Nonlinear Continuous Optimization and Numerical Methods

Garant předmětu:

Přednášející:

Cvičící:

Předmět zajišťuje:

katedra teoretické informatiky

Anotace:

Students will be introduced to nonlinear continuous optimization, principles of the most popular methods of optimization and applications of such methods to real-world problems. They will also learn the finite element method and the finite difference method used for solving ordinary and partial differential equations in engineering. They will learn to solve systems of linear algebraic equations that arise from discretization of the continuous problems by direct and iterative algorithms. They will also learn to implement these algorithms sequentially as well as in parallel.

Požadavky:

Basic knowledge of linear algebra (vectors, matrices, systems of linear algebraic equations, Gaussian elimination method), polynoms, differential calculus (derivative, integral).

Osnova přednášek:

1. Partial derivative, gradient, hessian.

2. Continuous optimization of the 1st and 2nd order.

3. Quasi-Newton method, conjugate gradient method.

4. Application of methods of nonlinear continuous optimization.

5. Introduction to ordinary and partial differential equations (taxonomy, the notion of the solution, physical interpretation).

6. Ordinary differential equations - boundary value problem (exact solution, finite difference method, finite differences).

7. Ordinary differential equations - boundary value problem (finite element method).

8. Partial differential equations - stationary cases (finite difference method).

9. Partial differential equations - stationary cases (finite element method).

10. Ordinary differential equations - initial value problem.

11. Partial differential equations - nonstationary problems.

12. Iterative methods (Gauss-Seidel method, conjugate gradient method).

13. Introduction to domain decomposition methods. Parallel solvers of sets of linear equations.

Osnova cvičení:

Cíle studia:

The module gives an introduction to continuous optimization with respect to the solution of complicated problems, e.g., data approximation or identification of model parameters. The second part deals with several parts of computational sciences, with the emphasis on the finite element method and the finite difference method which are massively used in all engineering branches, not only in academic community but also in industry.

Studijní materiály:

1. Kruis, J. ''Domain Decomposition Methods for Distributed Computing''. Saxe-Coburg Publications, 2007. ISBN 1874672237.

2. Petzold, L. R. ''Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations''. Society for Industrial and Applied Mathematics, 1998. ISBN 0898714125.

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