Course detail

High Performance Computations (in English)

FIT-VNVeAcad. year: 2026/2027

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.

Why is the course taught

Supercomputers are often used to solve large technical and scientific problems. Before writing the first line of code, the user should perfectly understand the problem, that is being solved.

The goal of this course is to familiarize the students with the physics behind the problems, that are often solved in practice. To be able to see connection between the equations that govern the problem (and then solve it using differential calculus) and the real system. The students should also understand the numerical methods that are being used in the often used software packages as "black boxes". To be able to choose a proper numerical method for a specific problem and not just pick one at random.

Commercial sofware

  • MATLAB (Simulink)

Free software

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

Ing. Václav Šátek, Ph.D.

Department

Department of Intelligent Systems (UITS)

Offered to foreign students

Of all faculties

Entry knowledge

  • Basic knowledge of the programming in procedural programming language (knowledge of software MATLAB/Simulink will be essential for solving homework problems).
  • Knowledge of secondary school level mathematics.

Rules for evaluation and completion of the course

Half-semester exam and final 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.
During the semester, there will be evaluated computer laboratories.

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

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Burden, R. L.: Numerical analysis, Cengage Learning, 2015
Corliss, G. F.: Automatic differentiation of algorithms, Springer-Verlag New York Inc., 2002
Duff, I. S.: Direct Methods for Sparse Matrices (Numerical Mathematics and Scientific Computation), Oxford University Press, 2017
Golub, G. H.: Matrix computations, Hopkins Uni. Press, 2013
Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Society for Industrial and Applied Mathematics, 2008
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.
Kunovský, J.: Modern Taylor Series Method, habilitation thesis, VUT Brno, 1995
LeVeque, R. J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-dependent Problems (Classics in Applied Mathematics), 2007
Meurant, G.: Computer Solution of Large Linear System, North Holland, 1999
Press, W. H.: Numerical recipes : the art of scientific computing, Cambridge University Press, 2007
Saad, Y.: Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics, 2003
Shampine, L. F.: Numerical Solution of ordinary differential equations, Chapman and Hall/CRC, 1994
Strang, G.: Introduction to applied mathematics, Wellesley-Cambridge Press, 1986
Strikwerda, J. C.: Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics, 2004

Recommended reading

Butcher, J. C.: Numerical Methods for Ordinary Differential Equations, 3rd Edition, Wiley, 2016.
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.
Lecture notes written in PDF format,
Source codes of all computer laboratories

Classification of course in study plans

  • Programme MIT-EN Master's 0 year of study, summer semester, elective

  • Programme IT-MGR-1H Master's

    specialization MGH , 0 year of study, summer semester, recommended course

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

Ing. Václav Šátek, Ph.D.
Ing. Petr Veigend, Ph.D.

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 parallel FFT
  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.

Exercise in computer lab

26 hours, compulsory

Teacher / Lecturer

Ing. Petr Veigend, Ph.D.
Ing. Václav Šátek, Ph.D.

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

Individual preparation for a lecture

26 hours, optionally

Teacher / Lecturer

Ing. Václav Šátek, Ph.D.
Ing. Petr Veigend, Ph.D.

Syllabus

Individual preparation for lectures. You should revise the topics covered during the previous lectures.

Individual preparation for excercises

26 hours, optionally

Teacher / Lecturer

Ing. Petr Veigend, Ph.D.
Ing. Václav Šátek, Ph.D.

Syllabus

Preparation for the computer labs. The topics and information are going to be available in advance.

Individual preparation for a final exam

24 hours, optionally

Teacher / Lecturer

Ing. Petr Veigend, Ph.D.
Ing. Václav Šátek, Ph.D.

Syllabus

To prepare for the final exam, you can:
  • revise what was discussed during the lectures
  • redo all of the examples
  • revise the topics covered during the computer labs or individual project