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Course, academic year 2017/2018
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Inverse Problems and Regularization - NMNV531
Czech title: Inverzní úlohy a regularizace
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2016
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: RNDr. Iveta Hnětynková, Ph.D.
Class: 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.
Literature -
Last update: RNDr. Iveta Hnětynková, Ph.D. (07.04.2015)

P. C. Hansen: Discrete Inverse Problems: Insight and Algorithms, Fundamentals of Algorithms, SIAM, 2010.

I. Hnětynková, M. Plešinger and 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 and 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: RNDr. Iveta Hnětynková, Ph.D. (07.04.2015)

Lectures are held in a lecture hall, practicals in a computer laboratory (Matlab enviroment).

Syllabus -
Last update: 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.

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