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Course, academic year 2023/2024
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Inverse Problems and Regularization - NMNV531
Title: Inverzní úlohy a regularizace
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Additional information:
Guarantor: doc. RNDr. Iveta Hnětynková, Ph.D.
Class: M Mgr. MMIB > Povinně volitelné
M Mgr. NVM
M Mgr. NVM > Povinně volitelné
Classification: Mathematics > Numerical Analysis
Annotation -
Last update: T_KNM (07.04.2015)
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.
Course completion requirements -
Last update: doc. RNDr. Iveta Hnětynková, Ph.D. (10.11.2022)

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.

Literature -
Last update: doc. RNDr. Václav Kučera, Ph.D. (15.01.2019)

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.

Teaching methods -
Last update: doc. RNDr. Iveta Hnětynková, Ph.D. (28.09.2020)

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.

Requirements to the exam -
Last update: doc. RNDr. Iveta Hnětynková, Ph.D. (28.09.2020)

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.

Syllabus -
Last update: doc. RNDr. Iveta Hnětynková, Ph.D. (07.04.2015)

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.

Entry requirements -
Last update: doc. RNDr. Iveta Hnětynková, Ph.D. (30.04.2018)

Previous knowledge of linear algebra and basic methods for matrix computations is expected.

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