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Course, academic year 2023/2024
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Saddle Point Problems and Their Solution - NMMO537
Title: Sedlobodové úlohy a jejich řešení
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: summer
E-Credits: 5
Hours per week, examination: summer 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
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Ing. Miroslav Rozložník, Dr.
Class: M Mgr. MMIB > Povinně volitelné
M Mgr. MOD
M Mgr. MOD > Povinně volitelné
M Mgr. NVM
M Mgr. NVM > Volitelné
Classification: Mathematics > Numerical Analysis
Annotation -
Last update: T_KNM (08.04.2015)
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.
Course completion requirements -
Last update: RNDr. Miloslav Vlasák, Ph.D. (17.05.2018)

It is necessary to give a presentation on the solution of saddle point problems in relation to the main specialization of the student.

Literature -
Last update: doc. RNDr. Václav Kučera, Ph.D. (29.10.2019)

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.)

Requirements to the exam -
Last update: RNDr. Miloslav Vlasák, Ph.D. (17.05.2018)

It is necessary to pass the exam oriented on the topics discussed during the course.

Syllabus -
Last update: T_KNM (08.04.2015)

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.

Entry requirements -
Last update: RNDr. Miloslav Vlasák, Ph.D. (17.05.2018)

Basic knowledge of following areas: linera algebra, matrix calculus, numerical mathematics and finite element method.

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