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Lecturer(s)
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Mlýnek Jaroslav, doc. RNDr. CSc.
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Knobloch Roman, RNDr. Ph.D.
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Course content
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Lectures: 1. Polynomial interpolation, Lagrange interpolation polynomial, Hermite interpolation polynomial. 2. Interpolation using spline functions. Linear spline, quadratic spline and cubic spline. 3. Linear regression, interpolation in two and more dimensions. 4. Solving systems of linear equations - the method of conjugate gradients. 5. Numerical solution of ordinary differential equations, predictor-corrector method. 6. Finite difference method for solving partial differential equations, transcript of boundary conditions. 7. Basic terms of functional analysis, Hilbert and Sobolev spaces. 8. Concept of classical solution, weak formulation of solutions of partial differential equations, formulation of boundary conditions. 9. Variational methods, Ritz and Galerkin methods. 10. Finite element approximation of solutions, one-dimensional finite element method, finite element metod in multiple dimensions. 11. Solving PDE using weak formulation, Laplace and Poisson equations. 12. Evolution algorithms - genetic algorithms. 13. Evolution algorithms - differential algorithms. 14. Reserve. Exercises: Practice topics according to lectures.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing), Written assignment presentation and defence
- Class attendance
- 56 hours per semester
- Preparation for credit
- 15 hours per semester
- Semestral paper
- 20 hours per semester
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Learning outcomes
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The introductory part of the course focuses on the approximation of functions and interpolation of functions using splines (quadratic and cubic spline, B-spline) and interpolation in two and more dimensions. Numerical solution of ordinary differential equations (predictor-corrector methods) is also discussed. The course includes methods for solving partial differential equations with boundary and initial conditions. The use of the finite difference method is described, including the transcription of boundary conditions. The course includes the introduction of the concept of classical solutions and an explanation of the principle of weak problem formulation. Subsequently, the principle of using the Ritz and Galerkin methods to find an approximation of the solution, the use of the finite element method is discussed. The final topic focuses on the basics of optimization problems using evolutionary algorithms.
Knowledge of fundamentals of numerical mathematics.
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Prerequisites
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Passing of mathematical lectures of first four semestrs.
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Assessment methods and criteria
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Oral exam, Written exam
Credit: Working out a semestral work. Exam: Written.
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Recommended literature
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Antia, H., M. Numerical methods for scientists and engineers. Hindustan Book Agency, 2012. ISBN 8185931305.
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Hynek, J. Genetické algoritmy a genetické programování. Grada, Praha, 2008.
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Křížek, M., Neittaanmäki, P. Finite Element Approximation of Variational Problems and Applications. Longman Scientific & Technical, 1990. ISBN 0-470-21539-9.
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Price, V., Storn, M., Lampien, A. Differential Evolution.. Springer, 2005. ISBN 978-3-540-20950-8.
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Ralston, A. Základy numerické matematiky. Praha, 1978.
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Rektorys, K.:. Variační metody v inženýrských problémech a v problémech matematické fyziky.. Praha, 1974.
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Stoer, J., Bulirsch, R. Introduction to Numerical Analysis. Springer, 2008. ISBN 0-387-95452-X.
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Taylor, A.:. Úvod do funkcionální analýzy. Praha, Academia, 1973.
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