Numerical analysis and optimisation
Presentation
Programme (detailed contents):
Numerical Analysis :
− Interpolation: Vand der Monde’s method, Lagrange’s method, Newton’s method and piece by piece interpolation,
− Numerical integration: trapezoidal rule, Simpson’s method and Gauss’ methods,
− The LU decomposition as well as the Cholesky factorization for solving linear systems,
− The fixed point method and Newton’s method for solving nonlinear systems,
Optimization :
- Introduction to numerical optimization, differential calculus
- Definition of a local/global minimum/maximum, definition and characterization of convexity
- Necessary optimality conditions of the first and the second order
- Gradient methods (constant step and steepest descent), Newton method, (linear and nonlinear) least square methods
Organisation:
This module is organized in course and labworks..
Main difficulties for students: No difficulties for the ones regularly working.
Objectives
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At the end of this module, the student will have understood and be able to explain (main concepts): Numerical Analysis : − Some polynomial interpolation technics, − Different methods for the numerical integration, − Numerical errors and the problem of numerical stability through the condition number, − The LU decomposition as well as the Cholesky factorization for solving linear systems, − The fixed point method and Newton’s method for solving nonlinear systems,
Optimization : Introduction to unconstrained numerical optimization. The differentiable case. - Concept of local extremum, inroduction to convexity - Necessary optimality conditions - Gradient methods, Newton method, least square problems
The student will be able to: Numerical Analysis : To be able to choose and to implement efficient numerical methods: to numerically compute an integral, to solve linear and nonlinear systems.
Optimization: To be able to choose and implement a suitable algorithm for solving a given unconstrained optimization problem. |
Needed prerequisite
- Precedent courses on the following subjects : linear algebra
- Differential Calculus from the 2nd year
Form of assessment
The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...
Bibliography
Polycopié remis.
Trefethen & Bau, Linear Numerical Algebra, SIAM, ISBN 0-89871-361-1
Lascaux & Theodor, Analyse numérique matricielle appliquée à l'art de l'ingénieur-2, ISBN 2-225-84122-5
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