Бакалавриат
2021/2022
Численные методы
Статус:
Курс обязательный (Прикладная математика)
Направление:
01.03.04. Прикладная математика
Кто читает:
Департамент прикладной математики
Когда читается:
4-й курс, 1 модуль
Формат изучения:
с онлайн-курсом
Охват аудитории:
для своего кампуса
Преподаватели:
Буровский Евгений Андреевич,
Загвоздина Ксения Олеговна,
Тутуров Алексей Александрович
Язык:
английский
Кредиты:
3
Контактные часы:
28
Course Syllabus
Abstract
Numerical computations historically play a crucial role in natural sciences and engineering. These days however, it’s not only traditional «hard sciences»: whether you do digital humanities or biotechnology, whether you design novel materials or build artificial intelligence systems, virtually any quantitative work involves some amount of numerical computing . These days, you hardly ever implement the whole computation yourselves from scratch. We rely on libraries which package tried-and-tested, battle-hardened numerical primitives. It is vanishingly rare however that a library contains a single pre-packaged routine which does all what you need. Numerical computing involves assembling these building blocks into computational pipelines. This kind of work requires a general understanding of basic numerical methods, their strengths and weaknesses, their limitations and their failure modes. And this is exactly what this course is about. It is meant to be an introductory, foundational course in numerical analysis, with the focus on basic ideas. We will review and develop basic characteristics of numerical algorithms (convergence, approximation, stability, computational complexity and so on), and will illustrate them with several classic problems in numerical mathematics. You will also work on implementing abstract mathematical constructions into working prototypes of numerical code. Upon completion of this course, you will have an overview of the main ideas of numerical computing, and will have a solid foundation for reading up on and working with more advanced numerical needs of your specific subject area. As prerequisites for this course, we assume a basic command of college-level mathematics (linear algebra and calculus, mostly), and a basic level of programming proficiency. ——- Contain the online component, https://www.coursera.org/learn/intro-to-numerical-analysis/
Learning Objectives
- Basic command of modern numerical methods of applied maths
- Basic use of common packages and libraries for numeric and scientific computing
- Basic use of common packages and libraries for numeric and scientific computing.
Expected Learning Outcomes
- Ability to choose an appropriate numerical method for a given problem.
- Experience working with modern software packages and libraries for numerical and scientific computing.
- Knowledge of mathematical foundations of numerical methods used in contemporary fundamental and applied research and development.
Course Contents
- Numerical mathematics.
- Matrix factorizations
- Eigenvalue problem.
- Systems of linear equations: direct methods.
- Variational and Projection methods for two-point boundary value problems.
- Systems of linear equations: iterative methods.
- Nonlinear equations and systems of equations.
- Finite difference schemes for 1D parabolic PDEs.
- Interpolation of functions.
- Finite difference schemes for hyperbolic PDEs
- Numerical differentiation
- Finite difference schemes for elliptic PDEs
- Numerical integration
- Nonlinear PDEs
- Economic methods for higher dimensional problems. Operator splitting.
- Initial value problem for ODEs
- Fredholm integral equations.
- Monte Carlo methods.
Assessment Elements
- Homework problems A
- In-class activity
- ExamThe final grade can be set based on the accumulated grade, without taking the exam, provided there is a non-zero grade on at least one problem type B from each problem set.
- Online course
- Homework problems B
- In-class activity
- Homework sets
- Third year 3-4 module intermediate grade
- ExamThe student's take-home assignment and answers to additional questions are graded on the 10-point scale. This grade, call it E, is used to compute the 'exam' grade which enters the resulting grade for the course via the following formula: exam = (E-7)*10/3. NOTE that the 'exam' grade can be both positive and negative.
Interim Assessment
- 2020/2021 4th module0.3 * Homework problems B + 0.1 * In-class activity + 0.2 * Online course + 0.2 * Homework problems A + 0.2 * Exam
- 2021/2022 1st module0.1 in-class activity + 0.25 homework + 0.2 exam + 0.45 (3rd year grade)
Bibliography
Recommended Core Bibliography
- Higham, N. J., & Dennis, M. R. (2015). The Princeton Companion to Applied Mathematics. Princeton: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1426583
- Вычислительные методы для инженеров : учеб. пособие для вузов, Амосов, А. А., 2003
Recommended Additional Bibliography
- William H. Press, Saul A. Teukolsky, William T. Vetterling, & Brian P. Flannery. (1992). Numerical Recipes in C: The Art of Scientific Computing. Second Edition. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.9CFCD6AE
- Wright, S. J., & Nocedal, J. (2015). Numerical optimization. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.92C9B6B0
- Численные методы : учеб. пособие для вузов, Калиткин, Н. Н., 2011
- Численные методы, Самарский, А. А., 1989