SubjectsSubjects(version: 944)
Course, academic year 2023/2024
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Analysis of matrix iterative methods - principles and interconnections - NMNV412
Title: Analýza maticových iteračních metod – principy a souvislosti
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:4/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. Ing. Zdeněk Strakoš, DrSc.
Class: M Mgr. MMIB > Povinně volitelné
M Mgr. MOD > Povinně volitelné
M Mgr. NVM > Povinné
Classification: Mathematics > Numerical Analysis
Is interchangeable with: NMNV407
Annotation -
Last update: doc. RNDr. Václav Kučera, Ph.D. (09.12.2018)
The course will deal with mathematical foundations of matrix iterative methods, in particular Krylov subspace methods, in connection with the areas of mathematics and computer science important for understanding basic principles and the state of the art. It will formulate open questions and explain existing misunderstandings going across the fields that prevent deeper understanding and the development of the theory as well as efficient use of the methods in applications.
Aim of the course -
Last update: prof. Ing. Zdeněk Strakoš, DrSc. (22.12.2022)

Using the Krylov subspace methods case study, the course aims at helping students in developing the ability of seeing the whole context, in asking themselves questions and

in seeking deep interconnections and overcoming narrowly specialized sights that restrict so much needed communication across the fields. Therefore the course will combine formulation and addressing questions in infinite dimensional Hilbert spaces using elements of linear functional analysis and spectral theory of operatots with traditional matrix approach. The course will also require self-study reading of selected publications followed by discussion.

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

J. Liesen, Z. Strakoš, Krylov Subspace Methods, Principles and Analysis, Oxford University Press, Oxford, 2013.

J. Málek, Z. Strakoš, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs, SIAM, Philadelphia, 2015.

Requirements to the exam -
Last update: prof. Ing. Zdeněk Strakoš, DrSc. (09.03.2021)

There will be oral exam consisting of discussion of topics of the course in the extent given by the course lectures.

Syllabus -
Last update: doc. RNDr. Václav Kučera, Ph.D. (19.12.2018)

The course will cover primarily projection methods and, in particular, Krylov subspace methods in relation to the problem of moments and related issues. The emphasis will be on interconnections between the relevant topics from various disciplines, including the elements of numerical solution of partial differential equations, approximation theory and functional analysis.

Tentative content:

1. Projection processes.

2. Krylov subspaces.

3. Basic methods.

4. Stieltjes moment problem.

5. Orthogonal polynomials, continued fractions, Gauss-Christoffel quadrature and model reduction .

6. Matrix representation and the method of conjugate gradients.

7. Vorobyev method of moments and non-symmetric generalizations.

8. Non-normality and spectral information.

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