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The course is devoted to the solution of large linear saddle-point systems that arise
in a wide variety of applications in computational science and engineering.
The aim is to discuss particular properties of such linear systems as well as a large selection
of algebraic methods for their solution with emphasis on iterative methods and preconditioning.
Last update: T_KNM (08.04.2015)
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It is necessary to give a presentation on the solution of saddle point problems in relation to the main specialization of the student. Last update: Vlasák Miloslav, RNDr., Ph.D. (17.05.2018)
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M. Benzi, G. H. Golub, J. Liesen. Numerical solution of saddle point problems. Acta Numerica, 2005, pp. 1- 137.
M. Rozložník: Saddle point problems, iterative solution and preconditioning: a short overview, Proceedings of the XV-th Summer School Software and Algorithms of Numerical Mathematics, I. Marek ed., University of West Bohemia, Pilsen, 97-108 (2003). (Iteračné riešenie rozsiahlych sústav sedlového bodu v matematickom modelovaní , Technical University of Liberec, Department of Modelling of Processes, Faculty of Mechatronics and Interdisciplinary Studies, Liberec, April 2004.) Last update: Kučera Václav, doc. RNDr., Ph.D. (29.10.2019)
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It is necessary to pass the exam oriented on the topics discussed during the course. Last update: Vlasák Miloslav, RNDr., Ph.D. (17.05.2018)
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1. Introductory remarks. Formulation of saddle-point problem.
2. Applications leading to saddle-point problems.
Augmented systems in the least squares problems.
Saddle point problems from the discretization of partial differential equations with constraints.
Kuhn-Karush-Tucker (KKT) systems in interior-point methods.
3. Properties of saddle point matrices.
The inverse of a saddle-point matrices.
Spectral properties of saddle-point matrices.
4. Solution approaches for saddle-point problems.
Schur complement reduction.
Null-space projection method.
5. Direct methods for symmetric indefinite systems.
Direct solution of saddle-point methods.
6. Iterative solution of saddle-point problems.
Stationary iteration methods.
/Krylov subspace methods. Preconditioned Krylov subspace methods.
7. Saddle-point preconditioners.
8. Implementation and numerical behavior of saddle-point solvers. 9. Polluted undeground water flow modelling in porous media. Last update: T_KNM (08.04.2015)
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Basic knowledge of following areas: linera algebra, matrix calculus, numerical mathematics and finite element method. Last update: Vlasák Miloslav, RNDr., Ph.D. (17.05.2018)
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