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Applied iterative methods

ДВС.1.06

Вибіркова дисципліна для ОП

First

2021/2022

8 Semester

5

LO 2. Be able to use basic principles and methods of mathematical, complex and functional analysis, linear algebra and number theory, analytical geometry, and differential equations, including partial differential equations, probability theory, mathematical statistics and random processes, numerical methods LO 5. Be able to develop and use in practice algorithms related to the approximation of functional dependencies, numerical differentiation and integration, solving systems of algebraic, differential and integral equations, solving boundary value problems, finding optimal solutions. PLO8. Develop discrete and continuous mathematical models. PLO15. Demonstrate the ability to self-study and continue professional development. PLO23.1. Know the main sections of the theory of calculations, theory of algorithms and theory of programming, mathematical logic, theory of probability and mathematical statistics, control theory. ..

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

Module 1 1. Mathematical apparatus of the theory of applied iterative methods. Eigenvalues ​​and eigenfunctions of difference operators. 2. Direct and iterative methods of solving mesh equations. Basic iterative methods (Jacobi, Seidel, upper relaxation, Richardson), convergence, conditions using. 3. Two-layer iterative methods. 4. Variable-triangular method. Algorithm, convergence, application to solving mesh equations. 5. Iterative methods of variable directions. Algorithm, convergence, application to solving mesh equations. 6. Iterative methods of the variational type. Methods of minimal correlations, minimal corrections, fastest descent, conjugate gradients. 7. Triangular iterative methods. Algorithms. Convergence. Application. 8. Three-layer iterative methods. Error estimation. Chebyshov's semi-weathering method. Three-layer methods of conjugate directions. 9. Iterative methods of solving the nonlinear heat conduction equation. Module 2 1. Examples of well-known problems that are solved by iterative methods. Approximation problem. Nonlinear case. Construction of a non-linear variant MNK — Gauss-Newton method, Levenberg-Marquardt method. 2. Ranking of search results by the PageRank algorithm. Communication with recommender systems. The big data problem. 3. Approaches to distributed processing and data storage. Vertical and horizontal scaling. History of development — distributed file systems. Basic ideas of HDFS and HADOOP. 4. The paradigm of MapReduce distributed computing. Implementation of the PageRank algorithm on MapReduce. 5. Further development of distributed computing. The main ideas of ApacheSpark. Implementation of iterative methods. A variant of the PageRank algorithm for Spark. 6. Implementation of other iterative methods for Spark. Levenberg-Marquardt method. Gradient method for linear regression. 7. Solving optimization problems in Spark.

1. Moskal-kov M.M., Rizhenko A.І., Voitsekhovs-kii S.O. ta іn. Praktikum z metodіv obchislen-. Kiїv. MAUP. 2008. 2. Samarskii A.A., Vabishchevich P.N. Vychislitel-naia teploperedacha. Librokom.–2014. 3. Samarskii A.A., Gulin A.V. Chislennye metody matematicheskoi fiziki. Al-ians, 2016. 4. Alexander A. Samarsky The theory of difference schemes. Marcel Dekker, Inc. New York, 2001. 5. Liashko S.І., Semenov V.V., Kliushin D.A. Spetsіal-nі pitannia optimіzatsії. Kiїv, VPТs “Kiїvs-kii unіversitet”, 2015. 6. Makarov V.L., Gavriliuk I.P. Metodi obchislen-. Kiїv, Vishcha shkola, 1995 7. Traub Dzh. Iteratsionnye metody resheniia uravnenii. M., Mir 1985. 8. Kheigenman L., Iang D. Prikladnye iteratsionnye metody. M., Mir, 1986. ..

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 each module). A student is admitted to the exam if he scored 36 or more points during the semester, and the projects were completed by at least 60%. To obtain an overall positive grade in the discipline, the grade for the exam cannot be less than 24 points.

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