Lecturer(s)


Černá Dana, doc. RNDr. Ph.D.

Finěk Václav, doc. RNDr. Ph.D.

Course content

1. Orthogonal bases and their properties, discrete Fourier transform, matrix representation. 2. Fast Fourier transform, its applications and modifications. 3. Orthogonal polynomials, GaussLegendre quadrature formula. 4. Numerical integration of multivariable functions: Simpson rule, Monte Carlo method. 5. Singularly pertubed problems: problem formulation, applications, analytical properties of the solution. 6. Singularly pertubed problems: finite difference method, Galerkin method, Shishkin mesh. 7. Integral equations: types of integral equations, analytical solutions, methods for solution. 8. Integral equations: Galerkin method, error estimates, choice of basis, wavelet basis. 9. Integral equations: sparse matrices, estimates of matrix entries, matrix compression. 10. Integrodifferential equations: Galerkin method, sparse structure and the condition number of the matrix, choice of basis. 11. Nonstationary integrodifferential equations: temporal and spatial discretization methods. 12. Applications of nonstationary integrodifferential equations: Lévy model for option pricing. 13. Numerical software, paralelization of numerical methods. 14. Computing on graphics processing units.

Learning activities and teaching methods

Lecture, Practicum
 Preparation for credit
 56 hours per semester
 Preparation for exam
 74 hours per semester
 Class attendance
 56 hours per semester
 Home preparation for classes
 14 hours per semester

Learning outcomes

The course is concerned with selected numerical methods, creation and optimization of programs for these methods, testing and evaluating the results, software review for mathematical computation, parallel computing and GPU computing.
The course gives students a knowledge of existing numerical software, its creation, utilization, testing and assessment of obtained results.

Prerequisites

Knowledge of programming.

Assessment methods and criteria

Student's performance analysis
Credit: Work with available software. Creation of own software. Exam: Written, composed of the theoretical and computational part.

Recommended literature


Norbert Hilber, Christoph Schwab, Oleg Reichmann, Christoph Winter. Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing. Berlin, Springer, 2013. ISBN 9783642354007.

PRESS, W.H.:. Numerical Recipes 3rd Edition, The Art of Scientific Computing. 2007. ISBN 9780521880688.

Roos, HansG., Stynes, Martin, Tobiska, Lutz. Methods for Singularly Perturbed Differential Equations. ISBN 9783540344667.

Zhongying Chen, Yuesheng Xu, Charles A. Micchelli. Multiscale Methods for Fredholm Integral Equations. ISBN 9781107103474.
