularly discussed. Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (10.11.2022)
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P. C. Hansen: Discrete Inverse Problems: Insight and Algorithms, Fundamentals of Algorithms, SIAM, 2010.
I. Hnětynková, M. Plešinger, Z. Strakoš: The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data, BIT Numerical Mathematics 49, pp. 669-696, 2009.
P. C. Hansen , J. G. Nagy, D. P. O'Leary: Deblurring Images: Matrices, Spectra, and Filtering, Fundamentals of Algorithms, SIAM, 2006.
P. C. Hansen: Rank-Deficient and Discrete Ill-Posed Problems, Mathematical Modeling and Computation, SIAM, 1998.
Last update: Kučera Václav, doc. RNDr., Ph.D. (15.01.2019)
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Lectures are held in a lecture hall, practicals in a computer laboratory (Matlab enviroment).
In case of distance learning, online communication platforms will be used. Texts, homework assignments, reading assignments and other instructions will be put on a course webpage at MOODLE2 UK. Lectures will take place every week and will have the form of short presentations and discussions at the ZOOM platform. Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (28.09.2020)
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The exam reflects all the material covered on lectures, practicals and reading assignments during the whole semester. The exam has oral form and can be passed using online communication platforms.
Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (28.09.2020)
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1. Inverse problems, their basic properties, examples.
2. Construction of the naive solution, need for regularization, influence of noise.
3. Basic direct and iterative regularization methods. Hybrid methods.
4. Regularization parameter selection approaches.
5. Propagation of noise in iterative regularization methods, noise level estimation without apriori information.
6. Special problems. Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (07.04.2015)
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Previous knowledge of linear algebra and basic methods for matrix computations is expected. Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (30.04.2018)
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