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Applied iterative methods

ДВС.1.06

Вибіркова дисципліна для ОП

First

2023/2024

8 Semester

5

LO 14. Demonstrate the ability to self-study and professional development. PLO 22.1 To know the main sections of applied mathematics and informatics, to the extent necessary for mastering general professional mathematical disciplines, applied disciplines and the use of their methods in the chosen profession. PLO 24.1. Be able to develop mathematical, numerical and structural-algorithmic models.

Distance form

1. Know: Algebra, mathematical analysis, theory of differential equations, equations of mathematical physics, numerical methods of mathematical physics and programming. 2. Be able to: create programs in at least one programming language, read and analyze mathematical texts, including in English, implement mathematical algorithms. 3. Possess elementary skills: working with a computer, searching for information on the Internet.

Mathematical apparatus of the theory of applied iterative methods. Eigenvalues ​​and eigenfunctions of difference operators. Direct and iterative methods of solving mesh equations. Basic iterative methods (Jacobi, Seidel, upper relaxation, Richardson), convergence, conditions of use. Two-layer iterative methods. Variable-triangular method. Algorithm, convergence, application to solving mesh equations. Iterative methods of variable directions. Algorithm, convergence, application to solving mesh equations. Iterative methods of the variational type. Methods of minimal discontinuities, minimal corrections, fastest descent, conjugate gradients. Triangular iterative methods. Algorithms. Convergence. Application. Three-layer iterative methods. Error estimation. Chebyshov's semi-weathering method. Three-layer methods of conjugate directions. Iterative methods of solving the nonlinear heat conduction equation. Module 2 Examples of well-known problems that are solved by iterative methods. Approximation problem. Nonlinear case. Construction of the nonlinear version of the MNC — the Gauss-Newton method, the Levenberg-Marquardt method. Ranking of search results by the PageRank algorithm. Communication with recommender systems. The big data problem. Approaches to distributed processing and data storage. Vertical and horizontal scaling. History of development — distributed file systems. Basic ideas of HDFS and HADOOP. The MapReduce distributed computing paradigm. Implementation of the PageRank algorithm on MapReduce. Further development of distributed computing. The main ideas of ApacheSpark. Implementation of iterative methods. A variant of the PageRank algorithm for Spark. Implementation of other iterative methods for Spark. Levenberg-Marquardt method. Gradient method for linear regression. Solving Spark optimization problems.

1. Boyd S., Vandenberghe L. Introduction to Applied Linear Algebra. Vectors, Matrices, and Least Squares. – Cambridge University Press, 2018 2. Moskal-kov M.M., Rizhenko A.І., Voitsekhovs-kii S.O. ta іn. Praktikum z metodіv obchislen-. Kiїv. MAUP. 2008. 3. Samarskii A.A., Vabishchevich P.N. Vychislitel-naia teploperedacha. Librokom.–2014. 4. Samarskii A.A., Gulin A.V. Chislennye metody matematicheskoi fiziki. Al-ians, 2016. 5. Alexander A. Samarsky The theory of difference schemes. Marcel Dekker, Inc. New York, 2001. 6. Liashko S.І., Semenov V.V., Kliushin D.A. Spetsіal-nі pitannia optimіzatsії. Kiїv, VPТs “Kiїvs-kii unіversitet”, 2015. 7. Makarov V.L., Gavriliuk I.P. Metodi obchislen-. Kiїv, Vishcha shkola, 1995 ..

Lectures, consultations, independent work

- semester assessment: 1. Control work: PH1.1, PH1.2, PH3.1 – 15 points / 9 points 2. Project-1: RN2.1, RN 4.1 – 15 points / 9 points 3. Project-2: RN2.1, RN 4.1 – 15 points / 9 points 4. Project-3: PH1.3, PH2.2, PH4.1 – 15 points / 9 points final assessment is conducted in the form of an exam. the maximum number of points that can be obtained by a student: 40 points; - learning outcomes that will be assessed: PH1.1, PH1.2, PH1.3, PH3.1; - form of implementation and types of tasks: written - types of tasks: 4 written tasks for 10 points each (2 theoretical questions for each module).

Ukrainian