Course detail

High Performance Computations

FIT-VNVAcad. year: 2019/2020

The course is aimed at practical methods of solving sophisticated problems encountered in science and engineering. Serial and parallel computations are compared with respect to a stability of a numerical computation. A special methodology of parallel computations based on differential equations is presented. A new original method based on direct use of Taylor series is used for numerical solution of differential equations. There is the TKSL simulation language with an equation input of the analysed problem at disposal. A close relationship between equation and block representation is presented. The course also includes design of special architectures for the numerical solution of differential equations.

Learning outcomes of the course unit

Ability to transform a sophisticated technical problem to a system of differential equations. Ability to solve sophisticated systems of differential equations using simulation language TKSL.
Ability to create parallel and quasiparallel computations of large tasks.

Prerequisites

Not applicable.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Vitásek, E.: Základy teorie numerických metod pro řešení diferenciálních rovnic. Academia, Praha 1994.
Čermák, L., Hlavička, R.: Numerické metody I, II, CERM, učební text FSI VUT Brno, 2008. (elektronicky dostupné z http://math.fme.vutbr.cz/Home/cermakl/soubory-ke-stazeni)
Kozubek, T., Brzobohatý, T., Jarošová, M., Hapla, V., Markopoulos, A.: Lineární algebra s MATLABem, učební text MI21 VŠB-TU Ostrava, 2012 (elektronicky dostupné z http://mi21.vsb.cz/sites/mi21.vsb.cz/files/unit/linearni_algebra_s_matlabem.pdf)
Přednášky ve formátu PDF
Zdrojové programy (TKSL, MATLAB, Simulink) jednotlivých počítačových cvičení
Kunovský, J.: Modern Taylor Series Method, habilitation thesis, VUT Brno, 1995
Hairer, E., Norsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I, vol. Nonstiff Problems. Springer-Verlag Berlin Heidelberg, 1987.
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, vol. Stiff And Differential-Algebraic Problems. Springer-Verlag Berlin Heidelberg, 1996.
Butcher, J. C.: Numerical Methods for Ordinary Differential Equations, 3rd Edition, Wiley, 2016.
Shampine, L. F.: Numerical Solution of ordinary differential equations, Chapman and Hall/CRC, 1994
Strang, G.: Introduction to applied mathematics, Wellesley-Cambridge Press, 1986
Meurant, G.: Computer Solution of Large Linear System, North Holland, 1999
Saad, Y.: Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, 2003
Burden, R. L.: Numerical analysis,  Cengage Learning, 2015
LeVeque, R. J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-dependent Problems (Classics in Applied Mathematics), 2007
Strikwerda, J. C.: Finite Difference Schemes and Partial Differential Equations,  Society for Industrial and Applied Mathematics, 2004
Golub, G. H.: Matrix computations, Hopkins Uni. Press, 2013
Duff, I. S.: Direct Methods for Sparse Matrices (Numerical Mathematics and Scientific Computation), Oxford University Press, 2017
Corliss, G. F.: Automatic differentiation of algorithms, Springer-Verlag New York Inc., 2002
Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Society for Industrial and Applied Mathematics, 2008
Press, W. H.: Numerical recipes : the art of scientific computing, Cambridge University Press, 2007
Brdička M., Samek L., Sopko B.: Mechanika kontinua, Academia, 2005
Vavřín, P.: Teorie automatického řízení I (Lineární spojité a diskrétní systémy). VUT, Brno, 1991.
Šebesta, V.: Systémy, procesy a signály I. VUTIUM, Brno, 2001.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Half Term Exam and Term Exam. The minimal number of points which can
be obtained from the final exam is 29. Otherwise, no points will
be assigned to a student.

Language of instruction

Czech

Work placements

Not applicable.

Aims

To provide overview and basics of practical use of parallel and quasiparallel methods for numerical solutions of sophisticated problems encountered in science and engineering.

Specification of controlled education, way of implementation and compensation for absences

During the semester, there will be evaluated computer laboratories. Any laboratory should be replaced in the final weeks of the semester.

Classification of course in study plans

  • Programme IT-MGR-2 Master's

    branch MBI , any year of study, summer semester, 5 credits, optional
    branch MPV , any year of study, summer semester, 5 credits, optional
    branch MGM , any year of study, summer semester, 5 credits, compulsory-optional
    branch MSK , any year of study, summer semester, 5 credits, optional
    branch MIS , any year of study, summer semester, 5 credits, optional
    branch MBS , any year of study, summer semester, 5 credits, optional
    branch MIN , any year of study, summer semester, 5 credits, compulsory-optional
    branch MMI , any year of study, summer semester, 5 credits, optional
    branch MMM , any year of study, summer semester, 5 credits, compulsory

  • Programme MITAI Master's

    specialization NADE , any year of study, summer semester, 5 credits, optional
    specialization NBIO , any year of study, summer semester, 5 credits, optional
    specialization NGRI , any year of study, summer semester, 5 credits, optional
    specialization NNET , any year of study, summer semester, 5 credits, optional
    specialization NVIZ , any year of study, summer semester, 5 credits, optional
    specialization NCPS , any year of study, summer semester, 5 credits, optional
    specialization NSEC , any year of study, summer semester, 5 credits, optional
    specialization NEMB , any year of study, summer semester, 5 credits, optional
    specialization NISD , any year of study, summer semester, 5 credits, optional
    specialization NIDE , any year of study, summer semester, 5 credits, optional
    specialization NISY , any year of study, summer semester, 5 credits, optional
    specialization NMAL , any year of study, summer semester, 5 credits, optional
    specialization NMAT , any year of study, summer semester, 5 credits, optional
    specialization NSEN , any year of study, summer semester, 5 credits, optional
    specialization NVER , any year of study, summer semester, 5 credits, optional
    specialization NSPE , any year of study, summer semester, 5 credits, optional
    specialization NHPC , 1. year of study, summer semester, 5 credits, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus


  1. Methodology of sequential and parallel computation (feedback stability of parallel computations)
  2. Extremely precise solutions of differential equations by the Taylor series method
  3. Parallel properties of the Taylor series method
  4. Basic programming of specialised parallel problems by methods using the calculus (close relationship of equation and block description)
  5. Parallel solutions of ordinary differential equations with constant coefficients, library subroutines for precise computations
  6. Adjunct differential operators and parallel solutions of differential equations with variable coefficients
  7. Methods of solution of large systems of algebraic equations by transforming them into ordinary differential equations
  8. The Bairstow method for finding the roots of high-order algebraic equations
  9. Fourier series and finite integrals
  10. Simulation of electric circuits
  11. Solution of practical problems described by partial differential equations
  12. Control circuits
  13. Conception of the elementary processor of a specialised parallel computation system.

Computer exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus


  1. Simulation system TKSL
  2. Exponential functions test examples
  3. First order homogenous differential equation
  4. Second order homogenous differential equation
  5. Time function generation
  6. Arbitrary variable function generation
  7. Adjoint differential operators
  8. Systems of linear algebraic equations
  9. Electronic circuits modeling
  10. Heat conduction equation
  11. Wave equation
  12. Laplace equation
  13. Control circuits

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