Course detail
High Performance Computations
FIT-VNVAcad. year: 2024/2025
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.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
During the semester, there will be evaluated computer laboratories. Any laboratory should be replaced in the final weeks of the semester.
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 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.
Study aids
Prerequisites and corequisites
Basic literature
Burden, R. L.: Numerical analysis, Cengage Learning, 2015
Butcher, J. C.: Numerical Methods for Ordinary Differential Equations, 3rd Edition, Wiley, 2016.
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
Šebesta, V.: Systémy, procesy a signály I. VUTIUM, Brno, 2001.
Vavřín, P.: Teorie automatického řízení I (Lineární spojité a diskrétní systémy). VUT, Brno, 1991. (CS)
Recommended reading
Hairer, E., Norsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I, vol. Nonstiff Problems. Springer-Verlag Berlin Heidelberg, 1987. (EN)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, vol. Stiff And Differential-Algebraic Problems. Springer-Verlag Berlin Heidelberg, 1996. (EN)
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) (CS)
Lecture notes in PDF format (EN)
Butcher, J. C.: Numerical Methods for Ordinary Differential Equations, 3rd Edition, Wiley, 2016. (EN)
Přednášky ve formátu PDF (CS)
Source codes (TKSL, MATLAB) of all computer laboratories (EN)
Vitásek, E.: Základy teorie numerických metod pro řešení diferenciálních rovnic. Academia, Praha 1994. (CS)
Zdrojové programy (TKSL, MATLAB, Simulink) jednotlivých počítačových cvičení (CS)
Elearning
Classification of course in study plans
- Programme MITAI Master's
specialization NGRI , 0 year of study, summer semester, elective
specialization NADE , 0 year of study, summer semester, elective
specialization NISD , 0 year of study, summer semester, elective
specialization NMAT , 0 year of study, summer semester, elective
specialization NSEC , 0 year of study, summer semester, elective
specialization NISY up to 2020/21 , 0 year of study, summer semester, elective
specialization NNET , 0 year of study, summer semester, elective
specialization NMAL , 0 year of study, summer semester, elective
specialization NCPS , 0 year of study, summer semester, elective
specialization NHPC , 1 year of study, summer semester, compulsory
specialization NVER , 0 year of study, summer semester, elective
specialization NIDE , 0 year of study, summer semester, elective
specialization NISY , 0 year of study, summer semester, elective
specialization NEMB , 0 year of study, summer semester, elective
specialization NSPE , 0 year of study, summer semester, elective
specialization NEMB , 0 year of study, summer semester, elective
specialization NBIO , 0 year of study, summer semester, elective
specialization NSEN , 0 year of study, summer semester, elective
specialization NVIZ , 0 year of study, summer semester, elective
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
- Methodology of sequential and parallel computation (feedback stability of parallel computations)
- Extremely precise solutions of differential equations by the Taylor series method
- Parallel properties of the Taylor series method
- Basic programming of specialised parallel problems by methods using the calculus (close relationship of equation and block description)
- Parallel solutions of ordinary differential equations with constant coefficients, library subroutines for precise computations
- Adjunct differential operators and parallel solutions of differential equations with variable coefficients
- Methods of solution of large systems of algebraic equations by transforming them into ordinary differential equations
- The Bairstow method for finding the roots of high-order algebraic equations
- Fourier series and finite integrals
- Simulation of electric circuits
- Solution of practical problems described by partial differential equations
- Control circuits
- Conception of the elementary processor of a specialised parallel computation system.
Exercise in computer lab
Teacher / Lecturer
Syllabus
- Simulation system TKSL
- Exponential functions test examples
- First order homogenous differential equation
- Second order homogenous differential equation
- Time function generation
- Arbitrary variable function generation
- Adjoint differential operators
- Systems of linear algebraic equations
- Electronic circuits modeling
- Heat conduction equation
- Wave equation
- Laplace equation
- Control circuits
Elearning