High Performance Computations

• 26 hrs lectures
• 26 hrs pc labs

• 60 pts final exam (written part)
• 20 pts mid-term test (written part)
• 20 pts labs

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 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.

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.

• 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.

• Lecture notes in PDF format
• Source codes (TKSL, MATLAB) of all computer laboratories

• 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
• Šebesta, V.: Systémy, procesy a signály I. VUTIUM, Brno, 2001.

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.

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

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