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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.
Last update: Kučera Václav, doc. RNDr., Ph.D. (09.12.2018)
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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.
Last update: Strakoš Zdeněk, prof. Ing., DrSc. (22.12.2022)
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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. Last update: Kučera Václav, doc. RNDr., Ph.D. (15.01.2019)
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There will be oral exam consisting of discussion of topics of the course in the extent given by the course lectures. Last update: Strakoš Zdeněk, prof. Ing., DrSc. (09.03.2021)
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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. Last update: Kučera Václav, doc. RNDr., Ph.D. (19.12.2018)
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