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Lecturer(s)
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Černá Dana, doc. RNDr. Ph.D.
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Finěk Václav, doc. RNDr. Ph.D.
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Course content
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1. Orthogonal bases and their properties, discrete Fourier transform, matrix representation. 2. Fast Fourier transform, its applications and modifications. 3. Orthogonal polynomials, Gauss-Legendre 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. Integro-differential equations: Galerkin method, sparse structure and the condition number of the matrix, choice of basis. 11. Nonstationary integro-differential equations: temporal and spatial discretization methods. 12. Applications of nonstationary integro-differential equations: Lévy model for option pricing. 13. Numerical software, paralelization of numerical methods. 14. Computing on graphics processing units.
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Learning activities and teaching methods
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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
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Learning outcomes
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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.
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Prerequisites
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Knowledge of programming.
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Assessment methods and criteria
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Student's performance analysis
Credit: Work with available software. Creation of own software. Exam: Written, composed of the theoretical and computational part.
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Recommended literature
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Norbert Hilber, Christoph Schwab, Oleg Reichmann, Christoph Winter. Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing. Berlin, Springer, 2013. ISBN 978-3-642-35400-7.
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PRESS, W.H.:. Numerical Recipes 3rd Edition, The Art of Scientific Computing. 2007. ISBN 9780521880688.
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Roos, Hans-G., Stynes, Martin, Tobiska, Lutz. Methods for Singularly Perturbed Differential Equations. ISBN 978-3-540-34466-7.
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Zhongying Chen, Yuesheng Xu, Charles A. Micchelli. Multiscale Methods for Fredholm Integral Equations. ISBN 978-1-107-10347-4.
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